Preface

A polynomial in mathematics is an expression consisting of coefficients and variables that involves only the operations of addition, subtraction, and multiplication with a non-negative integer power. Polynomials have significant applications in the description of many real-world problems. Various polynomial-type expressions are being used to describe the various chemical, biological, social, and economic problems. Further, polynomials are increasingly being used in numerical computations of large numbers of nonlinear problems. They are also used in calculus and numerical analysis to approximate other functions. In linear algebra during spectral analysis, eigen values are computed through characteristic polynomials, which further help in finding the radius of convergence of various matrices. In stability analysis, eigen values are computed through the help of minimal polynomials of the Jacobian matrix. Polynomials are powerful tools in approximation theory and advanced numerical analysis.

Chapter 1 discusses some characteristic functions and various valuable relations of polynomials.

Chapter 2 presents some problems of permutations and their applications, various relations, and results.

Chapter 3 discusses the effectiveness of basic sets of Gončarov polynomials and their different properties.

Chapter 4 reviews irreducible factors of polynomials as well as discusses the irreducibility of polynomials with specific requirements on their coefficients.

Chapter 5 describes the use of homogenous polynomials yield function and various results.

Chapter 6 examines the irreducibility of polynomials in non-binary fields.

Chapter 7 presents the efficiency of polynomial regression algorithms.

Chapter 8 describes the use of shifted Jacobi polynomials in some fractional order differential equations under initial and boundary conditions.

> **Kamal Shah** Department of Mathematics and Sciences, Prince Sultan University, Riyadh, Saudi Arabia

> > Department of Mathematics, University of Malakand, Chakdara, Pakistan

## **Chapter 1** Characteristic Polynomials

*Sándor Kovács, Szilvia György and Noémi Gyúró*

#### **Abstract**

In this chapter, we provide a short overview of the stability properties of polynomials and quasi-polynomials. They appear typically in stability investigations of equilibria of ordinary and retarded differential equations. In the case of ordinary differential equations we discuss the Hurwitz criterion, and its simplified version, the Lineard-Chippart criterion, furthermore the Mikhailov criterion and we show how one can prove the change of stability via the knowledge of the coefficients of the characteristic polynomial of the Jacobian of the given autonomous system. In the case of the retarded differential equation we use the Mikhailov criterion in order to estimate the length of the delay for which no stability switching occurs. These results are applied to the stability and Hopf bifurcation of an equilibrium solution of a system of ordinary differential equations as well as of retarded dynamical systems.

**Keywords:** Hurwitz stability, Schur stability, Mikhailov criterion, Hopf bifurcation

#### **1. Introduction**

As it is well-known, many systems of applied mathematics are modeled by retarded functional differential equations of type

$$
\dot{\mathfrak{x}} = f(\mathfrak{x}, \mathfrak{x}(\cdot - \mathfrak{x})) \tag{1}
$$

(cf. [1]), where *<sup>f</sup>* : <sup>Ω</sup> � <sup>Ω</sup> ! *<sup>d</sup>* is continuously differentiable and <sup>Ω</sup> <sup>⊂</sup> *<sup>d</sup>* is an open set. Here *τ* ≥0 represents the so-called delay or time lag. In order to have a solution in some interval 0, ð Þ*r* , *r*>0 one has to know the solution on ½ � �*τ*, 0 , which means that one has to attach a continuous initial function *<sup>ϕ</sup>* : ½ �! �*τ*, 0 *<sup>d</sup>* as an initial condition to the system (cf. [2]). Clearly, in the case of *τ* ¼ 0 one has to deal with the initial value problem for ordinary differential equations.

In order to examine the stability of an equilibrium *a*∈ Ω of system (1), i.e. the equilibrium solution

$$
\hat{a}: \mathbb{R} \to \mathbb{R}, \qquad \hat{a}(t) := a \tag{2}
$$

for which *f a*ð Þ¼ , *a* 0 holds, one has to discuss the spectral properties of the linearized system

$$
\dot{u} = Au + Bu(\cdot - \pi) \tag{3}
$$

where

$$A \coloneqq \partial\_{\mathcal{Y}} f(a, a) \in \mathbb{R}^{d \times d} \quad \text{and} \quad B \coloneqq \partial\_{\mathcal{Y}} f(a, a) \in \mathbb{R}^{d \times d}. \tag{4}$$

It may be supposed that system (3) has a solution of the form <sup>∍</sup> *<sup>t</sup>* <sup>↦</sup> *<sup>e</sup><sup>λ</sup><sup>t</sup>* � *<sup>s</sup>* where <sup>0</sup> 6¼ *<sup>s</sup>* <sup>∈</sup> *<sup>d</sup>*, that is

$$(\lambda I\_d - A - B e^{-\lambda \mathbf{r}})\mathbf{s} = \mathbf{0}.\tag{5}$$

This can only happen if and only if Δð Þ¼ *λ*; *τ* 0 holds, where

$$\Delta(\mathbf{z};\tau) \coloneqq \det(\mathbf{z}I\_d - A - Be^{-\mathbf{z}\tau}) \qquad (\mathbf{z} \in \mathbb{C}) \tag{6}$$

is called characteristic quasi-polynomial of the linear delay system (3).

The organization of the chapter is as follows. In the next section, we introduce and prove two criteria regarding stability of Δ ≔ Δð Þ �; 0 , i.e. we give conditions for which the zeros of Δ have negative real parts. Concretely, we deal with the Hurwitz criterion and with its simplified version, the Lineard-Chippart criterion, and the Mikhailov criterion. Furthermore, we show how one can check the conditions of Hopf bifurcation via knowledge of the coefficients of the characteristic polynomial. In the section that follows we examine the case when the delay *τ* is positive. We show how the Mikhailov criterion can be extended for quasi-polynomials and how the length of the delay can be estimated in order to have stability. Finally, we present a criterion for Hopf bifurcation.

### **2. The undelayed case:** *τ* ¼ **0**

If there is no delay present, i.e. *τ* ¼ 0 holds then we have to deal with the characteristic polynomial

$$\chi(\mathbf{z}) \coloneqq (-\mathbf{1})^d \det(\mathbf{A} - \mathbf{z}\mathbf{I}\_d) = \mathbf{z}^d + a\_{d-1}\mathbf{z}^{d-1} + a\_{d-2}\mathbf{z}^{d-2} + \dots + a\_1\mathbf{z} + a\_0 \qquad (\mathbf{z} \in \mathbb{C}) \tag{7}$$

where the coefficients of *χ<sup>A</sup>* are determined recursively by the Faddeev– LeVerrier algorithm (cf. [3, 4]) as follows

$$\begin{aligned} N\_0 &:= O & \mathbf{a}\_d &:= \mathbf{1} & (k=0), \\ N\_k &:= A N\_{k-1} + a\_{d-k+1} I\_d & a\_{d-k} &:= -\frac{\mathbf{1}}{k} \text{Tr}(A N\_k) & (k \in \{1, \dots, d\}). \end{aligned} \tag{8}$$

#### **2.1 The stability of the characteristic polynomial** *χ <sup>A</sup>*

The asymptotic stability of (3) is determined by the stability of the matrix *A*, i.e. by the stability of its characteristic polynomial *χA*. We are now supplying some criteria for the stability of the characteristic polynomial *χA*. Under stability, we mean the so-called Hurwitz stability, i.e. the zeros of *χ<sup>A</sup>* lie in the open left half of the complex plane. In this case *χ<sup>A</sup>* is called also Hurwitz polynomial.

There is a very simple but very important necessary condition for *χ<sup>A</sup>* to be Hurwitz (cf. [5, 6]).

Theorem 1.1 (Stodola). If the characteristic polynomial *χ<sup>A</sup>* in (7) is stable then all of its coefficients are positive, i.e. *ak* >0 holds where *k*∈f g 0 … , *d* � 1 .

**Proof:** The real and complex zeros of *χ<sup>A</sup>* my be written as *λk*, resp. *α<sup>l</sup>* � *iβl*, where *α<sup>l</sup>* and *β<sup>l</sup>* are both real. If the multiplicity of the real, resp. complex zeros are denoted by *σk*, resp. *τl*, where *k*∈f g 1, … ,*r* , resp. *l* ∈f g 1, … , *s* , then

*Characteristic Polynomials DOI: http://dx.doi.org/10.5772/intechopen.100200*

$$\sum\_{k=1}^{r} \sigma\_k + 2\sum\_{l=1}^{s} \tau\_l = d,\tag{9}$$

and we can split *χ<sup>A</sup>* into linear, resp. quadratic factors according to the real, resp. complex zeros as follows

$$\begin{split} \chi\_{A}(\mathbf{z}) &= a\_{0} + a\_{1}\mathbf{z} + \dots + a\_{d-1}\mathbf{z}^{d-1} + \mathbf{z}^{d} \\ &= \prod\_{k=1}^{r} (\mathbf{z} - \boldsymbol{\lambda}\_{k})^{\sigma\_{k}} \cdot \prod\_{l=1}^{s} (\mathbf{z} - \mathbf{a}\_{l} - i\boldsymbol{\beta}\_{l})^{\tau\_{l}} (\mathbf{z} - \mathbf{a}\_{l} + i\boldsymbol{\beta}\_{l})^{\tau\_{l}} \\ &= \prod\_{k=1}^{r} (\mathbf{z} - \boldsymbol{\lambda}\_{k})^{\sigma\_{k}} \cdot \prod\_{l=1}^{s} \left( \mathbf{z}^{2} - 2a\_{l}\mathbf{z} + \mathbf{a}\_{l}^{2} + \boldsymbol{\beta}\_{l}^{2} \right)^{\tau\_{l}}. \end{split} \tag{10}$$

Thus, the stability of *χ<sup>A</sup>* implies the sign conditions

$$
\lambda\_k < 0, \quad a\_l < 0 \qquad (k \in \{1, \ldots, r\}; \ l \in \{1, \ldots, s\}).\tag{11}
$$

This means that all coefficients of all factors in the product above are positive. By performing the multiplications one can see that the coefficients of *χ<sup>A</sup>* are positive. ■

In the case of *d* ¼ 1 and *d* ¼ 2 this criterion is sufficient and necessary. Indeed, in case of *d* ¼ 1 the characteristic polynomial has the form

$$\chi\_A(z) = z + a\_0 \qquad (z \in \mathbb{C}), \tag{12}$$

furthermore in case of *d* ¼ 2 we have

$$a\_1 = -\operatorname{Tr}(A), \qquad a\_0 = -\frac{1}{2}\left\{ \operatorname{Tr}(A^2) - \operatorname{Tr}(A)^2 \right\} = \det(A) \tag{13}$$

(cf. (8)), thus

$$\chi\_A(\xi\_\pm) = 0 \iff \xi\_\pm = \frac{\text{Tr}(A) \pm \sqrt{\text{Tr}(A)^2 - 4\text{det}(A)}}{2} \tag{14}$$

and *χ<sup>A</sup>* is stable if and only if Trð Þ *A* <0 and detð Þ *A* >0 hold, because if

$$\begin{aligned} \text{• } \operatorname{Tr}(A)^2 - 4\det(A) > 0 \text{, then we have } \xi\_- < 0, \\\\ \xi\_+ < 0 \quad \Longleftrightarrow \quad \operatorname{Tr}(A)^2 - 4\det(A) < \operatorname{Tr}(A)^2 \quad \Longleftrightarrow \quad \det(A) > 0; \end{aligned} \tag{15}$$

• Trð Þ *<sup>A</sup>* <sup>2</sup> � 4detð Þ¼ *<sup>A</sup>* 0, then the zeros *<sup>ξ</sup>*� and *<sup>ξ</sup>*<sup>þ</sup> are equal and real:

$$
\xi\_- = \xi\_+ = -\frac{\text{Tr}(\!A)}{2} < 0 \quad \Longleftrightarrow \quad \text{Tr}(A) > 0; \tag{16}
$$

• Trð Þ *<sup>A</sup>* <sup>2</sup> � 4detð Þ *<sup>A</sup>* <sup>&</sup>lt;0, then there are only complex zeros:

$$\Re(\xi\_{\pm}) = -\frac{\operatorname{Tr}(A)}{2} < 0 \quad \Longleftrightarrow \quad \operatorname{Tr}(A) > 0. \tag{17}$$

Unfortunately the criterion is for *d*>2 not sufficient. For example, the polynomial

$$p(\mathbf{z}) \coloneqq \mathbf{z}^4 + \mathbf{3}\mathbf{z}^3 + \mathbf{3}\mathbf{z}^2 + \mathbf{3}\mathbf{z} + \mathbf{3} = (\mathbf{z}^2 + \mathbf{1})(\mathbf{z} + \mathbf{1})(\mathbf{z} + \mathbf{2}) \qquad (\mathbf{z} \in \mathbb{C}) \tag{18}$$

has positive coefficients, but two of its zero, namely �*i* are not in the open left half-plane. In case of *d* ¼ 3 there is a result which can be proved in several ways. Pontryagin (cf. [7]) proves it in a circumstantial way. He uses that the zeros of a polynomial are continuous functions of the coefficients (cf. [8, 9]). Our presentation is based on the results of Suter (cf. [10]).

**Theorem 1.2** In case of *d* ¼ 3 the characteristic polynomial of *A* has the form

$$\chi\_A(z) \coloneqq z^3 - \operatorname{Tr}(A)z^2 + \operatorname{Tr}(\operatorname{adj}(A))z - \det(A) \qquad (z \in \mathbb{C}) \tag{19}$$

(cf. (8)) and *χ<sup>A</sup>* is stable if and only if

$$\operatorname{Tr}(A) < 0, \qquad \det(A) < 0, \qquad \operatorname{Tr}(A) \cdot \operatorname{Tr}(\operatorname{adj}(A)) < -\det(A) \tag{20}$$

hold

**Proof:** Using the Faddeev-LeVerrier-algorithm (cf. (8)) we have

$$\chi\_A(z) = z^3 + az^2 + bz + c \qquad (z \in \mathbb{C}), \tag{21}$$

where

$$a \coloneqq -\operatorname{Tr}(A), \qquad b \coloneqq \frac{\operatorname{Tr}(A)^2 - \operatorname{Tr}(A^2)}{2} = \operatorname{Tr}(\operatorname{adj}(A)), \qquad c \coloneqq -\det(A). \tag{22}$$

As a consequence of the fundamental theorem of algebra, we can split *χ<sup>A</sup>* into a linear and a quadratic factor

$$\chi\_A(\mathbf{z}) = (\mathbf{z} - a) \left( \mathbf{z}^2 + \beta \mathbf{z} + \gamma \right) = \mathbf{z}^3 + (\beta - a) \mathbf{z}^2 + (\gamma - a\beta) \mathbf{z} - a\gamma \qquad (\mathbf{z} \in \mathbb{C}).\tag{23}$$

In view of the above considerations for the first and the second-order polynomials we see that *χ<sup>A</sup>* is stable if and only if *α*< 0, *β* >0 and *γ* >0 hold. Thus, it is enough to show that the equivalence

$$a<0, \ \beta>0, \ \gamma>0 \quad \Longleftrightarrow \ a>0, \ c>0, \ ab-c>0. \tag{24}$$

holds. We prove this statement in two steps. **Step 1.** We prove that the positivity of the coefficients *a*, *b*,*c* entails

$$a < 0, \quad \gamma > 0 \quad \text{and} \quad \text{sgn}\,(\beta) = \text{sgn}\,(ab - c). \tag{25}$$

Indeed,

• from �*αγ* ¼ *c*> 0 it follows that *α* 6¼ 0, *γ* 6¼ 0. Hence the equivalence

$$a \rhd 0 \quad \Longleftrightarrow \quad \gamma < 0 \tag{26}$$

holds. The case *α*> 0, *γ* <0 cannot happen, because *γ* � *αβ* ¼ *b*> 0 would imply *β* <0, which is not possible due to *β* � *α* ¼ *a*>0.

$$a^2 + b > 0\tag{27}$$

*Characteristic Polynomials DOI: http://dx.doi.org/10.5772/intechopen.100200*

$$\begin{split} ab - c &= (\beta - a)(\chi - a\beta) + a\chi = a^2\beta - a\beta^2 + \beta\chi = \beta(a^2 - a\beta + \chi) = \\ &= \beta(a^2 + b). \end{split} \tag{28}$$

**Step 2.** It remains to prove that the equivalence (24) holds.


**Example 1.** If the matrix *A* ∈ <sup>3</sup>�<sup>3</sup> is antisymmetric, i.e. for suitable *a*, *b*,*c*∈

$$A = \begin{bmatrix} \mathbf{0} & a & b \\ -a & \mathbf{0} & c \\ -b & -c & \mathbf{0} \end{bmatrix} \tag{29}$$

holds, then its characteristic polynomial has the form

$$\chi\_A(\mathbf{z}) = \mathbf{z}^3 + \left(a^2 + b^2 + c^2\right)\mathbf{z} = \mathbf{z}\left[\mathbf{z}^2 + \left(a^2 + b^2 + c^2\right)\right] \qquad (\mathbf{z} \in \mathbb{C}).\tag{30}$$

This means that *χ<sup>A</sup>* and hence *A* is unstable.

In order to formulate the necessary and sufficient stability condition for the polynomial *χ<sup>A</sup>* with arbitrary degree *d*∈ , we shall first fix our terminology. Let us define the Hurwitz matrix of the characteristic polynomial *χ<sup>A</sup>* by

$$\mathcal{H}\_{\chi\_{A}} \coloneqq \begin{bmatrix} h\_{\vec{\eta}} \end{bmatrix}, \quad \text{where} \quad h\_{\vec{\eta}} \coloneqq \begin{cases} a\_{d - (2j - i)} & (0 \le 2j - i \le d), \\ 0 & (\text{elsewhere}) \end{cases} \quad (i, j \in \{1, \dots, d\}), \tag{31}$$

i.e.

$$\mathcal{H}\_{\mathcal{X}\_{\mathcal{A}}} \coloneqq \begin{bmatrix} a\_{d-1} & a\_{d-3} & a\_{d-5} & \dots & a\_{d-2d+3} & a\_{d-2d+1} \\ & a\_d & a\_{d-2} & a\_{d-4} & \dots & a\_{d-2d+4} & a\_{d-2d+2} \\ 0 & & a\_{d-1} & a\_{d-3} & \dots & a\_{d-2d+5} & a\_{d-2d+3} \\ & 0 & & a\_d & a\_{d-2} & \dots & a\_{d-2d+6} & a\_{d-2d+4} \\ & \vdots & & & & \vdots \\ & & & & & \ddots & & \vdots \\ 0 & 0 & \dots & \dots & a\_1 & 0 \\ & 0 & 0 & \dots & \dots & a\_2 & a\_0 \end{bmatrix} \in \mathbb{R}^{d \times d} \tag{32}$$

where *ad* ≔ 1 and *a*�*<sup>m</sup>* ≔ 0 if *m* >0.

**Theorem 1.3** (Routh-Hurwitz criterion). The characteristic polynomial *χ<sup>A</sup>* in (7) is stable if and only if all leading principal minors

$$\Delta\_k \coloneqq \det \begin{bmatrix} h\_{11} & \dots & h\_{1k} \\ \vdots & \ddots & \vdots \\ h\_{k1} & \dots & h\_{kk} \end{bmatrix} \qquad (k \in \{1, \dots, d\}) \tag{33}$$

of H*<sup>χ</sup><sup>A</sup>* are positive, i.e.

$$\Delta\_1 = a\_{d-1} > 0, \quad \Delta\_2 = \det\begin{bmatrix} a\_{d-1} & a\_{d-3} \\ a\_d & a\_{d-2} \end{bmatrix} > 0, \quad \dots, \quad \Delta\_d = a\_0 \Delta\_{d-1} > 0 \tag{34}$$

hold. □ For

• two-dimensional system we have

$$\mathcal{H}\_{\chi\_{\Lambda}} \coloneqq \begin{bmatrix} a\_1 & a\_{-1} \\ \mathbf{1} & a\_0 \end{bmatrix} = \begin{bmatrix} a\_1 & \mathbf{0} \\ \mathbf{1} & a\_0 \end{bmatrix}. \tag{35}$$

Thus, this criterion can be stated as

$$
\Delta\_1 = a\_1 > 0, \qquad \Delta\_2 = a\_1 - a\_0 > 0,\tag{36}
$$

or

$$a\_0 > 0, \qquad a\_1 > 0. \tag{37}$$

• third-dimensional system we have

$$\mathcal{H}\_{\chi\_A} \coloneqq \begin{bmatrix} a\_2 & a\_0 & \mathbf{0} \\ \mathbf{1} & a\_1 & \mathbf{0} \\ \mathbf{0} & a\_2 & a\_0 \end{bmatrix} . \tag{38}$$

Thus, this criterion is

$$
\Delta\_1 = a\_2 > 0, \qquad \Delta\_2 = a\_2 a\_1 - a\_0 > 0, \qquad \Delta\_3 = a\_0 \Delta\_2 > 0,\tag{39}
$$

or

$$a\_0 > 0, \qquad a\_2 > 0, \qquad a\_2 a\_1 > a\_0. \tag{40}$$

• fourth-dimensional system we have

$$
\mathcal{H}\_{\chi\_A} \coloneqq \begin{bmatrix} a\_3 & a\_1 & \mathbf{0} & \mathbf{0} \\ \mathbf{1} & a\_2 & a\_0 & \mathbf{0} \\ \mathbf{0} & a\_3 & a\_1 & \mathbf{0} \\ \mathbf{0} & \mathbf{1} & a\_2 & a\_0 \end{bmatrix} . \tag{41}
$$

Thus, this criterion can be stated as Δ<sup>1</sup> ¼ *a*<sup>3</sup> >0,

$$
\Delta\_2 = a\_3 a\_2 - a\_1 > 0, \quad \Delta\_3 = a\_3 a\_2 a\_1 - a\_3^2 a\_0 - a\_1^2 > 0, \quad \Delta\_4 = a\_0 \Delta\_3 > 0 \tag{42}
$$

or

$$a\_0 > 0, \qquad a\_3 > 0, \qquad a\_3 a\_2 > a\_1, \qquad a\_1 a\_2 a\_3 > a\_0 a\_3^2 + a\_1^2. \tag{43}$$

As the application of the above theorem, we mention the Orlando formula (cf. [11]) which establishes the useful relation between the Hurwitz determinants and the polynomial whose roots are sums of the roots of a given polynomial and which can be proved by mathematical induction (cf. [12]).

**Theorem 1.4** (Orlando-formula) If *ξ*1, … , *ξ<sup>d</sup>* are the roots of the characteristic polynomial *χ<sup>A</sup>* then the ð Þ *d* � 1 -th principal minor of the Hurwitz matrix can be expressed as

$$\Delta\_{d-1} = (-\mathbf{1})^{d(d-1)/2} \cdot \prod\_{\substack{i,j=1 \\ i$$

In case of

• *d* ¼ 2 this formula reduces to the well known Vieta formula in the quadratic equation

$$
\sigma\_1 = \Delta\_1 = -(\lambda\_1 + \lambda\_2);
\tag{45}
$$

• *d* ¼ 3 the formula in (44) reduces to

$$
\sigma\_2 \sigma\_1 - \sigma\_0 = \Delta\_2 = -(\lambda\_1 + \lambda\_2)(\lambda\_1 + \lambda\_3)(\lambda\_2 + \lambda\_3). \tag{46}
$$

We remark (cf. [12]) if criterion (34) is satisfied then *χ<sup>A</sup>* is stable which due to the form (10) has the consequence that all coefficients of *χ<sup>A</sup>* are positive:

$$a\_0 > 0, \qquad a\_1 > 0, \qquad \dots, \qquad a\_{d-1} > 0. \tag{47}$$

Clearly, if (47) holds then condition (34) is redundant: many of inequalities in (34) are unnecessary. For *χ<sup>A</sup>* to be a Hurwitz stable, a necessary and sufficient condition can be established which requires about half amount of computations needed in the criterion of Routh-Hurwitz (cf. [12, 13]).

**Theorem 1.5** (Liénard-Chipart) The following statements are equivalent:

1. the characteristic polynomial *χ<sup>A</sup>* in (7) is Hurwitz stable;

2.*a*<sup>0</sup> > 0, *a*<sup>2</sup> > 0, … ; Δ<sup>1</sup> >0, Δ<sup>3</sup> >0, … ; 3.*a*<sup>0</sup> > 0, *a*<sup>2</sup> > 0, … ; Δ<sup>2</sup> >0, Δ<sup>4</sup> > 0, … ;

$$\text{4.}\ a\_0 > 0,\ a\_1 > 0,\ a\_3 > 0 \ \dots \text{;}\ \Delta\_1 > 0,\ \Delta\_3 > 0,\ \dots \text{;}\ \Delta\_2 > 0$$

5.*a*<sup>0</sup> > 0, *a*<sup>1</sup> >0, *a*<sup>3</sup> >0 … ; Δ<sup>2</sup> >0, Δ<sup>4</sup> >0, … .

**Example 2.** For *α*, *β* ∈ the matrix

$$A \coloneqq \begin{bmatrix} \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1} \\\\ a\boldsymbol{\beta} - \mathbf{1} & -2 & -3 & -2 & -1 \end{bmatrix} \tag{48}$$

has the characteristic polynomial

$$\chi\_A(z) \coloneqq 1 - a\beta + 2z + 3z^2 + 2z^3 + z^4 \qquad (z \in \mathbb{C}).\tag{49}$$

It suffices to calculate

$$\Delta\_{4} = \det(\mathcal{H}\_{\chi\_{4}}) = (1 - a\beta)\Delta\_{3} = (1 - a\beta) \cdot \det\begin{bmatrix} 2 & 2 & 0\\ 1 & 3 & 1 - a\beta\\ 0 & 2 & 2 \end{bmatrix} \tag{50}$$

$$= (1 - a\beta) \cdot \{12 - 4(1 - a\beta) - 4\} = 4(1 - a\beta)(1 + a\beta).$$

Thus, *χ<sup>A</sup>* and hence *A* is stable if and only if

$$(\mathbf{1} - a\beta > \mathbf{0} \text{ and } \mathbf{1} + a\beta > \mathbf{0}), \quad \text{i.e.} \quad -\mathbf{1} < a\beta < \mathbf{1} \tag{51}$$

holds (cf. **Figure 1**).

There is a criterion of geometric character which is useful for the study of the stability of *χA*.

**Definition 1.** *The curve*

$$\Gamma\_A \coloneqq \{ \chi\_A(ioo) = (\Re(\chi\_A(ioo)), \Im(\chi\_A(ioo))) \in \mathbb{C} \, : \, \, o \in \mathbb{R} \}\tag{52}$$

is called Mikhailov hodograph or amplitude-phase curve (cf. [14]).

Some geometrical properties of the hodograph Γ*<sup>A</sup>* are in strong relationship to the stability of the characteristic polynomial *χA*. This polynomial has no zero on the imaginary axis if and only if the origin does not lie on the curve Γ*A*. In this case the function

$$\rho\_{\chi\_A}(oo) \coloneqq \arg(\chi\_A(ioo)) = \mathfrak{S}(\log(\chi\_A(ioo))) \qquad (o \in \mathbb{R}) \tag{53}$$

is continuous in every point of the real line. Moreover, we deal with the change

$$
\Delta\_{\chi\_A} := \Delta\_{\alpha \in (-\infty,\infty)} \varphi\_{\chi\_A}(\alpha) \tag{54}
$$

**Figure 1.** *Stability chart of the polynomial* <sup>∍</sup> *<sup>z</sup>* <sup>↦</sup> *<sup>z</sup>*<sup>4</sup> <sup>þ</sup> <sup>2</sup>*z*<sup>3</sup> <sup>þ</sup> <sup>3</sup>*z*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*<sup>z</sup>* <sup>þ</sup> <sup>1</sup> � *αβ.*

where Δ*<sup>χ</sup><sup>A</sup>* denotes the change of argument of the vector *φχ<sup>A</sup>* ð Þ *ω* in the complex plane as *ω* increases from �∞ to þ∞. Because

$$
\chi\_A(-i a) = \chi\_A(i a),\tag{55}
$$

therefore we have

$$
\Delta\_{\boldsymbol{\alpha}\in[0,\infty)}\boldsymbol{\uprho}\_{\boldsymbol{\chi}\_{\mathcal{A}}}(\boldsymbol{\alpha}) = \frac{1}{2} \cdot \Delta\_{\boldsymbol{\alpha}\in(-\infty,\infty)}\boldsymbol{\uprho}\_{\boldsymbol{\chi}\_{\mathcal{A}}}(\boldsymbol{\alpha}).\tag{56}
$$

This means that it is enough to know the behavior of the vector *φχ<sup>A</sup>* ð Þ *ω* for 0≤*ω*< ∞. The next theorem which is known as the Mikhailov criterion of stability is based on the principle of argument. Because the form as it is in the next theorem is a special case of the one formulated in the next section we omit its proof now.

**Theorem 1.6** (Mikhailov). The polynomial *χ<sup>A</sup>* is stable if and only if the following two conditions are fulfilled:

1. the curve Γ*<sup>A</sup>* does not cross the origin, i.e. the implication

$$\chi\_A(z) = 0 \quad \Rightarrow \quad \Re(z) \neq 0$$

is true, which means that *χ<sup>A</sup>* has no zeros on the imaginary axis;

2. the curve Γ*<sup>A</sup>* encircles the origin anticlockwise at an angle *dπ=*2 while *ω* changes from 0 to þ∞, i.e.

$$\Delta\_{\alpha \in [0,\infty)} \arg(\chi\_A(i\alpha)) = \frac{d\pi}{2}$$

holds.

**Example 3.** In case of *d* ¼ 2 we have for the characteristic polynomial

$$\chi\_A(z) \coloneqq a\_0 + a\_1 z + z^2 \qquad (z \in \mathbb{C}) \;:\tag{57}$$

$$\Re(\chi\_A(i\alpha)) = a\_0 - \alpha^2, \quad \Re(\chi\_A(i\alpha)) = a\_1\alpha,\tag{58}$$

from which

$$\begin{aligned} \sin\left(\arg(\chi\_A(i\omega))\right) &= \frac{\mathfrak{T}(\chi\_A(i\omega))}{\sqrt{\left[\mathfrak{R}(\chi\_A(i\omega))\right]^2 + \left[\mathfrak{T}(p(i\omega))\right]^2}} = \frac{a\_1\omega}{\sqrt{\left[a\_0 - \omega^2\right]^2 + a\_1^2\omega^2}} \\\\ \rightarrow \text{sgn}\left(a\_1\right)\mathfrak{0} \quad (\omega \to +\infty), \end{aligned} \tag{59}$$

resp. as *ω* ! þ∞

$$\cos\left(\arg(\chi\_A(i\omega))\right) = \frac{\Re(\chi\_A(i\alpha))}{\sqrt{\left[\Re(\chi\_A(i\alpha))\right]^2 + \left[\Im(\chi\_A(i\alpha))\right]^2}} = \frac{a\_0 - a^2}{\sqrt{\left[a\_0 - a^2\right]^2 + a\_1^2 a^2}} \to -1\tag{60}$$

follows. Thus,

$$\Delta\_{w \in [0, +\infty)} \arg(\chi\_A(i\omega)) = \begin{cases} 0 & (a\_1 < 0 \ \wedge \ a\_0 < 0), \\ -\pi & (a\_1 < 0 \ \wedge \ a\_0 > 0), \\ 0 & (a\_1 > 0 \ \wedge \ a\_0 < 0), \\ \pi & (a\_1 > 0 \ \wedge \ a\_0 > 0) \end{cases} \tag{61}$$

which means that *χ<sup>A</sup>* is stable if and only if *a*<sup>0</sup> >0 and *a*<sup>1</sup> >0 hold.

Often what is to be checked is not the stability of the characteristic polynomial *χ<sup>A</sup>* but the question as to whether every zero of the polynomial lies in the interior of the unit circle around the origin of the complex plane. In this case *χ<sup>A</sup>* is called Schur stable polynomial or simply Schur polynomial. This phenomenon plays a crucial role in the stability of discrete dynamical systems and in the asymptotic stability of periodic linear systems (cf. [15]). Regarding this problem there are two main treatments. The first way to investigate the Schur stability of *χ<sup>A</sup>* is to introduce the Möbius-transformation

$$w \coloneqq \frac{z+1}{z-1} \quad (-1 \neq z \in \mathbb{C}) \qquad \langle z \coloneqq \frac{w+1}{w-1} \quad (1 \neq w \in \mathbb{C}) \rangle \tag{62}$$

which takes the interior of the unit circle of the complex plane f g *z*∈ : j*z*j< 1 into the interior of the left half-plane f g *w* ∈ : ℜð Þ *w* <0 *:* Thus, if we want to know whether the polynomial *χ<sup>A</sup>* is Schur stable we perform the transformation

$$\varphi\_A(w) \coloneqq \left(w - 1\right)^d \cdot \chi\_A\left(\frac{w+1}{w-1}\right) \qquad (1 \ne w \in \mathbb{C}).\tag{63}$$

It is clear that *ψ<sup>A</sup>* is also a polynomial of degree *d* and *χ<sup>A</sup>* is Schur stable if and only if *ψ<sup>A</sup>* is Hurwitz stable. It is not difficult to calculate (cf. [2]) that in case of

• *d* ¼ 2 the polynomial *χ<sup>A</sup>* is Schur stable if and only if

$$-1 + |a\_1| < a\_0 < 1;\tag{64}$$

• *d* ¼ 3 the polynomial *χ<sup>A</sup>* is Schur stable if and only if the inequalities

$$\mathfrak{1} + a\_1 > |a\_0 + a\_2|, \quad \mathfrak{3} - a\_1 > |\mathfrak{3}a\_0 - a\_2|, \quad \mathfrak{1} - a\_1 > a\_0(a\_0 + a\_2) \tag{65}$$

hold.

The second way is the application of the so called Jury test which proof is based on the Rouché theorem (cf. [16]).

#### **2.2 Hopf bifurcation**

In what follows we shall examine the situation when system (1) with *τ* ≔ 0 exhibits Hopf bifurcation. In order to have this, we rewrite the version of (1) without delay in the parameter-dependent form

$$
\dot{\mathfrak{x}} = f \circ (\mathfrak{x}, p) \tag{66}
$$

where *p* represents a parameter of the given system. Hopf bifurcation occurs if and only if for the eigenvalues *μ*ð Þ� *p iν*ð Þ *p* of the Jacobi matrix *A* of *f* at the critical value *p*<sup>∗</sup>

• the eigenvalue crossing condition holds:

$$
\mu(p\_\*) = 0, \quad \nu(p\_\*) \neq 0, \quad \left(\sigma(A) \backslash \{\pm \nu(p\_\*)\}\right) \cap i\mathbb{R} = \mathfrak{D}; \tag{67}
$$

• the transversality condition *μ*<sup>0</sup> *p*<sup>∗</sup> 6¼ 0 is fulfilled.

In the case of the two-dimensional system there is a result about the fulfillment of the above two conditions.

**Lemma 1.** Let *I* ⊂ be an open interval and *β*, *γ* : *I* ! smooth functions. The roots of the characteristic polynomial

$$\chi\_A(z) \coloneqq z^2 + \beta z + \gamma \qquad (z \in \mathbb{C}) \tag{68}$$

fulfill the eigenvalue crossing condition and the transversality condition if and only if at the critical value *p* ¼ *p*<sup>∗</sup> ∈*I*

$$
\beta(p\_\* \,) = 0, \qquad \gamma(p\_\* \,) > 0 \qquad \text{and} \qquad \beta'(p\_\* \,) \neq 0 \tag{69}
$$

hold.

**Proof:**

**Step 1.** The polynomial *χ<sup>A</sup>* has purely imaginary zeros �*ωi* with *ω* 6¼ 0 if and only if

$$z^2 + \beta z + \gamma = (z - a\imath)(z + a\imath) = z^2 + a\imath^2 \qquad (z \in \mathbb{C}).\tag{70}$$

Thus, at the critical value *p* ¼ *p*<sup>∗</sup> ∈*I* the eigenvalue crossing condition holds exactly in case

$$
\beta(p\_\*) = 0 \quad \text{and} \quad \gamma(p\_\*) > 0. \tag{71}
$$

**Step 2.** Let denote by *ρ* the root of the equation

$$\mathbf{z}^2 + \beta \mathbf{z} + \boldsymbol{\chi} = \mathbf{0} \tag{72}$$

which at *p*<sup>∗</sup> takes the value *ωi*: *ρ p*<sup>∗</sup> <sup>¼</sup> *<sup>ω</sup>i*, and let us introduce the following function

$$\mathcal{F}(\mathbf{z}, p) := \mathbf{z}^2 + \beta(p)\mathbf{z} + \gamma(p). \tag{73}$$

Since

$$\mathcal{F}(\rho(p\_\*), p\_\*) = 0 \quad \text{and} \quad \partial\_1 \mathcal{F}(\rho(p\_\*), p\_\*) = 2a\dot{\imath} + \beta(p\_\*) = 2a\dot{\imath} \neq 0,\tag{74}$$

therefore we have

$$\begin{split} \rho'(p\_\* \ast) &= -\frac{\partial\_2 \mathcal{F}(oi, p\_\* \ast)}{\partial\_1 \mathcal{F}(oi, p\_\* \ast)} = -\frac{\beta'(h)z + \gamma'(h)}{2z + \beta(h)} \Big|\_{z} \mathbf{z} = ai \\\ &\quad h = p\_\* \ast \\\ &= -\frac{\beta'(p\_\* \ast)ai + \gamma'(p\_\* \ast)}{2ai + \beta(p\_\* \ast)} =: \frac{A + Bi}{C + Di} .\end{split} \tag{75}$$

Using the well known calculation

$$\frac{A+Bi}{C+Di} = \frac{A+Bi}{C+Di} \cdot \frac{C-Di}{C-Di} = \frac{(AC+BD)-(AD-BC)i}{C^2+D^2} = \frac{AC+BD}{C^2+D^2} + \frac{BC-AD}{C^2+D^2}i,\tag{76}$$

furthermore the first and third part of (69) the formula

$$\frac{\text{d}\Re\left(\rho\left(p\_{\ast}\right)\right)}{\text{d}h} = \Re\left(\rho'\left(p\_{\ast}\right)\right) = \frac{2\alpha^{2}\beta'\left(p\_{\ast}\right) + \gamma'\left(p\_{\ast}\right)\rho\left(p\_{\ast}\right)}{4\alpha^{2} + \left(\beta\left(p\_{\ast}\right)\right)^{2}} = -\frac{\beta'\left(p\_{\ast}\right)}{2} \neq \mathbf{0} \tag{77}$$

proves the lemma. ■

Because a matrix of order two can have no other eigenvalues besides the critical eigenvalues the crossing can happen only if for suitable *r*>0

$$\beta^2(h) - 4\gamma(h) < 0 \qquad \left(h \in \left(p\_\* - r, p\_\* + r\right)\right) \tag{78}$$

holds.

We have to remark that there are two forms of Hopf bifurcation: the standard one and the so called non-standard. Under standard Hopf bifurcation we mean the phenomenon when the critical eigenvalues of the Jacobian matrix *A* cross the imaginary axis from left to right and all other eigenvalues remain in the open left complex plane, whereas non-standard Hopf bifurcation means that the critical eigenvalues cross the imaginary axis from the right to the left and there is no restriction for the location of the other eigenvalues.

**Theorem 1.7.** If (78) holds then crossing can only happen


**Proof:** If condition (78) holds, then *χ<sup>A</sup>* has a pair of conjugate roots. From elementary mathematics, we know that


**Example 4.** Let be 0 <*a*, *b*∈ and consider the activator-inhibitor system of Schnackenberg-type

$$
\dot{\mathbf{x}} = \mathbf{a} - \mathbf{x} + \mathbf{x}^2 \mathbf{y}, \qquad \dot{\mathbf{y}} = \mathbf{b} - \mathbf{x}^2 \mathbf{y} \tag{79}
$$

(cf. [17]). System (79) has the unique equilibrium point

$$\left(\left(\mathbf{x}\_{\*},\boldsymbol{y}\_{\*}\right)\right) = \left(a+b,\frac{b}{\left(a+b\right)^{2}}\right). \tag{80}$$

The Jacobian of (79) at *x*<sup>∗</sup> , *y* <sup>∗</sup> � � takes the form *Characteristic Polynomials DOI: http://dx.doi.org/10.5772/intechopen.100200*

$$A \coloneqq \mathcal{J}(\mathbf{x}\_\*, \mathcal{Y}\_\*) = \begin{bmatrix} 2\mathbf{x}\_\*\mathcal{Y}\_\* - \mathbf{1} & \mathbf{x}\_\*^2 \\ -2\mathbf{x}\_\*\mathcal{Y}\_\* & -\mathbf{x}\_\*^2 \end{bmatrix} \tag{81}$$

whose eigenvalues are the zeros of its characteristic polynomial

$$\chi\_A(\mathbf{z}) = \det(\mathbf{z}I\_2 - A) = \mathbf{z}^2 - \operatorname{Tr}(A)\mathbf{z} + \det(A) = \mathbf{z}^2 + \left(\mathbb{1} - 2\mathbf{z}\_\*\mathbb{1}\_\* + \mathbf{z}\_\*^2\right)\mathbf{z} + \mathbf{z}\_\*^2$$

$$= \mathbf{z}^2 + \frac{a - b + (a + b)}{a + b}\mathbf{z} + (a + b)^2 =: \mathbf{z}^2 + \boldsymbol{\beta}\mathbf{z} + \boldsymbol{\gamma} \quad (\mathbf{z} \in \mathbb{C}). \tag{82}$$

It is easy to see that if we choose *b* ≕ *p* as parameter by fixed *a* then for every *p*ð Þ >0 we have *γ*ð Þ *p* >0 and

$$\rho'(p) = \frac{\left[-\mathbb{1} + \mathbb{3}(a+p)^2\right](a+p) - a + p - (a+p)^3}{\left(a+p\right)^2} = \frac{\mathbb{2}(a+p)^3 - 2a}{\left(a+p\right)^2} \neq 0. \tag{83}$$

This means that Hopf bifurcation occurs at the critical value *p*<sup>∗</sup> if and only if *p*<sup>∗</sup> is a positive real solution of the equation

$$\kappa(p) \coloneqq (\mathfrak{a} + p) \cdot \beta(\mathfrak{h}) = \mathfrak{a} - p + (\mathfrak{a} + p)^{\mathfrak{I}} = \mathbf{0} \tag{84}$$

extended in *p* to the whole real axis. For example,

• In case of *a* ¼ 1 the polynomial *κ* has one real root:

$$\kappa(p) = 0 \quad \Longleftrightarrow \quad p \in \left\{0, \frac{-3 - \sqrt{7}i}{2}, \frac{-3 + \sqrt{7}i}{2}\right\}.\tag{85}$$

Clearly, *κ*ð Þ¼ 1 8>0, therefore *κ* and so *β* assume on the positive half line positive values, which has a consequence that the characteristic polynomial and hence the equilibrium point *x*<sup>∗</sup> , *y* <sup>∗</sup> � � is stable, since a second order characteristic polynomial is stable if and only if its coefficients have the same (positive) sign.

• in case of *a* ¼ 0*:*1 the polynomial *κ* has three real roots:

$$\kappa(h) = 0 \quad \Longleftrightarrow \quad h \in \{p\_1 \coloneqq -1.18803\ldots, \ p\_2 \coloneqq 0.109149\ldots, \ p\_3 \coloneqq 0.778885\ldots\}.\tag{86}$$

Because

$$\kappa(\mathbf{0}) = a + a^3 > \mathbf{0}, \qquad \kappa(\mathbf{0}.5) = \mathbf{0}.184 < \mathbf{0}, \qquad \kappa(\mathbf{1}) = a \left(a^2 + 3a + 4\right) > \mathbf{0}, \tag{87}$$

the polynomial *κ* and so *β* changes its sign at *p*<sup>2</sup> from positive to negative, and at *p*<sup>3</sup> from negative to positive. This means in the light of the above that at the parameter value *p* ¼ *p*<sup>2</sup> standard Hopf bifurcation occurs: the roots migrate from the left open half plane to the right, *x*<sup>∗</sup> , *y* <sup>∗</sup> � � loses its stability; furthermore at the parameter value *p* ¼ *p*<sup>3</sup> non-standard Hop bifurcation takes place, i.e. the roots migrate from the right half plane to the left and as a consequence *x*<sup>∗</sup> , *y* <sup>∗</sup> � � becomes stable.

In the case of the three-dimensional system we post a result (cf. [18]), the proof of which is similar to the one in [19].

**Lemma 2.** Let be *I* ⊂ an open interval and *α*, *β*, *γ* : *I* ! smooth functions. The roots of the characteristic polynomial

$$\chi\_A(z) \coloneqq z^3 + az^2 + \beta z + \chi \qquad (z \in \mathbb{C}) \tag{88}$$

fulfill the crossing and the transversality conditions if and only if at the critical value *p* ¼ *p*<sup>∗</sup> ∈*I*

$$
\rho(p\_\* \Box > 0, \quad a(p\_\* \Box \neq 0, \quad \gamma(p\_\* \Box = a(p\_\* \Box) \beta(p\_\* \Box))\tag{89}
$$

furthermore

$$\frac{\mathbf{d}}{\mathbf{d}h} \{ \alpha(h)\beta(h) - \gamma(h) \}|\_{h=p\_\*} \neq \mathbf{0} \tag{90}$$

hold.

**Proof:**

**Step 1.** We show that the characteristic polynomial *χ<sup>A</sup>* has purely imaginary roots �*ωi* (*ω* 6¼ 0), if *αβ* � *γ* ¼ 0 and *β* >0 hold. Indeed, if *ξ*, *η*, *ζ* denote the roots of the polynomial *χA*, then using Orlando formula we have

$$
\beta = \xi \eta + \xi \zeta + \eta \zeta, \text{ resp. } a\beta - \chi = - (\xi + \eta)(\xi + \zeta)(\eta + \zeta). \tag{91}
$$

This means that

• if for some 0 6¼ *ω*∈ the equalities

$$\chi\_A(ai) = \mathbf{0} = \chi\_A(-ai) \tag{92}$$

hold, then one of the three zeros, like *ζ* is real, furthermore

$$
\xi = o\dot{\imath} = -\eta,\ \text{ resp. } \beta = o^2 + \zeta o\dot{\imath} - \zeta o\dot{\imath} = o^2 > 0;\tag{93}
$$

• *χ<sup>A</sup>* has exactly zeros with opposite sign but the same absolute value, if *αβ* � *γ* ¼ 0 holds, furthermore *χ<sup>A</sup>* has a complex root, since if *ξ* ¼ �*η* then

$$
\beta = -\xi^2 + \xi\zeta - \xi\zeta = -\xi^2 \le 0,\tag{94}
$$

which contradicts the fact that *β* >0.

It is clear that from conditions (89) it follows that the zeros of *χ<sup>A</sup>* are

$$-a \quad \text{and} \quad \pm \sqrt{\beta}i = \pm \sqrt{\frac{\gamma}{a}}i,\tag{95}$$

because

$$(\mathbf{z} + a) \left(\mathbf{z} - \sqrt{\beta}\mathbf{i}\right) \left(\mathbf{z} + \sqrt{\beta}\mathbf{i}\right) \equiv (\mathbf{z} + a) \left(\mathbf{z}^2 + \beta\right) \equiv \mathbf{z}^3 + a\mathbf{z}^2 + \beta\mathbf{z} + a\beta \tag{96}$$

$$\equiv \mathbf{z}^3 + a\mathbf{z}^2 + \beta\mathbf{z} + \chi.$$

**Step 2.** Let denote the roots of *χ<sup>A</sup>* by *ρ* which assumes at *p*<sup>∗</sup> the value *ωi*: *ρ p*<sup>∗</sup> � � <sup>¼</sup> *<sup>ω</sup>i*, furthermore let define

$$\mathcal{F}(z,h) := z^3 + a(h)z^2 + \beta(h)z + \gamma(h). \tag{97}$$

$$\mathcal{F}(\rho(p\_\*), p\_\*) = 0,\text{ and }\partial\_1 \mathcal{F}(oi, p\_\*) = \beta(p\_\*) - 3o^2 + 2a(p\_\*)oi \neq 0,\tag{98}$$

$$\begin{split} \rho'(p\_\*) &= -\frac{\partial\_2 \mathcal{F}(ai, p\_\*)}{\partial\_1 \mathcal{F}(ai, p\_\*)} = -\frac{a'(h)z^\sharp + \beta'(h)z + \gamma'(h)}{3z^2 + 2a(h)z + \beta(h)} \bigg|\_{h} = \alpha i \\\ &= \frac{a'(p\_\*)\alpha^2 - \beta'(p\_\*)ai - \gamma'(p\_\*)}{-3\alpha^2 + 2a(p\_\*)ai + \beta(p\_\*)}. \end{split} \tag{99}$$

$$\beta(h) = \alpha^2(h) \text{ and } \gamma(h) \notin \{0, \zeta(h)\} \tag{101}$$

$$
\zeta(h) \coloneqq \frac{3\alpha(h)\beta(h) - 2\alpha^2(h)}{9} \quad (h \in I), \tag{102}
$$

$$3\sqrt[3]{\gamma(h) - \zeta(h)} < 2a(h) \tag{103}$$

$$3\sqrt[3]{\chi(h) - \zeta(h)} > 2a(h) \tag{104}$$

**Proof:** Using the notations

$$a \coloneqq \frac{a}{3}, \quad b \coloneqq \frac{\beta}{3}, \quad \text{resp.} \quad A \coloneqq a^2 - b, \quad B \coloneqq 2a^2 - 3ab + \chi \tag{105}$$

one can see that if *A* ¼ 0 and 0 6¼ *B*∈ hold then the zeros of the polynomial *χ<sup>A</sup>* are as follows (cf. [17]):

$$\xi = -\sqrt[3]{B} - a, \qquad \eta = \frac{\sqrt[3]{B}}{2} - a + \frac{3\sqrt[3]{B^2}}{4}i, \qquad \zeta = \frac{\sqrt[3]{B}}{2} - a + \frac{3\sqrt[3]{B^2}}{4}i. \tag{106}$$

Thus, from *<sup>B</sup>* 6¼ 0 and ffiffiffi *B* <sup>p</sup><sup>3</sup> 6¼ <sup>2</sup>*<sup>a</sup>* it follows that *<sup>χ</sup><sup>A</sup>* has a pair of complex conjugate roots. Because of condition *γ* 6¼ 0 the third root could not be zero, furthermore this pair of complex conjugate zeros lies


**Example 5.** In [20] Liao, Zhou, and Tang proposed the following autonomous system of ordinary differential equations

$$
\dot{\mathbf{x}} = \mathbf{a}(\mathbf{y} - \mathbf{x}), \qquad \dot{\mathbf{y}} = d\mathbf{x} + c\mathbf{y} - \mathbf{x}\mathbf{z}, \qquad \dot{\mathbf{z}} = -b\mathbf{z} + \mathbf{x}\mathbf{y} \tag{107}
$$

where *a*, *b*,*c*, *d*∈ . If ð Þ *a*, *b*,*c*, *d* belongs to the set

$$\{(a, b, -1, d), \quad (a, b, c, c - a), \quad (a, b, c, 0)\}\tag{108}$$

then we get the Lorenz system (cf. [21]), the Chen system (cf. [22]) and the Lü system (cf. [23]). Yan showed (cf. [24]) that in the system (107) Hopf bifurcation may occur. In what follows we show that his calculations can be simplified as we know the coefficients of the characteristic polynomials of the Jacobian of system (107). If *b c*ð Þ þ *d* >0 holds then system (107) has three equilibria:

$$E\_0 \coloneqq (0,0,0), \qquad E\_{\pm} \coloneqq \left( \pm \sqrt{b(c+d)}, \pm \sqrt{b(c+d)}, c+d \right). \tag{109}$$

The Jacobian of (107) takes the form

$$f(x,y,z) \coloneqq \begin{bmatrix} -a & a & 0 \\ d - z & c & -x \\ y & x & -b \end{bmatrix} \qquad \left( (x,y,z) \in \mathbb{R}^3 \right). \tag{110}$$

Hence the corresponding Jacobians are:

$$A\_0 \coloneqq J(E\_0) \coloneqq \begin{bmatrix} -a & a & \mathbf{0} \\ d & c & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & -b \end{bmatrix},\tag{111}$$

resp.

$$A\_{\pm} \coloneqq f(E\_{\pm}) \coloneqq \begin{bmatrix} -a & a & 0 \\ d - (c+d) & c & \mp\sqrt{b(c+d)} \\ \pm\sqrt{b(c+d)} & \pm\sqrt{b(c+d)} & -b \end{bmatrix}.\tag{112}$$

The eigenvalues of *A*<sup>0</sup> are the roots of the characteristic polynomial

$$\chi\_{A\_0}(\xi) := \det(\xi I\_3 - J\_0) = \xi^3 + a\xi^2 + \beta\xi + \gamma \qquad (\xi \in \mathbb{C}), \tag{113}$$

where *γ* ≔ � detð Þ¼� *A*<sup>0</sup> *ab c*ð Þ þ *d* , *β* ≔ Tr adj ð Þ¼ ð Þ *A*<sup>0</sup> *ab* � *bc* � *ac* � *ad* and *α* ≔ � Trð Þ¼ *A*<sup>0</sup> *a* þ *b* � *c*.

We remark that only a special parameter configuration was investigated in [24], namely *a* ¼ *c*, *d* ¼ �2*c c*ð Þ >0 . On the other hand one can observe more due to Lemma 2:


$$(y - a\theta)(d\_\* \, ) = 0 \quad \Longleftrightarrow \quad d\_\* \, = \frac{ab + b^2 - ac - bc}{a}$$

(in case of *<sup>a</sup>* <sup>¼</sup> 0 the system would be two dimensional) and *<sup>β</sup>*ð Þ¼� *<sup>d</sup>*<sup>∗</sup> *<sup>b</sup>*<sup>2</sup> , this contradicts the first condition in (89).

3. the parameter *c* can be chosen as a bifurcation parameter only under some restrictions, because in the case of *a* 6¼ �*b*

$$(\gamma - a\theta)(c\_\*\,) = 0 \quad \Longleftrightarrow \quad c\_\* \in \left\{ a, \, \frac{ab + b^2 - ad}{a + b} \right\}$$

and

$$\beta(c\_\*) \in \{-a(a+d), \ -b^2\}$$

will be positive only in case if *a* and *a* þ *d* have opposite signs as it is in case *a* ¼ *c*, *d* ¼ �2*c c*ð Þ > 0 proposed in [24]. If *a* ¼ �*b* and *a* 6¼ 0 then

$$(y - a\beta)(c\_\*\,) = \mathbf{0} \quad \Longleftrightarrow \quad c\_\*\, = \frac{ab + b^2 - ac - bc}{a}$$

(in case of *<sup>a</sup>* <sup>¼</sup> 0 the system would be two dimensional) and *<sup>β</sup>*ð Þ¼� *<sup>c</sup>* <sup>∗</sup> *<sup>b</sup>*<sup>2</sup> which contradicts the first condition in (89). If *a* ¼ *c* then

$$(\gamma - a\beta)(b\_\*\,\_{\*}) = \mathbf{0} \quad \Longleftrightarrow \quad d = \mathbf{0}.$$

In this case we have

$$\frac{\mathbf{d}}{\mathbf{d}b}(\boldsymbol{\gamma} - a\boldsymbol{\beta})(\boldsymbol{b}\_\*) = \mathbf{0},$$

which contradicts the transversality condition.

It is easy to see the following: by fixing the parameters *b*,*c*, resp. *d* ¼ �2*c c*ð Þ >0 (cf. [24]) the parameter *a* will be chosen as bifurcation parameter then at value *a*<sup>∗</sup> ≔ *c* Hopf bifurcation takes place, because with substitution *d* ¼ �2*c* we have

$$\beta(a\_\* \,) = cb - bc - cc - c(-2c) = c^2 > 0, \qquad a(a\_\* \,) = b \neq 0,\tag{114}$$

resp.

$$(y - a\theta)(a\_\* \,) = [abc - (a+b-c)(ab-bc+ac)]\_{a=c} = bc^2 - bc^2 = 0 \tag{115}$$

and

$$\frac{\mathbf{d}}{\mathbf{d}a} \{ \chi - a\beta \} (a \,\_{\ast} \,) = \frac{\mathbf{d}}{\mathbf{d}a} (abc - (a+b-c)(ab-bc+ac)) \Big|\_{a=\pm}$$

$$\mathbf{d} = [bc - (ab - bc + ac) - (a+b-c)(b+c)]\_{a=\pm} = bc - c^2 - b(b+c) = -c^2 - b^2 \neq 0.$$

The eigenvalues of the matrix *A*� are the zeros of the characteristic polynomial

$$\chi\_{A\_{\pm}}(\xi) := \det(\xi I\_3 - A\_{\pm}) = \xi^3 + a\xi^2 + \beta\xi + \gamma \qquad (\xi \in \mathbb{C}) \tag{117}$$

where

$$y \coloneqq -\det(I) = 2ab(c+d), \quad \beta \coloneqq \operatorname{Tr}(\operatorname{adj}(I)) = b(a+d), \quad a \coloneqq -\operatorname{Tr}(I) = a+b-c. \tag{118}$$

If

$$a(\mathfrak{A}a+d)(a+b+d) \neq \mathbf{0},\tag{119}$$

(116)

then at

$$\mathcal{L}\_{\ast} = \frac{a^2 + ab - ad + bd}{3a + d} \tag{120}$$

Hopf bifurcation takes place, because

$$\beta(c\_\* \,) = b(a+d) > 0, \qquad a(c\_\* \,) = \frac{2a(a+b+d)}{3a+d}, \qquad (\gamma - a\beta)(c\_\* \,) = 0,\tag{121}$$

resp.

$$\frac{\mathbf{d}}{\mathbf{d}c}(\mathbf{y} - a\theta)(\mathbf{c}\_\*) = -2ab - b(\mathbf{a} + d) = -b(3a + d) \neq \mathbf{0}.\tag{122}$$

#### **3. The delayed case:** *τ* **> 0**

When modeling and analyzing processes and behaviors which come from a natural environment, it often happens that we need a bit of distance in time to see the changes of the considered quantities (which are the variables of our model). For example, when we think about the epidemiological models, it is a well-founded thought that we need some time while susceptibles become infectious, and hence it is reasonable to assume that the migration of the individuals from the class of susceptibles into the infected is subject to delay.

*Characteristic Polynomials DOI: http://dx.doi.org/10.5772/intechopen.100200*

Another expressive example is the modeling of the processes of the human body or the brain, like emotions: love, hate, etc. A bit of time has to pass for the brain to process the signals coming from various places, and only after this delay, the mood could change. These changes can be described and analyzed with delayed differential equations. One type of these systems is the so called Romeo and Juliet model, where the changes of Romeo's and Juliet's love and hate in time are described as a system of two linear ordinary differential equations. In this chapter we are going to consider this model with general coefficients and investigate the stability of the linear system.

We are going to consider the following linear system:

$$\begin{aligned} \dot{\mathbf{x}} &= a\_1 \mathbf{x} + A\_1 \mathbf{x} (\cdot - \boldsymbol{\tau}) + a\_2 \mathbf{y} + A\_2 \mathbf{y} (\cdot - \boldsymbol{\tau}), \\ \dot{\mathbf{y}} &= a\_3 \mathbf{x} + A\_3 \mathbf{x} (\cdot - \boldsymbol{\tau}) + a\_4 \mathbf{y} + A\_4 \mathbf{y} (\cdot - \boldsymbol{\tau}), \end{aligned} \tag{123}$$

where *Ai*, *α<sup>i</sup>* ∈ , (*i*∈f g 1, 2, 3, 4 ), *τ* >0, with initial conditions Φ ¼ ð Þ Φ1, Φ<sup>2</sup> in the Banach space

$$\{\Phi \in \mathcal{C}([-\tau, 0], \mathbb{R}\_+^2) \, : \, \Phi\_1(\theta) = \mathfrak{x}(\theta), \Phi\_2(\theta) = \mathfrak{y}(\theta)\},\tag{124}$$

where Φ*i*ð Þ*θ* >0, (*θ* ∈½ � �*τ*, 0 , *i* ∈f g 1, 2 ). Straightforward calculation shows that the characteristic function of the above system (with regard to the trivial equilibrium point) takes the form:

$$\begin{split} \Delta(\mathbf{z}, \mathbf{r}) & \equiv \mathbf{z}^2 - (a\_1 + a\_4)\mathbf{z} + a\_1 a\_4 - a\_2 a\_3 + \dots + e^{-\mathbf{z}\tau} \cdot (-(A\_1 + A\_4)\mathbf{z} + a\_1 A\_4 \\ & + a\_4 A\_1 - a\_2 A\_3 - a\_3 A\_2) + e^{-2\mathbf{z}\tau} (A\_1 A\_4 - A\_2 A\_3) \quad (\mathbf{z} \in \mathbb{C}). \end{split} \tag{125}$$

In [25] the authors treat delay differential equations, which characteristic function for arbitrary *z*∈ has the form

$$
\Delta(z,\tau) = z^2 + a\_1 z + a\_0 + (b\_1 z + b\_0)e^{-x\tau} + c e^{-2x\tau} \tag{126}
$$

where *a*0, *a*1, *b*0, *b*1, *c* are arbitrary real constants. It can be seen that the characteristic functions (125) and (126) has the same form. In [25] the authors assume that *c* ¼ 0 to simplify the analysis. Furthermore, they say that due to the continuous dependence of eigenvalues of the model parameters (cf. [26]) their results are valid for sufficiently small *c* parameters, too. Nevertheless, in the literature a lot of models and systems are investigated in which the coefficient of *e*�2*z<sup>τ</sup>* of the characteristic function is not equal to zero, and maybe not sufficiently small. Therefore, the aim of this section is to show that we can analyze the stability of the system in the case where *c* ¼6 0, too.

In this section we assume that *τ* >0 holds and investigate the qualitative behavior of the linearized system (3), more precisely we study the stability of characteristic function

$$\Delta(z,\tau) := p(z) + q(z)e^{-\mathbf{z}\tau} + r(z)e^{-2\tau\tau} \tag{127}$$

where *p*, *q* and *r* are polynomials with real coefficients and degð Þ*r* < degð Þ*q* . Under stability of Δ we mean that the zeros of Δ lie in the open left half of the complex plane. Using the Mikhailov criterion we give for special *p*, *q* and *r* fulfilling the above condition an estimate on the length of delay *τ* for which no stability switching occurs. Then for special parameters we compare our results with other methods. It follows then a delay independent stability analysis. Finally, a formula for Hopf bifurcation is calculated in terms of *p*, *q* and *r*. If we assume that the characteristic function has the form as in (126), then we can give conditions easily

on the parameters *a*1, *a*0, *b*1, *b*0,*c* and an upper bound *τ*<sup>1</sup> such that with *τ* <*τ*<sup>1</sup> the system is asymptotically stable. In other words stability change may happen only for *τ* ≥*τ*1.

In what follows, the Mikhailov stability criterion will be proved, which is the implication of the argument principle (cf. [27, 28]). The treatment is based on [29].

**Theorem 1.9** (Mikhailov criterion). Consider the quasi-polynomial

$$M(z) \coloneqq \mathbf{Q}(z) + \sum\_{k=1}^{p} R\_k(z) e^{-s\tau\_k} \qquad (z \in \mathbb{C}), \tag{128}$$

where the order of the polynomials *Q* and *Rk* is less than or equal to *d*∈ , and they are defined as

$$\mathbf{Q}(\mathbf{z}) \coloneqq q\_d \mathbf{z}^d + \dots + q\_0, \quad \mathbf{R}\_k(\mathbf{z}) \coloneqq r\_{k\_d} \mathbf{z}^d + \dots + r\_{k\_0} \qquad (\mathbf{z} \in \mathbb{C}) \tag{129}$$

where *qi* , *rki* ∈ for *i* ¼ 1, … , *d*, *k* ¼ 1, … , *p*, *qd* >0 and

$$\max\_{k \in \{1, \dots, p\}} (\deg(R\_k)) < d,\tag{130}$$

furthermore *τ<sup>k</sup>* ≥0 for *k* ¼ 1, … , *p*. If *M* defined by (128) has no zeros on the imaginary axis, then *M* is stable if and only if

$$\Delta \coloneqq \Delta\_{w \in \left[0, +\infty\right)} \arg(M(i\alpha)) = \frac{d\pi}{2} \tag{131}$$

holds where Δ denotes the change of argument of the vector *M i*ð Þ *ω* anticlockwise in the complex plane as *ω* increases from 0 to þ∞.

**Proof:** In order to prove the theorem, we will apply the argument principle (cf. [28]) to *M* on the Γ-contour (cf. **Figure 2**) where Γ ≕ *C*1∪*C*<sup>2</sup> denotes the

**Figure 2.** Γ*-contour on the complex plane.*

positive oriented curve in the complex plane which consists of the interval ½ � �*ρ*, *ρ* (*ρ*>0) on the imaginary axis, i.e.

$$\mathcal{C}\_1 \coloneqq \{ \dot{\mathfrak{s}} \in \mathbb{C} \, : \, \mathfrak{s} \in [-\rho, \rho] \}\tag{132}$$

and the semicircle *C*<sup>2</sup> of the radius *ρ* in the right-hand half-plane:

$$\mathcal{C}\_2 \coloneqq \left\{ \rho e^{i\phi} \in \mathbb{C} \, : \, \phi \in \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \right\}. \tag{133}$$

Since

$$\max\_{k \in \{1, \dots, p\}} (\deg(R\_k)) < \deg(Q), \tag{134}$$

there is only a finite number of roots of *M* in the right-half plane. On *C*<sup>2</sup> the characteristic equation can be written for every *z*∈ as follows:

$$\begin{split} M(\mathbf{z}) &= q\_d \rho^d (\cos(d\phi) + i \sin(d\phi)) + \dots + q\_0 \\ &\quad + \sum\_{k=1}^p \left\{ r\_{k(d-1)} \rho^{d-1} (\cos \left( (d-1)\phi \right) + i \sin \left( (d-1)\phi \right) ) e^{-\rho r\_k \ell^\phi} + \dots + r\_{k0} e^{-\rho r\_k \ell^\phi} \right\}. \end{split} \tag{135}$$

Now from the summation, we can write a typical term as

$$\begin{split} r\_{k(d-1)} \rho^{d-1} e^{i(d-1)\phi} e^{-\rho \tau\_k \dot{e}^{i\phi}} &= r\_{k(d-1)} \rho^{d-1} e^{i(d-1)\phi} e^{-\rho \tau\_k \cos\left(\phi\right) - i\rho \tau\_k \sin\left(\phi\right)} \\ &= r\_{k(d-1)} \rho^{d-1} e^{-\rho \tau\_k \cos\left(\phi\right)} e^{i((d-1)\phi - \rho \tau\_k \sin\left(\phi\right))} \end{split} \tag{136}$$

Therefore,

$$\begin{split} M(\mathbf{z}) &= q\_d \rho^d (\cos(d\phi) + i \sin(d\phi)) + \dots + q\_0 + \sum\_{k=1}^p e^{-\rho \tau\_k \cos(\phi)} \\ &\cdot \left[ r\_{k(d-1)} \rho^{d-1} [\cos \left( (d-1)\phi - \rho \tau\_k \sin \left( \phi \right) \right) + i \sin \left( (d-1)\phi - \rho \tau\_k \sin \left( \phi \right) \right)] \right. \\ &\left. + r\_{k(d-2)} \rho^{d-2} [\cos \left( (d-2)\phi - \rho \tau\_k \sin \left( \phi \right) \right) + i \sin \left( (d-2)\phi - \rho \tau\_k \sin \left( \phi \right) \right)] \right] \\ &\left. + \dots + r\_{k\_0} [\cos \left( \rho \tau\_k \sin \left( \phi \right) \right) - i \sin \left( \rho \tau\_k \sin \left( \phi \right) \right)] \right]. \end{split} \tag{137}$$

Hence the argument or phase *θ* of the vector *M z*ð Þ on *C*<sup>2</sup> may be written

$$\tan\left(\theta\right) = \frac{\sin\left(\theta\right)}{\cos\left(\theta\right)} =: \frac{A}{B},\tag{138}$$

where

$$\begin{aligned} A & \coloneqq \sin\left(\theta\right) = q\_d \rho^d \sin\left(d\phi\right) + \dots + q\_1 \rho \sin\left(\phi\right) \sum\_{k=1}^p e^{-\rho \tau\_k \cos\left(\phi\right)} \\ & \cdot \left\{ r\_{k(d-1)} \rho^{d-1} \sin\left((d-1)\phi - \rho \tau\_k \sin\left(\phi\right)\right) \\ & + \dots + r\_{k1} \rho \sin\left(\phi - \rho \tau\_k \sin\left(\phi\right)\right) - r\_{k\_0} \sin\left(\rho \tau\_k \sin\left(\phi\right)\right) \right\} \end{aligned} \tag{139}$$

and

$$B \coloneqq \cos\left(\theta\right) = q\_d \rho^d \cos\left(d\phi\right) + \dots + q\_1 \rho \cos\left(\phi\right) + q\_0 + \sum\_{k=1}^p e^{-\rho \tau\_k \cos\left(\phi\right)}\tag{140}$$

$$\cdot \left\{r\_{k(d-1)} \rho^{d-1} \cos\left((d-1)\phi - \rho \tau\_k \sin\left(\phi\right)\right) + \dots + r\_{k\_0} \cos\left(\rho \tau\_k \sin\left(\phi\right)\right)\right\}.$$

Dividing the numerator and denominator by *ρ<sup>d</sup>* gives

$$\tan\left(\theta\right) =: \frac{A\_1}{B\_1},\tag{141}$$

where

$$A\_1 \coloneqq q\_d \sin\left(d\phi\right) + \dots \; + \frac{q\_1}{\rho^{d-1}} \sin\left(\phi\right) + \sum\_{k=1}^p e^{-\rho \tau\_k \cos\left(\phi\right)}$$

$$\cdot \left\{ \frac{r\_{k(d-1)}}{\rho} \sin\left((d-1)\phi - \rho \tau\_k \sin\left(\phi\right)\right) \right. \tag{142}$$

$$+ \dots + \frac{r\_{k\_1}}{\rho^{d-1}} \sin\left(\phi - \rho \tau\_k \sin\left(\phi\right)\right) - \frac{r\_{k\_0}}{\rho^d} \sin\left(\rho \tau\_k \sin\left(\phi\right)\right) \right\}$$

and

$$\begin{aligned} B\_1 &:= q\_d \cos \left( d\phi \right) + \dots + \frac{q\_1}{\rho^{d-1}} \cos \left( \phi \right) + \frac{q\_0}{\rho^d} + \sum\_{k=1}^p e^{-\rho \tau\_k \cos \left( \phi \right)} \\ &\cdot \left\{ \frac{r\_{k(d-1)}}{\rho} \cos \left( (d-1)\phi - \rho \tau\_k \sin \left( \phi \right) \right) + \dots + \frac{r\_{k\_0}}{\rho^d} \cos \left( \rho \tau\_k \sin \left( \phi \right) \right) \right\}. \end{aligned} \tag{143}$$

Now since

$$|\cos\left(a\right)| \le \mathbf{1} \text{ and } |\sin\left(a\right)| \le \mathbf{1} \qquad (a \in \mathbb{R}) \tag{144}$$

and since

$$\cos\left(a\right) \ge 0 \qquad \left(a \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\right),\tag{145}$$

we have

$$\tan\left(\theta\right) = \frac{\sin\left(d\phi\right)}{\cos\left(n\phi\right)} = \tan\left(d\phi\right) \tag{146}$$

or

$$
\theta = d\phi + m\pi, \qquad m \in \mathbb{Z} \tag{147}
$$

as *ρ* ! ∞. Therefore, the change in argument of *M z*ð Þ on *C*<sup>2</sup> is given by

$$
\Delta\_{\rm C2} \arg(M) = \frac{d\pi}{2} + m\pi - \left(-\frac{d\pi}{2} + m\pi\right) = d\pi. \tag{148}
$$

Now from the argument principle we can write

$$
\Delta\_{\text{C1}} \text{arg}(\mathbf{M}) + \Delta\_{\text{C2}} \text{arg}(\mathbf{M}) = 2\pi \mathbf{N},\tag{149}
$$

where *N* is the total number of zeros of *M* inside Γ. Therefore

$$
\Delta\_{\mathcal{C}\_1} \text{arg}(\mathcal{M}) = 2\pi \mathcal{N} - d\pi. \tag{150}
$$

If we reverse the direction of integration along *C*<sup>1</sup> and note the symmetry about the real axis, we have

$$\Delta\_{\boldsymbol{\alpha}\in[0,+\infty)}\arg(\mathcal{M}(i\boldsymbol{\alpha})) = \frac{1}{2}(d\boldsymbol{\pi} - 2\boldsymbol{N}) = \frac{d\boldsymbol{\pi}}{2} - \boldsymbol{\pi}\boldsymbol{N}.\tag{151}$$

in case of stability we have *N* ¼ 0. Hence, stability requires

$$\Delta\_{o\ell \in [0, +\infty)} \arg(M(i\alpha)) = \frac{d\pi}{2}. \quad \blacksquare \tag{152}$$

As a consequence, we have the following.

**Lemma 3.** Let *p*, *q* and *r* be polynomials, with condition

$$\deg(p) > \max\{\deg(q), \deg(r)\},\tag{153}$$

and assume that the quasi-polynomial in (127) has no roots on the imaginary axis. Then Δð Þ �, *τ* is stable, i.e. all of its roots have negative real part if and only if

$$\left[\arg \Delta(i o , \tau)\right]\_{o=0}^{w=+\infty} = \frac{\pi}{2} \cdot \deg(p(i o)),\tag{154}$$

i.e. the argument of Δð Þ *iω*, *τ* increases *π=*2 � degð Þ *p i*ð Þ *ω* as *ω* increases from 0 to <sup>þ</sup>∞. □

**Theorem 1.10** If for the delay parameter *τ* in the characteristic function (126)

$$
\tau < \frac{a\_1 - |b\_1|}{|b\_0| + 2|c|}, \qquad a\_0 + b\_0 + c > 0 \tag{155}
$$

hold, then the characteristic function, and hence the trivial equilibrium point of system (123) is asymptotically stable.

**Proof:** Substituting *z* ¼ *iω ω*ð Þ >0 into (126), we get

$$
\Delta(i o \sigma, \tau) = p(i o) + q(i o)e^{-i o \tau} + r(i o)e^{-2i o \tau}. \tag{156}
$$

Hence using the characteristic function (126), where

$$p(z) \equiv z^2 + a\_1 z + a\_0, \quad q(z) \equiv b\_1 z + b\_0, \quad r(z) \equiv c \tag{157}$$

we have for *ω*>0 that

$$\Delta\_R(i\nu,\tau) \equiv -\alpha^2 + a\_0 + b\_0 \cos\left(\alpha\tau\right) + b\_1 \alpha \sin\left(\alpha\tau\right) + c \cos^2(\alpha\tau) - c \sin^2(\alpha\tau),\tag{158}$$

and also

$$\Delta\_I(i\nu,\tau) \equiv a\_1\nu + b\_1\nu\cos\left(\nu\tau\right) - b\_0\sin\left(\nu\tau\right) - 2c\sin\left(\nu\tau\right)\cos\left(\nu\tau\right). \tag{159}$$

It could be seen that

$$
\Delta\_{\mathbb{R}}(\mathbf{0}, \boldsymbol{\tau}) = a\_0 + b\_0 + \boldsymbol{c} > \mathbf{0} \quad \text{and} \quad \Delta\_{\mathbb{I}}(\mathbf{0}, \boldsymbol{\tau}) = \mathbf{0}, \tag{160}
$$

furthermore

$$\lim\_{\alpha \to +\infty} \Delta\_{\mathbb{R}}(i\alpha, \mathfrak{r}) = -\infty. \tag{161}$$

Therefore, we have to show that Δ*I*ð Þ *iω*, *τ* >0 for each *ω*>0. If it holds, then

$$\arg \left( \Delta(io, \pi) \right) = \pi,\tag{162}$$

and hence by using the Mikhailov criterion we have stability. Substituting *w* ≔ *ωτ* into (159) and multiplying the result by *τ*, we get

$$\tau \Delta\_I \left( i \frac{w}{\tau}, \tau \right) = a\_1 w + b\_1 w \cos \left( w \right) - \tau b\_0 \sin \left( w \right) - 2\pi c \sin \left( w \right) \cos \left( w \right). \tag{163}$$

By using straightforward estimations we obtain that

$$
\pi \Delta\_I \left( i \frac{\alpha}{\tau}, \tau \right) > (a\_1 - |b\_1| - \pi(|b\_0| + 2|c|)) \alpha,\tag{164}
$$

and hence Δ*I*ð Þ *iω*, *τ* >0 fulfills for *ω* >0, if the first condition of (155) is satisfied, i.e. if

$$x < \frac{a\_1 - |b\_1|}{|b\_0| + 2|c|}\tag{165}$$

## fulfills. ■

We show now a simple example in order to demonstrate the above theorem and what the conditions say. First of all, we are going to see that the conditions of the previous theorem are sufficient, but not necessary.

**Example 6.** Let us consider the following system of two linear delay differential equations.

$$\dot{\mathbf{x}} = -\frac{1+\sqrt{7}}{2}\mathbf{x} + \mathbf{x}(\cdot - \tau) + \frac{-4+\sqrt{7}}{2}\mathbf{y} - \frac{1}{2}\mathbf{y}(\cdot - \tau), \quad \dot{\mathbf{y}} = \mathbf{x} + \mathbf{x}(\cdot - \tau) + \frac{-3+\sqrt{7}}{2}\mathbf{y}. \tag{166}$$

The characteristic polynomial of (166) is

$$\Delta(\mathbf{z},\tau) = \mathbf{z}^2 + 2\mathbf{z} + \mathbf{1} + (\mathbf{z}-\mathbf{1})e^{-\mathbf{z}\tau} + \frac{\mathbf{1}}{2}e^{-2\mathbf{z}\tau} \qquad (\mathbf{z}\in\mathbb{C}, \tau\ge 0). \tag{167}$$

Since *<sup>a</sup>*<sup>0</sup> <sup>þ</sup> *<sup>b</sup>*<sup>0</sup> <sup>þ</sup> *<sup>c</sup>* <sup>¼</sup> <sup>1</sup> <sup>2</sup> >0, i.e. the second condition in Theorem 1.10 fulfills, we know from that theorem that if

$$
\pi < \frac{a\_1 - |b\_1|}{|b\_0| + 2|c|} = \frac{1}{2},
\tag{168}
$$

then the trivial solution of (166) is asymptotically stable. As earlier mentioned, the conditions of Theorem 1.10 are sufficient, but not necessary, which can be easily seen, if we study the phase portrait of the system (166) with the following different values of the parameter *τ*: firstly with *τ* ¼ 0*:*48, then with *τ* ¼ 0*:*6 and finally with *τ* ¼ 0*:*94. The **Figures 3** and **4** represent the solutions of system (166) with different values of the parameter *τ*.

**Figure 3.**

In **Figure 3** (above) the parameter *τ* is less than half, so the parameters fulfill the condition of Theorem 1.10, and the origin is asymptotically stable. In **Figure 3** (bottom) the parameter value shows that the theorem does not give a necessary condition, because here the value of the parameter is bigger than half, but the origin is still asymptotically stable. But if we increase more the value of the parameter *τ*, the quasi-polynomial and hence the origin changes to unstable.

The previous example shows that it would be useful to give the largest bound in the theorem, because if we have a larger bound, then we can guarantee the stability of the quasi-polynomial for higher value of the parameter *τ*. In this sense, we can compare our result in Theorem 1.10 with another result in the literature. In this chapter, we compare the conditions of the theorem coming from [30, 31]. In [31] Stépán considered the system

$$
\dot{\mathbf{x}} = -a\_{11}\mathbf{x} - a\_{12}\mathbf{y} + b\_{11}\mathbf{x}(\cdot - \tau), \qquad \dot{\mathbf{y}} = -a\_{21}\mathbf{x} - a\_{22}\mathbf{y}b\_{22}\mathbf{y}(\cdot - \tau) \tag{169}
$$

where *a*11, *a*12, *a*21, *a*22, *b*11, *b*<sup>22</sup> >0 and *τ* ≥0. The characteristic function of system (169) is the quasi-polynomial

$$\begin{split} \Delta\_{\mathbb{S}}(\mathbf{z}, \boldsymbol{\tau}) & \coloneqq \mathbf{z}^2 + (\mathfrak{a}\_{11} + \mathfrak{a}\_{22})\mathbf{z} + \mathfrak{a}\_{11}\mathfrak{a}\_{22} - \mathfrak{a}\_{12}\mathfrak{a}\_{21} - ((\mathfrak{b}\_{11} + \mathfrak{b}\_{22})\mathbf{z} + \mathfrak{a}\_{22}\mathfrak{b}\_{11} \\ & + \mathfrak{a}\_{11}\mathfrak{b}\_{22}) \cdot \mathbf{e}^{-\mathfrak{a}\tau} + \mathfrak{b}\_{11}\mathfrak{b}\_{22}\mathbf{e}^{-2\mathfrak{a}\tau} \qquad (\mathbf{z} \in \mathbb{C}, \tau \ge \mathbf{0}). \end{split} \tag{170}$$

In [30] Freedman and Rao worked with the system

$$
\dot{\mathbf{x}} = -D\_1 \mathbf{x} - D\_2 \mathbf{y} + B\_1 \mathbf{x} (\cdot - \tau), \qquad \dot{\mathbf{y}} = -F\_1 \mathbf{x} - F\_2 \mathbf{y} + E\_2 \mathbf{y} (\cdot - \tau) \tag{171}
$$

where *B*1, *D*1, *D*2, *E*2, *F*1, *F*<sup>2</sup> >0 are constants and the characteristic function of (171) is the quasi-polynomial

$$\begin{split} \Delta\_{\rm FR}(\mathbf{z}, \tau) & \coloneqq \mathbf{z}^2 + (D\_1 + F\_2)\mathbf{z} + D\_1 F\_2 - D\_2 F\_1 - ((B\_1 + E\_2)\mathbf{z} + B\_1 F\_2 \\ & + D\_1 E\_2) \cdot \mathbf{e}^{-\mathbf{z}\tau} + B\_1 E\_2 \mathbf{e}^{-2\mathbf{z}\tau} \qquad (\mathbf{z} \in \mathbb{C}, \tau \ge \mathbf{0}). \end{split} \tag{172}$$

Similarly to papers [30, 31] we gave an upper bound for *τ* under which the quasipolynomial is Hurwitz stable.

Let us write these conditions for *τ* using the notations of Stépán. In [31] we can find the condition

$$\sigma < \frac{a\_{11} + a\_{22} - b\_{11} - b\_{22}}{b\_{11}(b\_{22} + 0.22a\_{22}) + b\_{22}(b\_{11} + 0.22a\_{11})} =: C\_S,\tag{173}$$

in [30] the condition

$$\sigma \quad \quad \quad \le \quad \frac{D\_1 + F\_2 - B\_1 - E\_2}{2(B\_1 F\_2 + D\_1 E\_2 + B\_1 E\_2)} = \frac{a\_{11} + a\_{22} - b\_{11} - b\_{22}}{2(b\_{11} b\_{22} + a\_{11} b\_{22} + a\_{22} b\_{11})} =: \mathbf{C}\_{FR}.\tag{174}$$

Furthermore let us denote the right hand side of the condition in **Theorem 1.10** by *CGyK*

$$\tau < \frac{a\_1 - |b\_1|}{|b\_0| + 2|c|} = \frac{a\_{11} + a\_{22} - |b\_{11} + b\_{22}|}{2b\_{11}b\_{22} + |a\_{11}b\_{22} + a\_{22}b\_{11}|} =: C\_{GyK}.\tag{175}$$

Since *a*11, *a*22, *b*11, *b*<sup>22</sup> >0, the numerators of *CFR* and *CGyK* are equal and

$$|2b\_{11}b\_{22} + |a\_{11}b\_{22} + a\_{22}b\_{11}| = 2b\_{11}b\_{22} + a\_{11}b\_{22} + a\_{22}b\_{11} < 2(b\_{11}b\_{22} + a\_{11}b\_{22} + a\_{22}b\_{11}),\tag{176}$$

because

$$a\_{11}b\_{22} + a\_{22}b\_{11} < 2(a\_{11}b\_{22} + a\_{22}b\_{11})\tag{177}$$

is true following from the positivity of these constants. This means that *CGyK* >*CFR*.

Repeatedly, following from the positivity of the constants *a*11, *a*22, *b*<sup>11</sup> and *b*<sup>22</sup> we get that the numerators of *CGyK* and *CS* are equal, but

$$b\_{11}(b\_{22} + 0.22a\_{22}) + b\_{22}(b\_{11} + 0.22a\_{11}) < 2b\_{11}b\_{22} + a\_{11}b\_{22} + a\_{22}b\_{11},\tag{178}$$

hence *CGyK* <*CS*. On one hand our result in Theorem 1.10 is applicable in general cases, because we have no additional constraints on the sign of the coefficients of the system (123), which means that in this sense our result is better. On the other hand we can increase the upper bound *CS* a little bit in the following way. In [31] Stépán used the estimation

$$
\sin\left(x\right) > -0.22x \qquad (x>0),
\tag{179}
$$

but actually this estimation is not sharp for positive *x*, cf. **Figure 5**.

If we find a tangent line of the sine function at a certain point *x*0, such that this line passes through the origin, i.e. the equation of this line is *y* ¼ *ax* with a certain *a*<0, then we can get a better estimation than (179), namely

$$
\sin\left(\mathbf{x}\right) \ge a\mathbf{x} \qquad \left(\mathbf{x} > \mathbf{0}\right). \tag{180}
$$

We can easily determine the constant *a*<0 in the following way: the equation of the searched tangent line at *x*<sup>0</sup> is

$$f = f'(\mathbf{x}\_0)(\mathbf{x} - \mathbf{x}\_0) + f(\mathbf{x}\_0) = f'(\mathbf{x}\_0)\mathbf{x} + f(\mathbf{x}\_0) - f'(\mathbf{x}\_0)\mathbf{x}\_0,\tag{181}$$

where *f* is the sine function. We would like to find an *x*<sup>0</sup> such that

$$f(\mathbf{x}\_0) - f'(\mathbf{x}\_0)\mathbf{x}\_0 = \sin\left(\mathbf{x}\_0\right) - \cos\left(\mathbf{x}\_0\right) \cdot \mathbf{x}\_0 = \mathbf{0} \tag{182}$$

**Figure 5.** *The estimation of the sine function on the interval* ½ � 0, 5*:*5 *and* ½ � 4, 4*:*9 *.*

fulfills, which is true if and only if *x*<sup>0</sup> is a solution of the equation tan ð Þ¼ *x x*. Let *x*<sup>0</sup> be the solution of this equation, then we find a better linear lower estimation for the function sine on the positive half-line

$$
\sin\left(\pi\right) \ge \cos\left(\pi\_0\right) \cdot \pi \qquad \left(\pi > 0\right),
\tag{183}
$$

i.e. *a* ≔ cosð Þ *x*<sup>0</sup> ≈ � 0*:*21724.

The proposition of the last theorem of this section is similar to one of Theorem 1.10. In the proof of Theorem 1.10 we used first-order Taylor polynomials to approximate the functions sine and cosine to obtain the estimation (164). But if we use higher-order polynomials, we can get a better result, i.e. a better estimation for *τ*, such that if *τ* satisfies the conditions, then the quasi-polynomial (126) is stable.

**Theorem 1.11.** If the coefficients of the characteristic function (125) fulfill the conditions

$$a\_0 \left(2 - 6\sqrt{105}\right)a\_0 < 59b\_0, \qquad -\frac{4}{21}a\_1 \le b\_1 < 0, \qquad \frac{32}{3\sqrt{105} - 31}c < b\_0 < -32c,\tag{184}$$

and for the delay parameter *τ*

$$
\pi < -\kappa + \sqrt{\kappa^2 - \xi} =: \mathbb{C}\_2 \tag{185}
$$

holds, where

$$\kappa := \frac{B}{2A}, \qquad \xi := \frac{C}{4A} \tag{186}$$

with

$$A \coloneqq -\left(b\_0^2 + 124b\_0c + 64c^2\right), \qquad B \coloneqq 6(a\_1(b\_0 + 32c) - 4b\_1(b\_0 + 2c)), \qquad C \coloneqq 45b\_1^2. \tag{187}$$

then the quasi-polynomial (126) is Hurwitz-stable.

**Proof:** Firstly, let us make the same steps as in the proof of Theorem 1.10 and study the imaginary part of Δð Þ *iω*, *τ* :

$$\Delta\_I(i\nu,\tau) = -a\_1\nu + b\_1\nu\cos\left(\nu\tau\right) - b\_0\sin\left(\nu\tau\right) - 2c\sin\left(\nu\tau\right)\cos\left(\nu\tau\right). \tag{188}$$

To show that Δ*I*ð Þ *iω*, *τ* >0 for each *ω*>0 we apply the estimations

$$
\sin\left(\mathbf{x}\right) < \mathbf{x} - \frac{\mathbf{x}^3}{6} + \frac{\mathbf{x}^5}{120}, \quad \cos\left(\mathbf{x}\right) > 1 - \frac{\mathbf{x}^2}{2} \qquad (\mathbf{x} > \mathbf{0}).\tag{189}
$$

Then we have

$$
\Delta\_I(i\alpha, \tau) > \alpha \cdot P\_\tau(\alpha), \tag{190}
$$

with

$$P\_{\tau}(\omega) := \omega \left( -\omega^4 \frac{\tau^5}{120} (b\_0 + 32c) + \omega^2 \frac{\tau^2}{6} (\tau (b\_0 + 8c) - 3b\_1) + a\_1 + b\_1 - \tau (b\_0 + 2c) \right). \tag{191}$$

If the conditions (184) are fulfilled, then the coefficient of *ω*<sup>4</sup> is positive in the polynomial *Pτ*, moreover if *τ* satisfies the condition (185) too, then the discriminant of *P<sup>τ</sup>* is negative. Hence with conditions (184) and (185) the inequality Δ*I*ð Þ *iω*, *τ* >0 is valid for all *ω*>0. Similarly to the previous proof we get by applying the Mikhailov criterion that the quasi-polynomial is asymptotically stable.

In the following example, we are going to show that in some cases the result of Theorem 1.11 is better than the result in [31].

**Example 7.** Let us consider the following system of delay differential equations

$$
\dot{\mathbf{x}} = -\mathbf{0}.7\mathbf{x} - \mathbf{y} + \mathbf{0}.01\mathbf{x}(\cdot - \tau), \qquad \dot{\mathbf{y}} = -\mathbf{0}.2\mathbf{x} - \mathbf{0}.4\mathbf{y} + \mathbf{0}.07\mathbf{y}(\cdot - \tau). \tag{192}
$$

The characteristic function of (192) is

$$\Delta(z;\tau) \coloneqq z^2 + 1.1\mathbf{z} + \frac{227}{10000} - e^{-\mathbf{z}\tau} \left(\frac{2}{25}\mathbf{z} + \frac{53}{1000}\right) + e^{2\mathbf{z}\tau} \frac{7}{10000} \quad (\mathbf{z} \in \mathbb{C}, \ \tau \ge 0). \tag{193}$$

Let us see what condition gives [31] for the parameter *τ*. The condition

$$(a\_{11} - b\_{11})(a\_{22} - b\_{22}) > a\_{12}a\_{21} \tag{194}$$

fulfills, hence if

$$\tau < \frac{a\_{11} + a\_{22} - b\_{11} - b\_{22}}{b\_{11}(b\_{22} + 0.22a\_{22}) + b\_{22}(b\_{11} + 0.22a\_{11})} \approx 78.1,\tag{195}$$

then the quasi-polynomial (193) is asymptotically stable. Furthermore simple calculations show that following from Theorem 1.11, the system is asymptotically stable for all *τ* ≤ 170*:*07, which is greater, than (195). In **Figure 6** the phase portrait of the system (192) could be seen with some values of the parameter *τ*.

#### **3.1 Stability investigation, independently of the delay**

We are going to consider the general form of the characteristic function (126):

$$
\Delta(z,\tau) = p(z) + q(z)e^{-x\tau} + r(z)e^{-2x\tau} \qquad (z \in \mathbb{C}) \tag{196}
$$

where degð Þ*r* < degð Þ*q* . We assume that if there is not any delay in the system, i.e. *τ* ¼ 0, then the trivial equilibrium point is asymptotically stable, which is equivalent to the assumption that the polynomial Δð Þ �, 0 is Hurwitz stable. We

**Figure 6.** *The solutions of system (192) with τ* ¼ 0*:*5 *and τ* ¼ 170*.*

know from [32] that with this assumption the system (and also its trivial equilibrium point) is delay-independently asymptotically stable if and only if for every *τ* > 0 the quasi-polynomial Δð Þ �, *τ* has no non-zero real root on the imaginary axis.

In this chapter we will add a condition to the polynomial *p*, *q* and *r* such that the mentioned property on the root of Δð Þ �, *τ* fulfills. In the computations we will follow the idea of [33].

Firstly, let us multiply the equality <sup>Δ</sup>ð Þ¼ *<sup>i</sup>ω*, *<sup>τ</sup>* 0 by *eiωτ*, then we can see that the equivalence

$$
\Delta(i a \omicron, \tau) = \mathbf{0} \iff \mathsf{e}^{i a \tau} \Delta(i a \omicron, \tau) = \mathbf{0} \tag{197}
$$

is valid. Let us introduce the notations

$$\begin{aligned} p(i\omega) &= p\_R(i\omega) + ip\_I(i\omega), & q(i\omega) &= q\_R(i\omega) + iq\_I(i\omega), \\ r(i\omega) &= r\_R(i\omega) + ir\_I(i\omega), & \Delta(i\omega,\tau) &= \Delta\_R(i\omega,\tau) + i\Delta\_I(i\omega,\tau). \end{aligned} \tag{198}$$

With these notations the characteristic function (196) can be written at *z* ¼ *iω* in the form

$$e^{i\alpha\tau}\Delta(i o \sigma, \tau) = e^{i\alpha\tau}p(i o) + q(i o) + r(i o)e^{-i\alpha\tau} = \left(p\_R(i o) + i p\_I(i o)\right)(\cos\left(\alpha\tau\right)$$

$$+ i\sin\left(\alpha\tau\right) + \left(q\_R(i o) + i q\_I(i o)\right) + \left(r\_R(i o) + i r\_I(i o)\right)(\cos\left(\alpha\tau\right)$$

$$-i\sin\left(\alpha\tau\right) = \Delta\_R(i o \sigma, \tau) + i\Delta\_I(i o \sigma, \tau). \tag{199}$$

Let

$$\alpha := \cos\left(\frac{\alpha \pi}{2}\right) \text{ and } \jmath := \sin\left(\frac{\alpha \pi}{2}\right), \tag{200}$$

then with straightforward calculations we can make the following transformations: 

$$\begin{split} \Delta\_{R}(i\alpha,\tau) &= \cos\left(\alpha\tau\right) \left(p\_{R} + r\_{R}\right) + q\_{R} + \sin\left(\alpha\tau\right) \left(r\_{l} - p\_{l}\right) \\ &= \left(\cos^{2}\left(\frac{\alpha\tau}{2}\right) - \sin^{2}\left(\frac{\alpha\tau}{2}\right)\right) \left(p\_{R} + r\_{R}\right) \\ &+ \left(\cos^{2}\left(\frac{\alpha\tau}{2}\right) + \sin^{2}\left(\frac{\alpha\tau}{2}\right)\right) q\_{R} + 2\sin\left(\frac{\alpha\tau}{2}\right) \cdot \cos\left(\frac{\alpha\tau}{2}\right) \left(r\_{l} - p\_{l}\right) \\ &= \mathbf{x}\left(\mathbf{x}\left(q\_{R} + p\_{R} + r\_{R}\right) + \mathbf{y}\left(r\_{l} - p\_{l}\right)\right) + \mathbf{y}\left(\mathbf{x}\left(q\_{R} - p\_{R} - r\_{R}\right) + \mathbf{y}\left(r\_{l} - p\_{l}\right)\right) \\ &= \mathbf{x}\left(\mathbf{A}\_{\mathbf{x}}\cdot\mathbf{x} + \mathbf{A}\_{\mathbf{y}}\cdot\mathbf{y}\right) + \mathbf{y}\left(\mathbf{B}\_{\mathbf{x}}\cdot\mathbf{x} + \mathbf{B}\_{\mathbf{y}}\cdot\mathbf{y}\right) =: \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{y}. \end{split} \tag{201}$$

Furthermore we can write the imaginary part in the same way, too:

$$\begin{split} \Delta\_{l}(i\alpha,\tau) &= \cos\left(a\tau\right) \left(p\_{I} + r\_{I}\right) + q\_{I} + \sin\left(a\sigma\right) \left(r\_{R} - p\_{R}\right) \\ &= \left(\cos^{2}\left(\frac{a\sigma}{2}\right) - \sin^{2}\left(\frac{a\sigma}{2}\right)\right) \left(p\_{I} + r\_{I}\right) + \left(\cos^{2}\left(\frac{a\sigma}{2}\right) + \sin^{2}\left(\frac{a\sigma}{2}\right)\right) q\_{I} \\ &+ 2\sin\left(\frac{a\sigma}{2}\right) \cdot \cos\left(\frac{a\sigma}{2}\right) \left(r\_{R} - p\_{R}\right) \\ &= \mathbf{x}\left(\mathbf{x}\left(q\_{I} + p\_{I} + r\_{I}\right) + \mathbf{y}\left(r\_{R} - p\_{R}\right)\right) + \mathbf{y}\left(\mathbf{x}\left(q\_{I} - p\_{I} - r\_{I}\right) + \mathbf{y}\left(r\_{R} - p\_{R}\right)\right) \\ &= \mathbf{x}\left(\mathbf{C}\_{\mathbf{x}} \cdot \mathbf{x} + \mathbf{C}\_{\mathbf{y}} \cdot \mathbf{y}\right) + \mathbf{y}\left(\mathbf{D}\_{\mathbf{x}} \cdot \mathbf{x} + \mathbf{D}\_{\mathbf{y}} \cdot \mathbf{y}\right) =: \mathbf{C}\mathbf{x} + \mathbf{D}\mathbf{y}. \end{split} \tag{202}$$

*Characteristic Polynomials DOI: http://dx.doi.org/10.5772/intechopen.100200*

Hence

$$e^{i\alpha \tau} \Delta(i\alpha, \tau) = \mathbf{0} \quad \Longleftrightarrow \quad \begin{bmatrix} A(\alpha \tau) & B(\alpha \tau) \\ C(\alpha \tau) & D(\alpha \tau) \end{bmatrix} \cdot \begin{bmatrix} \mathbf{x} \\ \mathbf{y} \end{bmatrix} = \mathbf{0}.\tag{203}$$

Since the coefficients *A*, *B*, *C* and *D* in the above matrix are expressed as the linear combination of *<sup>x</sup>* and *<sup>y</sup>*, we can write expressions *eiωτ*Δ*R*ð Þ *<sup>i</sup>ω*, *<sup>τ</sup>* and *<sup>e</sup>iωτ*Δ*I*ð Þ *<sup>i</sup>ω*, *<sup>τ</sup>* as follows

$$\begin{aligned} e^{i\alpha \pi} \Delta\_R(i\alpha, \mathfrak{r}) &= A\_\mathfrak{x} B\_\mathfrak{x} \mathfrak{x}^2 + \left( A\_\mathfrak{x} B\_\mathfrak{y} + A\_\mathfrak{y} B\_\mathfrak{x} \right) \mathfrak{x} \mathfrak{y} + A\_\mathfrak{y} B\_\mathfrak{y} \mathfrak{y}^2\\ &= c\_0 \mathfrak{x}^2 + c\_1 \mathfrak{x} \mathfrak{y} + c\_2 \mathfrak{y}^2 \end{aligned} \tag{204}$$

and

$$\begin{split} e^{i\alpha \boldsymbol{\tau}} \Delta\_I(i\alpha, \boldsymbol{\tau}) &= \mathbf{C}\_\mathbf{x} D\_\mathbf{x} \mathbf{x}^2 + \left( \mathbf{C}\_\mathbf{x} D\_\mathbf{y} + \mathbf{C}\_\mathbf{y} D\_\mathbf{x} \right) \mathbf{x} \mathbf{y} + \mathbf{C}\_\mathbf{y} D\_\mathbf{y} \mathbf{y}^2 \\ &=: d\_0 \mathbf{x}^2 + d\_1 \mathbf{x} \mathbf{y} + d\_2 \mathbf{y}^2. \end{split} \tag{205}$$

Then, by dividing the equalities *<sup>e</sup><sup>i</sup>ωτ*Δ*R*ð Þ¼ *<sup>i</sup>ω*, *<sup>τ</sup>* 0 and *eiωτ*Δ*I*ð Þ¼ *<sup>i</sup>ω*, *<sup>τ</sup>* 0 by *<sup>y</sup>*2, and introducing a new variable *u* ≔ *x=y*, we obtain

$$\Delta\_R(i\alpha,\tau) = \frac{y^2(c\_0u^2 + c\_1u + c\_2)}{e^{i\alpha\tau}}, \quad \Delta\_I(i\alpha,\tau) = \frac{y^2(d\_0u^2 + d\_1u + d\_2)}{e^{i\alpha\tau}}.\tag{206}$$

Thus, since *<sup>e</sup><sup>i</sup>ωτ=y*<sup>2</sup> ¼6 0, the equation <sup>Δ</sup>ð Þ¼ *<sup>i</sup>ω*, *<sup>τ</sup>* 0 has no real non-zero root for any given *τ* >0 if and only of the polynomials *f* and *g* have no common real non-zero root, where *f u*ð Þ, resp. *g u*ð Þ denote the expressions for Δ*R*, resp. Δ*I*. This is equivalent to that *res f* ð Þ¼ , *g det R f* ð Þ ½ � , *g* ¼6 0 or if *res f* ð Þ¼ , *g det R f* ð Þ¼ ½ � , *g* 0, then discr½ � *f* <0 and discr½ � *g* <0, where the resultant of the polynomials *f* and *g* is defined as

$$R[f, \mathfrak{g}] = \det\begin{bmatrix} c\_0 & c\_1 & c\_2 & \mathbf{0} \\ \mathbf{0} & c\_0 & c\_1 & c\_2 \\ d\_2 & d\_1 & d\_0 & \mathbf{0} \\ \mathbf{0} & d\_2 & d\_1 & d\_0 \end{bmatrix} = (c\_0 d\_0 - c\_2 d\_2)^2 + (c\_2 d\_1 - c\_1 d\_0)(c\_0 d\_1 - c\_1 d\_2),\tag{207}$$

and the discriminant of a polynomial *F u*ð Þ <sup>≔</sup> *au*<sup>2</sup> <sup>þ</sup> *bu* <sup>þ</sup> *<sup>c</sup>* is discr½ � *<sup>F</sup>* <sup>≔</sup> *<sup>b</sup>*<sup>2</sup> � <sup>4</sup>*ac:* Hence we have proved the following statement.

**Theorem 1.12.** The characteristic function (196) has not a non-zero root on the imaginary axis if and only if the polynomial Δð Þ �, 0 is Hurwitz stable, and *res f* ð Þ¼ , *g det R f* ð Þ ½ � , *g* 6¼ 0 or if *res f* ð Þ¼ , *g det R f* ð Þ¼ ½ � , *g* 0, then discr½ � *f* < 0 and discr½ � *g* < 0, where

$$f(u) = c\_0 u^2 + c\_1 u + c\_2 \quad \text{and} \quad g(u) = d\_0 u^2 + d\_1 u + d\_0,\tag{208}$$

where *c*0, *c*1, *c*2, *d*0, *d*<sup>1</sup> and *d*<sup>2</sup> are defined in (204) and (205). **Example 8.** Let us consider again system (166), i.e.

$$\dot{\mathbf{x}} = A\mathbf{x} + \mathbf{x}(\cdot - \tau) + B\mathbf{y} - \frac{1}{2}\mathbf{y}(\cdot - \tau), \qquad \dot{\mathbf{y}} = \mathbf{x} + \mathbf{x}(\cdot - \tau) + \mathbf{C}\mathbf{y} \tag{209}$$

with

$$A \coloneqq -\left(1 + \sqrt{7}\right)/2, \quad B \coloneqq \left(-4 + \sqrt{7}\right)/2, \quad C \coloneqq \left(-3 + \sqrt{7}\right)/2. \tag{210}$$

Straightforward calculations show that for all *ω*∈

$$\text{res}(f, \mathbf{g}) = 8\alpha^4 \left( \mathbf{0}.\mathbf{5} - \alpha^2 \right) \left( \alpha^2 - \mathbf{0}.\mathbf{25} \right) \cdot \mathbf{0} = \mathbf{0},\tag{211}$$

$$\text{discr}(f) = -16\alpha^2 < 0, \quad \text{discr}(\mathbf{g}) = 16\alpha^2 \left(\alpha^2 - 0.5\right)^2 \ge 0. \tag{212}$$

Thus, the discriminant of *g* is not negative, therefore the stability of system (209) changes at some value *τ* <sup>∗</sup> of the delay (as we have seen in the previous example).

**Figure 7** also shows the changing of the stability of system (209), with *τ* ¼ 0 the origin is asymptotically stable, but with *τ* ¼ 2 the origin changes to unstable. The solutions of system (209) with different values of the parameter *τ* can be seen on **Figure 7**. The stability of system (209) changes if the value of the parameter *τ* increases.

#### **3.2 Hopf bifurcation**

In this subsection we are going to see for which value of the delay *τ* could change the stability of the system (123). For this purpose we are going to give conditions on the coefficients *p*, *q* and *r* to obtain the value of the delay at which stability switch may occur. Let us assume that for *ω*<sup>∗</sup> >0 the conditions of Theorem 1.12 do not fulfill, i.e. for the polynomials *f* and *g* (defined in (206)) the resultant is equal to 0 and the discriminants of *f* or *g* is nonnegative. Furthermore let us assume that *τ* <sup>∗</sup> is

**Figure 7.** *Example: solutions with τ* ¼ 0 *and τ* ¼ 2*.*

#### *Characteristic Polynomials DOI: http://dx.doi.org/10.5772/intechopen.100200*

a solution of <sup>Δ</sup> *<sup>i</sup>ω*<sup>∗</sup> ð Þ¼ , *<sup>τ</sup>* 0. Let us denote by *<sup>z</sup>*3ð Þ*<sup>τ</sup>* the root of the quasi-polynomial (196) that assumes *<sup>z</sup>*<sup>3</sup> *<sup>τ</sup>* <sup>∗</sup> ð Þ¼ *<sup>i</sup>ω*<sup>∗</sup> and the characteristic function <sup>Δ</sup>ð Þ �; *<sup>τ</sup>* as a function of the parameter *τ* by

$$I(z,\tau) \coloneqq p(z) + q(z)e^{-x\tau} + r(z)e^{-2x\tau} \qquad (z \in \mathbb{C}, \ \tau > 0). \tag{213}$$

Thus, we can determine the derivative of *z*<sup>3</sup> at *τ* <sup>∗</sup> by the Implicit Function Theorem (cf. [34]):

$$z\_3'(\tau^\*) = -\frac{\partial\_{\mathbf{r}}I(i o \nu^\*, \tau^\*)}{\partial\_{\mathbf{z}}I(i o \nu^\*, \tau^\*)} = \frac{\mathcal{E}(\mathbf{z}, \mathbf{r})}{\mathcal{D}(\mathbf{z}, \mathbf{r})} \Big|\_{\mathbf{z} = i o \nu^\*, \tau = \tau^\*} \tag{214}$$

where <sup>E</sup>ð Þ *<sup>z</sup>*, *<sup>τ</sup>* <sup>≔</sup> *zq z*ð Þþ <sup>2</sup>*zr z*ð Þ*e*�*z<sup>τ</sup>* and <sup>D</sup>ð Þ *<sup>z</sup>*, *<sup>τ</sup>* <sup>≔</sup> *<sup>p</sup>*<sup>0</sup> ð Þ*<sup>z</sup> <sup>e</sup>z<sup>τ</sup>* <sup>þ</sup> *<sup>q</sup>*<sup>0</sup> <sup>ð</sup> ð Þ� *<sup>z</sup> <sup>τ</sup>q z*ð ÞÞ þ *<sup>r</sup>*<sup>0</sup> ð Þ ð Þ� *<sup>z</sup>* <sup>2</sup>*zr z*ð Þ *<sup>e</sup>*�*z<sup>τ</sup> :* To investigate and prove the occurrence of the Hopf bifurcation we have to see the sign of the real part of the above fraction. But since *p*, *q* and *r* are almost arbitrary polynomials, the fraction could be too complicated, that is why we introduce the following notation:

$$\frac{a+ib}{c+id} \coloneqq \frac{\mathcal{E}(z,\tau)}{\mathcal{D}(z,\tau)}\Big|\_{z=i\nu^\*, \tau=\tau^\*} \quad \text{with} \quad \Re\left(\frac{a+ib}{c+id}\right) = \frac{ac+bd}{c^2+d^2}.\tag{215}$$

Since *<sup>c</sup>*<sup>2</sup> <sup>þ</sup> *<sup>d</sup>*<sup>2</sup> <sup>&</sup>gt; 0, it is enough to consider the sign of *ac* <sup>þ</sup> *bd*. We are going to use the notations introduced in (198) and along the lines of these we introduce the following notations, too:

$$p'(i\omega) = p'\_R(i\omega) + ip'\_I(i\omega), \; q'(i\omega) = q'\_R(i\omega) + iq'\_I(i\omega), \; r'(i\omega) = r'\_R(i\omega) + ir'\_I(i\omega). \tag{216}$$

(For sake of simplicity replacing *<sup>ω</sup>* by *<sup>ω</sup>*<sup>∗</sup> we write *P iω*<sup>∗</sup> ð Þ <sup>≕</sup> *<sup>P</sup>* for *P*∈ *pI*, *pR*, *qI*, *qR*,*rI*,*rR* .)

Computing the exact value of *a*, *b*, *c* and *d* we have:

$$\begin{aligned} a &= 2a^\* \left( r\_R \sin \left( \boldsymbol{w}^\* \boldsymbol{\tau}^\* \right) - r\_I \cos \left( \boldsymbol{w}^\* \boldsymbol{\tau}^\* \right) - \frac{1}{2} q\_I \right), \\ b &= 2a^\* \left( r\_R \cos \left( \boldsymbol{w}^\* \boldsymbol{\tau}^\* \right) + r\_I \sin \left( \boldsymbol{w}^\* \boldsymbol{\tau}^\* \right) + \frac{1}{2} q\_R \right), \\ c &= \left( p\_R' + r\_R' - 2 \boldsymbol{\tau}^\* r\_R \right) \cos \left( \boldsymbol{w}^\* \boldsymbol{\tau}^\* \right) + \left( -p\_I' + r\_I' - 2 \boldsymbol{\tau}^\* r\_I \right) \sin \left( \boldsymbol{w}^\* \boldsymbol{\tau}^\* \right) + q\_R' - \boldsymbol{\tau}^\* q\_R, \\ d &= \left( p\_I' + r\_I' - 2 \boldsymbol{\tau}^\* r\_I \right) \cos \left( \boldsymbol{w}^\* \boldsymbol{\tau}^\* \right) + \left( p\_R' - r\_R' + 2 \boldsymbol{\tau}^\* r\_R \right) \sin \left( \boldsymbol{w}^\* \boldsymbol{\tau}^\* \right) + q\_I' - \boldsymbol{\tau}^\* q\_I. \end{aligned} \tag{217}$$

Thus,

$$\begin{split} ac + bd &= 2a^\* \left\{ \frac{1}{2} \left( q\_R q\_I' - q\_I q\_R' \right) + \left[ r\_R q\_I' - r\_I q\_R' + \frac{1}{2} \left( q\_R \left( p\_I' + r\_I' \right) - q\_I \left( p\_R' + r\_R' \right) \right) \right] \right. \\ &\left. \cdot \cos \left( a^\* \,\tau^\* \right) + \left[ r\_I q\_I' + r\_R q\_R' + \frac{1}{2} \left( q\_R \left( p\_R' - r\_R' \right) + q\_I \left( p\_I' - r\_I' \right) \right) \right] \cdot \sin \left( a^\* \,\tau^\* \right) \\ &+ \cos^2 \left( a^\* \,\tau^\* \right) \left[ r\_R \left( p\_I' + r\_I' \right) - r\_I \left( p\_R' + r\_R' \right) \right] + \sin^2 \left( a^\* \,\tau^\* \right) \left[ r\_I \left( p\_R' - r\_R' \right) - r\_R \left( p\_I' - r\_I' \right) \right] \right. \\ &\left. \cdot + \sin \left( 2a^\* \,\tau^\* \right) \left( r\_R p\_R' + r\_I p\_I' \right) \right] . \end{split} \tag{218}$$

Using the elementary identities

$$\mathfrak{R}(z) \equiv \mathfrak{R}(\overline{z}) \qquad \mathfrak{T}(z) \equiv -\mathfrak{T}(\overline{z}), \qquad \mathfrak{R}(iz) \equiv -\mathfrak{T}(z) \qquad \mathfrak{T}(iz) \equiv \mathfrak{R}(z), \tag{219}$$

furthermore the Euler formula *<sup>e</sup>*�*iz* � cosð Þ� *<sup>z</sup> <sup>i</sup>*sin ð Þ*<sup>z</sup>* , we can simplify the enumerator of ℜ *z*<sup>0</sup> <sup>3</sup>ð Þ*<sup>τ</sup>* as follows *ac* <sup>þ</sup> *bd* <sup>¼</sup> *<sup>ω</sup>*<sup>∗</sup> � <sup>ℑ</sup>ð Þ <sup>A</sup> , where

$$\mathfrak{A} \coloneqq \overline{q}q' + \mathfrak{Z}\overline{r}r' + (\mathfrak{Z}\overline{r}q' + \overline{q}p')e^{i\alpha^\circ \,\,\overline{r}^\circ} + \overline{q}r'e^{-i\alpha^\circ \,\,\overline{r}^\circ} + \mathfrak{Z}\overline{r}p'e^{2i\alpha^\circ \,\,\overline{r}^\circ}.\tag{220}$$

Therefore, Hopf bifurcation occurs if sgn ð Þ¼� ℑð Þ A 1 holds.

#### **4. Summary**

The location of zeros of polynomials and quasi-polynomials as well is crucial in the point of view of the stability of ordinary and retarded differential equations. Namely, if the zeros of the characteristic polynomial of the linearized matrix lie in the open left half of the complex plane, then the constant solution of the particular equation is asymptotically stable. The main task of our work was to depict different methods which allow the investigation of the stability of characteristic (quasi-) polynomials, too. The second objective of this work was in the case of retarded differential equations to treat a method how to estimate the length of the delay for which no stability switching occurs. As an application, we showed a method to detect Hopf bifurcation in ordinary and retarded dynamical systems.

#### **Author details**

Sándor Kovács<sup>1</sup> \*, Szilvia György<sup>2</sup> and Noémi Gyúró<sup>2</sup>


\*Address all correspondence to: alex@ludens.elte.hu

<sup>© 2022</sup> The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Characteristic Polynomials DOI: http://dx.doi.org/10.5772/intechopen.100200*

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#### **Chapter 2**

## Some Proposed Problems on Permutation Polynomials over Finite Fields

*Mritunjay Kumar Singh and Rajesh P. Singh*

#### **Abstract**

From the 19th century, the theory of permutation polynomial over finite fields, that are arose in the work of Hermite and Dickson, has drawn general attention. Permutation polynomials over finite fields are an active area of research due to their rising applications in mathematics and engineering. The last three decades has seen rapid progress on the research on permutation polynomials due to their diverse applications in cryptography, coding theory, finite geometry, combinatorics and many more areas of mathematics and engineering. For this reason, the study of permutation polynomials is important nowadays. In this chapter, we propose some new problems in connection to permutation polynomials over finite fields by the help of prime numbers.

**Keywords:** finite field, permutation polynomial

#### **1. Introduction to permutation polynomials**

In this section, we collect some basic facts about permutation polynomials over a finite field that will be frequently used throught the chapter. First it will be convenient to define permutation polynomial over a finite field.

**Definition 1.** A polynomial *f x*ð Þ∈*q*½ � *x* is said to be a permutation polynomial over *<sup>q</sup>* for which the associated polynomial function *c*↦*f c*ð Þ ia a permutation of *q*, that is, the mapping from *<sup>q</sup>* to *<sup>q</sup>* defined by *f* is one–one and onto.

Finite fields are polynomially complete, that is, every mapping from *<sup>q</sup>* into *<sup>q</sup>* can be represented by a unique polynomial over *q*. Given any arbitrary function *ϕ* : *<sup>q</sup>* ! *q*, the unique polynomial *g* ∈ *q*½ � *x* with degð Þ*g* < *q* representing *ϕ* can be

found by the formula *g x*ð Þ¼ <sup>P</sup> *c*∈*<sup>q</sup> <sup>ϕ</sup>*ð Þ*<sup>c</sup>* <sup>1</sup> � ð Þ *<sup>x</sup>* � *<sup>c</sup> <sup>q</sup>*�<sup>1</sup> � �, see ([1], Chapter 7).

Two polynomials represent the same function if and only if they are the same by reduction modulo *xq* � *<sup>x</sup>*, according to the following result.

**Lemma 1.** *[1] For f*, *g* ∈ *q*½ � *x we have f*ð Þ¼ *α g*ð Þ *α for all α*∈ *<sup>q</sup> if and only if f x*ð Þ� *g x*ð Þ mod *<sup>x</sup>* ð Þ ð Þ *<sup>q</sup>* � *<sup>x</sup> .*

Due to the finiteness of the field, the followings are the equivalent conditions for a polynomial to be a permutation polynomial.

**Definition 2.** The polynomial *f* ∈*q*½ � *x* is a permutation polynomial of *<sup>q</sup>* if and only if one of the following conditions holds:

i. the function *f* : *c*↦*f c*ð Þ is onto;

ii. the function *f* : *c*↦*f c*ð Þ is one-to-one;

iii. *f x*ð Þ¼ *a* has a solution in *<sup>q</sup>* for each *a*∈*q*;

iv. *f x*ð Þ¼ *a* has a unique solution in *<sup>q</sup>* for each *a*∈*q*.

#### **1.1 Criteria for permutation polynomials**

Some well-known criteria for being permutation polynomials are the following.

*1.1.1 First criterion for permutation polynomials*

The first and in some way most useful, criterion was proved by Hermite for *q* prime and by Dickson for general *q*. This criterion has special name what is called Hermite's criterion.

**Theorem 3** (Hermite's criterion). *[1] A polynomial f x*ð Þ∈ *q*½ � *x is a permutation polynomial of <sup>q</sup> if and only if following two conditions hold:*

i. *f x*ð Þ *has exactly one root in q;*

ii. *for each integer t with* 1≤*t*≤ *q* � 2 *and t not divisible by p, the residue f x*ð Þ*<sup>t</sup> mod x* ð Þ ð Þ *<sup>q</sup>* � *<sup>x</sup> has degree* <sup>≤</sup> *<sup>q</sup>* � <sup>2</sup>*.*

For the detailed proof, one can see [1]. Above theorem is mainly used to show negative result. The following is a useful corollary for this purpose.

**Corollary 4.** *There is no permutation polynomial of degree d dividing q* � 1 *over q.*

*Proof*. We note that deg *f q*�1 *d* � � <sup>¼</sup> *<sup>q</sup>* � 1. The proof follows from the last condition of Hermite's criterion.

**Remark 5.** Hermite's criterion is interesting theoretically but difficult to use in practice.

*1.1.2 Second criterion for permutation polynomials*

**Theorem 6.** *[1] Let f* ∈*q*½ � *x . Write*

$$D(f) = \left\{ \frac{f(b) - f(a)}{b - a} : a \neq b \in \mathbb{F}\_q \right\}.$$

*Then f x*ð Þ *is a permutation polynomial of <sup>q</sup> if and only if* 0 ∉ *D f* ð Þ*.*

*1.1.3 Third criterion for permutation polynomials*

**Theorem 7.** *[1] The polynomial f* ∈*q*½ � *x is a permutation polynomial of <sup>q</sup> if and only if*

$$\sum\_{c \in \mathbb{F}\_q} \chi(f(c)) = \mathbf{0}$$

*for all nontrivial additive characters χ of q.*

*1.1.4 Fourth criterion for permutation polynomials*

**Theorem 8.** *[1] Let the trace map Tr* : *qn* ! *<sup>q</sup> be defined as Tr x*ð Þ¼ *<sup>x</sup>* <sup>þ</sup> *xq* <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *xqn*�<sup>1</sup> *. Then the polynomial f* ∈ *q*½ � *x is a permutation polynomial of <sup>q</sup> if and only if for every nonzero η*∈*q,*

$$\sum\_{\mathbf{x}\in\mathbb{F}\_q} \zeta^{\mathrm{Tr}(\mathfrak{y}^\zeta(\mathbf{x}))} = \mathbf{0},$$

*where ζ* ¼ *e* 2*πi <sup>p</sup> is a primitive p-th root of unity.*

In what follows, we will discuss some well known classes of permutation polynomials which are commonly used.

#### **1.2 Some well-known classes of permutation polynomials**

In this subsection, several basic results on permutation polynomials are presented. Many times, we see that one of these general classes are obtained by simplifying complicated classes of permutation polynomials for proving their permutation nature.

**Theorem 9.** *[1] Every linear polynomial, that is, polynomial of the form ax* þ *b*, *a* 6¼ 0 *over finite field is a permutation polynomial.*

**Theorem 10.** *[1] The monomial x<sup>n</sup> is a permutation polynomial over <sup>q</sup> if and only if* gcdð Þ¼ *n*, *q* � 1 1*.*

**Theorem 11.** *Let g x*ð Þ *and h x*ð Þ *be two polynomials over q. Then f x*ð Þ¼ *ghx* ð Þ ð Þ *is a permutation polynomial over <sup>q</sup> if and only if both g x*ð Þ *and h x*ð Þ *permute q.*

#### **1.3 Open problems on permutation polynomials**

Very little is known concerning which polynomials are permutation polynomials, despite the attention of numerous authors. There are so many open problems and conjectures on permutation polynomials over finite fields but here we are listing few of them.

**Open Problem 12.** [2] Find new classes of permutation polynomials of *q*.

Although several classes of permutation polynomials have been found in recent years, but, an explicit and unified characterization of permutation polynomials is not known and seems to be elusive today. Therefore, it is both interesting and important to find more explicit classes of permutation polynomials.

**Open Problem 13.** [2] Find inverse polynomial of known classes of permutation polynomials over *q*.

The construction of permutation polynomials over finite fields is an old and difficult problem that continues to attract interest due to their applications in various area of mathematics. However, the problem of determining the compositional inverse of known classes of permutation polynomial seems to be an even more complicated problem. In fact, there are very few known permutation polynomials whose explicit compositional inverses have been obtained, and the resulting expressions are usually of a complicated nature except for the classes of the permutation linear polynomials, monomials, Dickson polynomials.

**Open Problem 14.** [2] Find *Nd*, where *Nd* ¼ *Nd*ð Þ*q* denote the number of permutation polynomials of degree *d* over *q*.

To date, there is no method for counting the exact number of permutation polynomials of given degree. However, Koyagin and Pappalardi [3, 4], found the asymptotic formula for the number of permutations for which the associated permutation polynomial has degree smaller than *q* � 2.

#### **1.4 Applications of permutation polynomials**

The study of permutation polynomials would not complete without mentioning their applications in other area of mathematics and engineering. It is a major subject in the theory and applications of finite fields. The study of permutation polynomials over the finite fields is essentially about relations between the algebraic and combinatoric structures of finite fields. Nontrivial permutation polynomials are usually the results of the intricate and sometimes mysterious interplay between the two structures. Here we mention some applications of permutation polynomials.

#### **1.5 Coding theory**

In coding theory, error correcting codes are fundamental to many digital communication and storage systems, to improve the error performance over noisy channels. First proposed in the seminal work of Claude Shannon [5], they are now ubiquitous and included even in consumer electronic systems such as compact disc players and many others. Permutation polynomials have been used to construct error correcting codes. Laigle-Chapuy [6] proposed a conjecture equivalent to a conjecture related to cross-correlation functions in coding theory. In [7], Chunlei and Helleseth derived several classes of *p*-ary quasi-perfect codes using permutation polynomials over finite fields. In 2005, Carlet, Ding and Yuan [8] obtained Linear codes using planar polynomials over finite fields.

#### **1.6 Cryptography**

The advent of public key cryptography in the 1970's has generated innumerable security protocols which find widespread application in securing digital communications, electronic funds transfer, email, internet transactions and the like. In recent years, permutation polynomials over finite fields has been used to design public key cryptosystem. Singh, Saikia and Sarma [9–15] designed efficient multivariate public key cryptosystem using permutation polynomials over finite fields. The same authors used a group of linearized permutation polynomials to design an efficient multivariate public key cryptosystem [16].

Permutation polynomials with low differential uniformity are important candidate functions to design substitution boxes (S-boxes) of block ciphers. S-boxes can be constructed from permutation polynomials over even characteristics [17] with desired cryptographic properties such as low differential uniformity and play important role in iterated block ciphers.

#### **1.7 Finite geometry**

Permutation polynomial *f x*ð Þ∈*q*½ � *x* is called a complete permutation polynomial if *f x*ð Þþ *x* is also a permutation polynomial and an orthomorphism polynomial if *f x*ð Þ� *x* is also a permutation polynomial. Orthomorphism polynomials can be used in check digit systems to detect single errors and adjacent transpositions whereas complete permutation polynomials to detect single and twin errors. For more details on complete mappings and orthomorphisms over finite fields, we refer to the reader [3–19]. In addition, complete permutation polynomials are very useful in the study of orthogonal latin squares and orthomorphism polynomials are useful

in close connection to hyperovals in finite projective plane. In 1968, planar functions were introduced by Dembowski and Ostrom [20] in context of finite geometry to describe projective planes with specific properties. Since 1991, planar functions have attracted interest also from cryptography as functions with optimal resistance to differential cryptanalysis.

#### **2. Some proposed problems**

Let *<sup>q</sup>* denotes finite fields with *<sup>q</sup>* <sup>¼</sup> <sup>2</sup>*<sup>m</sup>* elements. Nowaday permutation polynomials are an interesting subject for study not for only research purposes but also for their various applications in many areas of mathematics and engineering. We refer [21] to the reader for recent advances and contributions to the area.

The rising applications of permutation polynomials in mathematics and engineering from last decade propels us to do new research. Recently, permutation polynomials with few terms over finite fields paying more attention due to their simple algebraic form and some extraordinary properties. We refer to the reader [22–25] for some recent developments. This motivates us to propose some new problems. In this chapter, by the help of prime numbers, we constructed several new polynomials that have no root in *μ*2*m*þ<sup>1</sup> and two of them are generalizations of known ones. The constructed polynomials here may lay a good foundation for finding new classes of permutation polynomials.

Throughout the chapter, for a positive integer *d*, the set of *d*-th roots of unity in the algebraic closure *<sup>q</sup>* of *<sup>q</sup>* is denoted by *μd*. That is,

$$\mu\_d = \{ \mathfrak{x} \in \overline{\mathbb{F}\_q} : \mathfrak{x}^d = \mathbf{1} \}.$$

For every element *x*∈*q*, we denote *x*<sup>2</sup>*<sup>m</sup>* by *x* in analogous to the usual complex conjugation. Clearly, *xx*, *x* þ *x*∈*q*. Define the unit circle of *<sup>q</sup>* as

$$\mu\_{2^{m}+1} = \{ \mathfrak{x} \in \mathbb{F}\_q : \mathfrak{x}^{2^{m}+1} = \mathfrak{x}\overline{\mathfrak{x}} = 1 \}.$$

The permutation polynomial of the form *xr h x<sup>q</sup>*�<sup>1</sup> *d* are interesting and have been paid attention, where *h x*ð Þ∈*q*½ � *<sup>x</sup>* with *<sup>d</sup>* dividing *<sup>q</sup>* � 1 and 1≤*r*<sup>≤</sup> *<sup>q</sup>*�<sup>1</sup> *<sup>d</sup>* . The permutation behavior of this type of polynomials are investigated by Park and Lee [26] and Zieve [27].

**Lemma 2** ([26, 27]). *Let r*, *d*> 0 *with d dividing q* � 1 *and h x*ð Þ∈ *q*½ � *x . Then f x*ð Þ¼ *xr h x<sup>q</sup>*�<sup>1</sup> *d permutes <sup>q</sup> if and only if*

$$\begin{aligned} \text{i. } \gcd\left(r, \frac{q-1}{d}\right) &= 1 \text{ and} \\\\ \text{ii. } x^r h(\mathbf{x})^{\frac{q-1}{d}} &\text{permutes } \mu\_d. \end{aligned}$$

In view of Lemma 2, the permutation property of *xr h xq*�<sup>1</sup> *d* is decided by

whether *x<sup>r</sup> h x*ð Þ*<sup>q</sup>*�<sup>1</sup> *<sup>d</sup>* permutes *μd*. In the process to prove that *xr h x*ð Þ*<sup>q</sup>*�<sup>1</sup> *<sup>d</sup>* permutes *μd*, first we need to prove that *h x*ð Þ has no root in *μ<sup>d</sup>* [22]. Thus the polynomials which have no roots in *μ<sup>d</sup>* are interesting and can be used to construct new classes of permutation polynomials. Therefore, it is is both interesting and important to find more polynomials that have no roots in *μ<sup>d</sup>* which play key role in showing the

permutation property of *xr h x*ð Þ*<sup>q</sup>*�<sup>1</sup> *<sup>d</sup>* . For more recent progresses about this type of constructions, we refer [23, 25]. In next section, we also need the following definition.

**Definition 15.** Two polynomials are said to be conjugate to each other if one is obtained by raising 2*m*-th power and multiplying them by the highest degree term of the other.

Next, we propose some new problems by reviewing various recent contributions. The polynomials that have no roots in *μ*2*m*þ<sup>1</sup> play important role in theory of finite fields because these polynomials may give rise to a new class of permutation polynomials.

Let *<sup>p</sup>*<sup>∈</sup> 1, 2, … , 2*<sup>m</sup>* f g � <sup>1</sup> , and let the binary representation of *<sup>p</sup>* be

$$p = \sum\_{k=0}^{m-1} p\_k \ 2^k$$

with *pk* ∈f g 0, 1 . Define the weight of *p* by

$$w(p) = \sum\_{k=0}^{m-1} p\_k \cdot$$

We define a polynomial function over 2*<sup>m</sup>* as

$$L\_p(\mathbf{x}) = \sum\_{k=0}^{m-1} p\_k \cdot \mathbf{x}^{\mathbf{z}^k}.$$

For example,

$$L\_{11}(\mathbf{x}) = \mathbf{1} + \mathbf{x} + \mathbf{x}^3$$

$$L\_{13}(\mathbf{x}) = \mathbf{1} + \mathbf{x}^2 + \mathbf{x}^3$$

$$L\_{19}(\mathbf{x}) = \mathbf{1} + \mathbf{x} + \mathbf{x}^4.$$

We observe that there is a good connection between prime numbers and polynomials that have no roots in *μ*2*m*þ<sup>1</sup> in the sense that most of these polynomials can be derived from prime numbers. In this way, for the prime numbers 11, 13 and 19 we get the polynomials *L*11ð Þ *x* , *L*13ð Þ *x* and *L*19ð Þ *x* respectively that have no roots in *μ*2*m*þ1. This result is obtained by Gupta and Sharma in [22]. More precisely,

**Lemma 3** ([22]). *Let m* <sup>&</sup>gt; <sup>0</sup> *be integer. Then each of the polynomials* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*3, 1 <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> *and* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>4</sup> *have no roots in <sup>μ</sup>*2*m*þ<sup>1</sup>*.*

Similarly, for the primes 59 and 109, we obtain the same polynomials as in [25] of Xu Guangkui et al.

**Lemma 4** ([25]). *Let m* <sup>&</sup>gt;<sup>0</sup> *be integer. Then each of the polynomials* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>4</sup> <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> *and* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> <sup>þ</sup> *<sup>x</sup>*<sup>6</sup> *have no roots in <sup>μ</sup>*2*m*þ<sup>1</sup>*.*

It is not necessary that all polynomials are obtained from prime numbers. For example, the polynomials 1 <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>4</sup> by Gupta and Sharma in [22] and 1 <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>4</sup> <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> by Xu Guangkui et al. [25] are obtained corresponding to the number 25 and 55 respectively. In this respect, we propose the following problem.

**Problem 16.** *Which prime numbers will give polynomials that have no roots in μ*2*m*þ<sup>1</sup>?*.*

The generalization of Lemma 2.2 of [22] corresponding to the polynomials 1 þ *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> and 1 <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> are given by the following lemma.

*Some Proposed Problems on Permutation Polynomials over Finite Fields DOI: http://dx.doi.org/10.5772/intechopen.99351*

**Lemma 5.** *For sufficiently large positive integers m and n, each of the polynomials* <sup>1</sup> <sup>þ</sup> *xn* <sup>þ</sup> *<sup>x</sup>*2*n*�<sup>1</sup> *and* <sup>1</sup> <sup>þ</sup> *xn* <sup>þ</sup> *<sup>x</sup>*2*n*þ<sup>1</sup> *have no roots in <sup>μ</sup>*2*m*þ1*.*

*Proof*. Suppose *α* ∈*μ*2*m*þ<sup>1</sup> satisfies the equation

$$\mathbf{1} + a^n + a^{2n-1} = \mathbf{0}.\tag{1}$$

Raising both sides of (1) to the 2*m*-th power and multiplying by *α*2*n*�1, we get

$$1 + a^{n-1} + a^{2n-1} = 0.\tag{2}$$

Adding (1) and (2), we get

$$a^{n-1} + a^n = \mathbf{0}$$

Since *α* 6¼ 0, which gives *α* ¼ 1. But *α* ¼ 1 does not satisfy (1), a contradiction. Hence 1 <sup>þ</sup> *<sup>x</sup><sup>n</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>2</sup>*n*�<sup>1</sup> has no roots in *<sup>μ</sup>*2*m*þ1. Similarly, we can show that the polynomial 1 <sup>þ</sup> *xn* <sup>þ</sup> *<sup>x</sup>*<sup>2</sup>*n*þ<sup>1</sup> has no roots in *<sup>μ</sup>*2*m*þ1.

In particular, we get the following lemma by Gupta and Sharma [22].

**Lemma 6** ([22]). *Let m* <sup>&</sup>gt;<sup>0</sup> *be integer. Then each of the polynomials* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> *and* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> *have no roots in <sup>μ</sup>*2*m*þ<sup>1</sup>*.*

Based on the Lemma 5, we propose the following problem.

**Problem 17.** *Let h*1ð Þ¼ *<sup>x</sup>* <sup>1</sup> <sup>þ</sup> *xn* <sup>þ</sup> *<sup>x</sup>*<sup>2</sup>*n*�<sup>1</sup> *and h*2ð Þ¼ *<sup>x</sup>* <sup>1</sup> <sup>þ</sup> *xn* <sup>þ</sup> *<sup>x</sup>*<sup>2</sup>*n*þ<sup>1</sup>*. Characterize n and r such that the polynomials xr h*1ð Þ *x* <sup>2</sup>*m*�<sup>1</sup> *and xr h*2ð Þ *x* <sup>2</sup>*m*�<sup>1</sup> *permutes <sup>μ</sup>*2*m*þ<sup>1</sup>*.*

By the help of prime numbers below 1000, we obtain the following polynomials that have no roots in *μ*2*m*þ1. Most of these polynomials are directly or indirectly associated with prime numbers in the sense that corresponding to either each polynomial or their conjugate polynomial, a prime number can be obtained. The proof of the following lemmas can be done in similar fashion as in [22].

**Lemma 7.** *For a positive integer m, each of the polynomials* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>6</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>4</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup> <sup>þ</sup> *<sup>x</sup>*<sup>9</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup> <sup>þ</sup> *<sup>x</sup>*<sup>9</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> <sup>þ</sup> *<sup>x</sup>*<sup>6</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>4</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>4</sup> <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>6</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> <sup>þ</sup> *<sup>x</sup>*<sup>6</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>4</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>4</sup> <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>4</sup> <sup>þ</sup> *<sup>x</sup>*<sup>6</sup> <sup>þ</sup> *<sup>x</sup>*<sup>9</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> <sup>þ</sup> *<sup>x</sup>*<sup>6</sup> <sup>þ</sup> *<sup>x</sup>*9, 1 <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>9</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup> <sup>þ</sup> *<sup>x</sup>*<sup>9</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>4</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>9</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>9</sup> *have no roots in μ*2*m*þ1*.*

**Lemma 8.** *For a positive integer m, each of the polynomials* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> <sup>þ</sup> *<sup>x</sup>*<sup>6</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>6</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> <sup>þ</sup> *<sup>x</sup>*<sup>6</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>4</sup> <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> <sup>þ</sup> *<sup>x</sup>*<sup>6</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>4</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> <sup>þ</sup> *<sup>x</sup>*<sup>6</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup> <sup>þ</sup> *<sup>x</sup>*<sup>9</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>4</sup> <sup>þ</sup> *<sup>x</sup>*<sup>6</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup> <sup>þ</sup> *<sup>x</sup>*<sup>9</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>4</sup> <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup> <sup>þ</sup> *<sup>x</sup>*<sup>9</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>4</sup> <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> <sup>þ</sup> *<sup>x</sup>*<sup>6</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup> <sup>þ</sup> *<sup>x</sup>*<sup>9</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup> <sup>þ</sup> *<sup>x</sup>*<sup>9</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>6</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup> <sup>þ</sup> *<sup>x</sup>*<sup>9</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>4</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup> <sup>þ</sup> *<sup>x</sup>*<sup>9</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup> <sup>þ</sup> *<sup>x</sup>*<sup>9</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>4</sup> <sup>þ</sup> *<sup>x</sup>*<sup>6</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>9</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> <sup>þ</sup> *<sup>x</sup>*<sup>6</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>9</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>4</sup> <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>9</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>4</sup> <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> <sup>þ</sup> *<sup>x</sup>*<sup>6</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>9</sup>*, have no roots in <sup>μ</sup>*2*m*þ<sup>1</sup>*.*

**Lemma 9.** *For a positive integer m, each of the polynomials* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>4</sup> <sup>þ</sup> *<sup>x</sup>*<sup>6</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup> <sup>þ</sup> *<sup>x</sup>*<sup>9</sup>*,* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>5</sup> <sup>þ</sup> *<sup>x</sup>*<sup>6</sup> <sup>þ</sup> *<sup>x</sup>*<sup>7</sup> <sup>þ</sup> *<sup>x</sup>*<sup>8</sup> <sup>þ</sup> *<sup>x</sup>*<sup>9</sup> *have no roots in <sup>μ</sup>*2*m*þ<sup>1</sup>*.*

The above list of polynomials are not complete. However, computational experiments shows that there should be more polynomials. A complete determination of all polynomials with few terms over finite fields seems to be out of reach for the time bing.

Now, we are in condition to propose the following problem in connection to above three lemmas.

**Problem 18.** *Find new classes of permutation polynomials corresponding to polynomials obtained in Lemmas 7, 8 and 9.*

### **Classification**

AMS 2020 MSC: 11T06.

#### **Author details**

Mritunjay Kumar Singh<sup>1</sup> \* and Rajesh P. Singh<sup>2</sup>


\*Address all correspondence to: mmathbhu2012@gmail.com

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Some Proposed Problems on Permutation Polynomials over Finite Fields DOI: http://dx.doi.org/10.5772/intechopen.99351*

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[4] S. Konyagin, F. Pappalardi, Enumerating permutation polynomials over finite fields by degree ii, Finite Fields Appl. 12 (1) (2006) 26–37.

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[17] K. Nyberg, Differentially uniform mappings for cryptography, in: Workshop on the Theory and Application of of Cryptographic Techniques, Springer, 1993, pp. 55–64.

[18] R. P. Singh, A. Saikia, B. Sarma, Little dragon two: An efficient multivariate public key cryptosystem, Int. J. Netw. Secur. Appl. 2 (2) (2010) 1–10.

[19] R. P. Singh, A. Saikia, B. K. Sarma, Poly-dragon: An efficient multivariate public key cryptosystem, J. Math. Cryptol. 4 (4) (2011) 349–364.

[20] P. Dembowski, T. G. Ostrom, Planes of ordern with collineation groups of order *n*2, Math. Z. 103 (3) (1968) 239–258.

[21] X. Hou, Permutation polynomials over finite fields—A survey of recent advances, Finite Fields and Their Applications 32 (2015) 82–119.

[22] R. Gupta, R. Sharma, Some new classes of permutation trinomials over finite fields with even characteristic, Finite Fields and Their Applications 41 (2016) 89–96.

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#### **Chapter 3**

## Effectiveness of Basic Sets of Goncarov and Related Polynomials

*Jerome A. Adepoju*

#### **Abstract**

The Chapter presents diverse but related results to the theory of the proper and generalized Goncarov polynomials. Couched in the language of basic sets theory, we present effectiveness properties of these polynomials. The results include those relating to simple sets of polynomials whose zeros lie in the closed unit disk *U* ¼ f g *z* : j j *z* ≤1 *:* They settle the conjecture of Nassif on the exact value of the Whittaker constant. Results on the proper and generalized Goncarov polynomials which employ the q-analogue of the binomial coefficients and the generalized Goncarov polynomials belonging to the Dq- derivative operator are also given. Effectiveness results of the generalizations of these sets depend on whether *q*< 1 or *q*>1. The application of these and related sets to the search for the exact value of the Whittaker constant is mentioned.

**Keywords:** Basic sets, Simple sets, Effectiveness, Whittaker constant, Goncarov polynomials, Dq operator

#### **1. Introduction**

The Chapter is on the effectiveness properties of the Goncarov and related polynomials of a single complex variable. It is essentially a compendium of certain results which seem diverse but related to the theory of the proper and generalized Goncarov polynomials.

Our first set of results deals with simple sets of polynomial [1], whose zeros lie in the closed unit disk *U*. It is a complement of a theorem of Nassif [1] which resolved his conjecture on the value of the Whittaker constant [2]. We provide also the relation between this problem and the theory of the proper Goncarov polynomials.

Next are results on a generalization of the problem where the polynomials are of the form

$$p\_0(\mathbf{z}) = \mathbf{1}; \quad p\_n(\mathbf{z}) = \sum\_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix} a\_n^{n-k} z^k; \ n \ge \mathbf{1},\tag{1}$$

and the points ð Þ *an* <sup>∞</sup> <sup>0</sup> are given complex numbers with *<sup>n</sup> k* � � the q-analogue of the binomial coefficient *<sup>n</sup> k* � �. From the results reported, it is shown that the location of the points ð Þ *ak* <sup>∞</sup> <sup>0</sup> that leads to favorable effectiveness results depends on whether *q*<1 or *q*>1. The relation of this problem to the generalized Goncarov polynomials belonging to the Dq-derivative operator is also recorded.

It is shown that applying the results of Buckholtz and Frank [3] on the generalized Goncarov polynomials *Qn* f g ð Þ *z*; *z*0, *z*1, … , *zn*�<sup>1</sup> belonging to the Dq-derivative operator when *<sup>q</sup>*>1, leads to the result that, when the points ð Þ *zk* <sup>∞</sup> <sup>0</sup> lie in the unit disk *U*, the resulting polynomials fail to be effective.

Consequently, we provide some results on the polynomials *Qn* f g ð Þ *z*; *z*0, *z*1, … , *zn*�<sup>1</sup> when

$$|z\_k| \le q^{-k}; \quad k \ge 0,\tag{2}$$

with the obtained results justifying the restriction (2) on the points ð Þ *zk* <sup>∞</sup> 0 .

Finally, we provide other relevant and related results on the properties of the generalized Goncarov polynomials *Qn* f g ð Þ *z*; *z*0, *z*1, … , *zn*�<sup>1</sup> belonging to the Dqderivative operator. For a comprehensive and easy reading, background results are provided in the Preliminaries of sections 2.1–2.5.

#### **2. Preliminaries**

We record here some background information for easy reading of the contents of the presentation.

#### **2.1 Basic sets and effectiveness**

A sequence *pn*ð Þ*<sup>z</sup>* � � of polynomials is said to be basic if any polynomial and, in particular, the polynomials 1, *z*, *z*2, … , *zn*, … , can be represented uniquely by a finite linear combination of the form.

$$z^n = \sum\_{k=0} \pi\_{n,k} p\_k(z); n \ge 0. \tag{3}$$

The polynomials *pn*ð Þ*<sup>z</sup>* � � are linearly independent.

In the representation (3), let *f z*ð Þ¼ <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>*anzn* be an analytic function about the origin. Substituting (3) into *f z*ð Þ, we have

$$f(\mathbf{z}) = \sum\_{n=0}^{\infty} a\_n \mathbf{z}^n = \sum\_{n=0}^{\infty} a\_n \sum\_{k=0} \pi\_{n,k} p\_k(\mathbf{z}) \dots$$

Formally rearranging the terms, we obtain the series

$$\sum\_{k=0}^{\infty} p\_k(z) \left[ \sum\_{n=0}^{\infty} a\_n \pi\_{n,k} \right].$$

We write

$$\prod\_{k} (f) = \sum\_{n=0}^{\infty} a\_n \pi\_{n,k}; k \ge 0.1$$

Hence, we obtain the series

$$\sum\_{k=0}^{\infty} \prod\_{k} (f) p\_k(z),$$

*Effectiveness of Basic Sets of Goncarov and Related Polynomials DOI: http://dx.doi.org/10.5772/intechopen.99411*

which is called the basic series associated with the function *f z*ð Þ and the correspondence is written as

$$f(\mathbf{z}) \sim \sum\_{k=0}^{\infty} \prod\_{k} (f) p\_k(\mathbf{z}). \tag{4}$$

The coefficients f g Π*k*ð Þ*f* is the basic coefficients of *f z*ð Þ relative to the basic set *pk*ð Þ*<sup>z</sup>* � � and is a linear functional in the space of functions f g *f z*ð Þ .

If *pn*ð Þ*z* is of degree *n* then the set is called a simple set and is necessarily a basic set.

The basic series (4) is said to represent *f z*ð Þ in a disk j j *z* ≤*r* where *f z*ð Þ analytic, if the series is converges uniformly to *f z*ð Þ in j j *<sup>z</sup>* <sup>≤</sup>*<sup>r</sup>* or that the basic set *pn*ð Þ*<sup>z</sup>* � � represents *f z*ð Þ in j j *z* ≤*r*.

When the basic set *pn*ð Þ*<sup>z</sup>* � � represents in j j *<sup>z</sup>* <sup>≤</sup>*<sup>r</sup>* every function analytic in j j *z* ≤*R*, *R*≥*r*, then the basic set is said to be effective in j j *z* ≤ *r* for the class *H R*ð Þ of functions analytic in j j *z* ≤*R*.

When *R* ¼ *r*, the basic set represents, in j j *z* ≤*r*, every function which is analytic there and we say that the basic set is effective in j j *z* ≤*r*.

To obtain conditions for effectiveness, we form the Cannon sum

$$\omega\_n(r) = \sum\_{k=0} |\pi\_{n,k}| M\_k(r), \tag{5}$$

where

$$\mathcal{M}\_k(r) = \max\_{|x|=r} |p\_k(x)|. \tag{6}$$

From (3), we have that *wn*ð Þ*<sup>r</sup>* <sup>≥</sup> *rn*, so that, if we write

$$\lambda(r) = \lim\_{n \to \infty} \sup \{ w\_n(r) \}^{\frac{1}{n}},\tag{7}$$

$$
\lambda(r) \ge r^n. \tag{8}
$$

The function *<sup>λ</sup>*ð Þ*<sup>r</sup>* is called the Cannon function of the set *pn*ð Þ*<sup>z</sup>* � � in j j *<sup>z</sup>* <sup>≤</sup>*r*.

Theorems about the effectiveness of basic sets are due to Cannon and Whittaker (cf. [2, 4, 5]).

A necessary and sufficient condition for a Cannon set *pn*ð Þ*<sup>z</sup>* � � to be effective, in j j *z* ≤*r*, is

$$
\lambda(r) = r.\tag{9}
$$

#### **2.2 Mode of increase of basic sets**

The mode of increase of a basic set *pn*ð Þ*<sup>z</sup>* � � is determined by the order and type of the set. If *pn*ð Þ*<sup>z</sup>* � � is a Cannon set, its order is defined, Whittaker [2], by

$$w = \lim\_{r \to \infty} \limsup\_{n \to \infty} \frac{\log w\_n(r)}{n \log n}. \tag{10}$$

where *wn*ð Þ*r* is given by (5). The type *γ* is defined, when 0 < *w* < ∞, by

$$\gamma = \lim\_{r \to \infty} \frac{e}{w} \left[ \lim\_{n \to \infty} \sup \{ w\_n(r) \} ^{\frac{1}{n}} n^{-w} \right]^{\frac{1}{w}}.\tag{11}$$

The order and type of a set define the class of entire functions represented by the set.

#### **Theorem 2.2.1 (Cannon [6]).**

The necessary and sufficient conditions for the Cannon set of polynomials to be effective for all entire functions of increase less than order *p* type *q* is

$$\lim\_{n \to \infty} \sup \left[ \left( \frac{\epsilon p q}{\mathbf{n}} \right)^{\sharp} \{ w\_n(r) \}^{\sharp} \right] \le \mathbf{1} \text{ for all } r > \mathbf{0}. \tag{12}$$

#### **2.3 Zeros of simple sets of polynomials**

The relation between the order of magnitude of the zeros of polynomials belonging to simple sets and the mode of increase of the sets has led to many convergence results, just as that between the order of magnitude of the zeros and the growth of the coefficients has. In the case of the zeros and mode of increase, the approach to achieve effectiveness is to determine the location of the zeros while that between the zeros and the coefficients is to determine appropriate bounds (cf. Boas [7], Nassif [8], Eweida [9]).

#### **2.4 Properties of the Goncarov polynomials**

We record in what follows certain properties of the proper and generalized Goncarov polynomials together with the definitions of the q-analogues and the Dq-derivative operator.

The proper Goncarov polynomials f g *Gn*ð Þ *z*, *z*0, *:* … *zn*�<sup>1</sup> associated with the sequence f g *zn* <sup>∞</sup> <sup>0</sup> of points in the plane are defined through the relations, Buckholtz ([10], p. 194),

$$G\_0(z) = 1,$$

$$\frac{z^n}{n!} = \sum\_{k=0}^n \frac{z^{n-k}}{(n-k)!} G\_k(z, z\_0, \dots, z\_{k-1}); n \ge 1. \tag{13}$$

These polynomials generate any function *f z*ð Þ analytic at the origin through the Goncarov series

$$f(\mathbf{z}) \sim \sum\_{k=0}^{\infty} f^k(\mathbf{z}\_k) \mathbf{G}\_k(\mathbf{z}, \mathbf{z}\_0, \dots, \mathbf{z}\_{k-1}),\tag{14}$$

which represents *f z*ð Þ in a disk j j *z* ≤*r*, if it uniformly converges to *f z*ð Þ in j j *z* ≤ *r*. In this case, if *f* ð Þ*<sup>k</sup>* ð Þ¼ *zk* 0, *<sup>k</sup>*≥0, the Goncarov series (14) vanishes and *<sup>f</sup>* � 0. A consideration of *g z*ð Þ¼ sin *<sup>π</sup>* <sup>4</sup> ð Þ <sup>1</sup> � *<sup>z</sup>* , for which *<sup>g</sup>*ð Þ *<sup>n</sup>* ð Þ �<sup>1</sup> *<sup>n</sup>* f g <sup>¼</sup> 0 and P<sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>*g*ð Þ *<sup>n</sup>* ð Þ �<sup>1</sup> *<sup>n</sup>* f g*Gn*ð Þ¼ *<sup>z</sup>*, 1, �1, *::* 0 cf. Nassif [8], shows that the Goncarov series does not always represent the associated function and hence certain restrictions have to be imposed on the points ð Þ *zk* <sup>∞</sup> <sup>0</sup> and on the growth of the function *f z*ð Þ.

*Effectiveness of Basic Sets of Goncarov and Related Polynomials DOI: http://dx.doi.org/10.5772/intechopen.99411*

Concerning the case where the points ð Þ *zk* <sup>∞</sup> <sup>0</sup> lie in the unit disk U, the Whittaker constant W (cf. Whittaker, Buckholtz, [2, 10]), is defined as the supremum of the number c with the following property:

If *f z*ð Þ is an entire function of exponential type less than c and if each of *f*, *f* 0 , *f* }, *::* … has a zero in U then *f z*ð Þ� 0.

Buckholtz [10] obtained an exact determination of the constant W. In fact, if we write

$$H\_n = \max |G\_k(0; z\_0, \dots, z\_{n-1}),\tag{15}$$

where the maximum is taken over all sequences ð Þ *zk <sup>n</sup>*�<sup>1</sup> <sup>0</sup> whose terms lie in *U*, Buckholtz ([10], Lemma 3) proved that lim *<sup>n</sup>*!<sup>∞</sup>*H* 1*=n <sup>n</sup>* exists and is equal to sup1≤*n*<sup>&</sup>lt; <sup>∞</sup>*H* 1*=n <sup>n</sup> :*

Moreover, if we put

$$\lim\_{n \to \infty} H\_n^{1\downarrow\_n} = H = \sup\_{1 \le n < \infty} H\_n^{1\downarrow\_n},\tag{16}$$

Buckholtz ([10], formula 2) further showed that

$$W = \frac{1}{H}.\tag{17}$$

Employing an equivalent definition of the polynomials f g *Gn*ð Þ *z*; *z*0, … , *zn*�<sup>1</sup> as originally given by Goncarov [11] in the form

$$G\_n(z, z\_0, \dots, z\_{n-1}) = \int\_{x\_0}^{x} ds\_1 \int\_{x\_1}^{t\_1} ds\_2, \dots, \int\_{x\_{n-1}}^{t\_{n-1}} ds\_n; n \ge 1,\tag{18}$$

and differentiating with respect to z, we can obtain

$$G\_n^{(k)}(z, z\_0, \dots, z\_{n-1}) = G\_{n-k}(z, z\_k, \dots, z\_{n-1}); \ 1 \le k \le n-1. \tag{19}$$

Writing

$$G\_n(0, z\_0, \dots, z\_{n-1}) = F\_n(z\_0, \dots, z\_{n-1}) \text{ } n \ge 1,\tag{20}$$

then (18) yields, among other results,

$$G\_n(z; z\_0, \dots, z\_{n-1}) = F\_n(z; z\_0, \dots, z\_{n-1}) - F\_n(z; z\_1, \dots, z\_{n-1}); \ n \ge 1,\tag{21}$$

and

$$F\_n(0, z\_1, \ldots, z\_{n-1}) = 0; \ n \ge 1. \tag{22}$$

Applying (21) and (22) to (19) we obtain

$$F\_k^{(k)}(z\_0, z\_1, \dots, z\_{n-1}) = -F\_{n-k}(z\_0, \dots, z\_{n-1}) \tag{23}$$

for 1≤*k*≤*n* � 1, where the differentiation is with respect to the first argument. Expanding *Fn*ð Þ *z*0, … , *zn*�<sup>1</sup> in powers of *z*0, in the form

$$F\_n(z\_0, \dots, z\_{n-1}) = \sum\_{k=0}^n \frac{z\_0^k}{k!} F\_n^{(k)}(0, z\_1, \dots, z\_{n-1}),$$

we arrive through (22) and (23) to the formulae of Levinson [12],

$$F\_n(z\_0, \dots, z\_{n-1}) = -\sum\_{k=1}^n \frac{z\_0^k}{k!} F\_{n-k}(z\_k, \dots, z\_{n-1}).\tag{24}$$

Also, differentiating (18) with respect to *zk*, we obtain with Macintyre ([13], p. 243),

$$\frac{\partial}{\partial \mathbf{s}\_k} \left( G\_n^{(k)}(\mathbf{z}, \mathbf{z}\_0, \dots, \mathbf{z}\_{n-1}) \right) = -G\_k(\mathbf{z}, \mathbf{z}\_0, \dots, \mathbf{z}\_{k-1}) G\_{n-k-1}(\mathbf{z}\_k, \mathbf{z}\_{k-1}, \dots, \mathbf{z}\_{n-1}) \tag{25}$$

for 0≤ *k*≤*n* � 1.

#### **2.5 The** *q***-analogues and** *Dq* **derivatives**

Let *q* be a positive number different from 1. The *q*–analogue of the positive integer *n* is given by

$$
\hat{\mathbf{p}}\left[\mathbf{n}\right] = \frac{q^n - 1}{q - 1}.\tag{26}
$$

Also, the *q*-analogue of *n*! is

$$\left[n\right]! = \left[n\right]\left[n-1\right]\dots\left[2\right]\left[1\right]; n \ge 1; \left[0\right]! = 1,\tag{27}$$

and the *q*-analogue of the binomial coefficient *<sup>n</sup> k* � � is

$$
\begin{bmatrix} \binom{n}{k} \end{bmatrix} = \frac{[n]!}{[k]![n-k]!} \; ; \; 0 \le k \le n. \tag{28}
$$

Moreover, the *Dq*– derivative operator, corresponding to the number q is defined as follows: *If f z*ð Þ is any function of z, then

$$D\_q(\ f(z)) = \frac{f(qz) - f(z)}{z(q-1)},\tag{29}$$

so that when *f z*ð Þ¼ *<sup>z</sup><sup>n</sup>*, then according to (26), we have *Dq zn* ð Þ¼ ½ � *<sup>n</sup> <sup>z</sup><sup>n</sup>*�<sup>1</sup> and if *f z*ð Þ¼ <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>*anz<sup>n</sup>*�<sup>1</sup> is any function analytic at the origin then

$$D\_{\mathfrak{g}}f(\mathbf{z}) = \sum\_{n=1}^{\infty} [n] a\_n \mathbf{z}^{n-1}. \tag{30}$$

In [3] we have a generalization of the Goncarov polynomials as in (13) belonging to the operator D such that for *f z*ð Þ¼ <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>*anz<sup>n</sup>*,

$$D(f(\mathbf{z})) = \sum\_{n=1}^{\infty} d\_n a\_n \mathbf{z}^{n-1} \tag{31}$$

associated with the sequence ð Þ *zk* <sup>∞</sup> <sup>0</sup> , where *en* <sup>¼</sup> ð Þ *<sup>d</sup>*1*d*<sup>2</sup> … , *dn* �<sup>1</sup> ,*e*<sup>0</sup> <sup>¼</sup> 1 and ð Þ *dn* <sup>∞</sup> 1 is a non-decreasing sequence of numbers to obtain

*Effectiveness of Basic Sets of Goncarov and Related Polynomials DOI: http://dx.doi.org/10.5772/intechopen.99411*

$$\begin{cases} \begin{aligned} p\_0(z) &= 1, \\ e\_n z^n = \sum\_{k=0}^n e\_{n-k} z\_k^{n-k} P\_k(z, (z\_0, \dots, z\_{k-1}); n \ge 1. \end{aligned} \tag{32} \end{cases} \tag{32}$$

When *dn* <sup>¼</sup> *<sup>n</sup>*, the relations (32) reduce to (6), hence the polynomials *pn*ð Þ*<sup>z</sup>* � � reduce to the proper Goncarov polynomials f g *Gn*ð Þ *z*; *z*0, … , *zn*�<sup>1</sup> . Comparing (30) and (32), Nassif [14] investigated the class of generalized Goncarov polynomials *Qn* f g ð Þ *z*; *z*0, … , *zn*�<sup>1</sup> belonging to the Dq- derivative operator when *dn* ¼ ½ � *n* and *en* <sup>¼</sup> <sup>1</sup> ½ � *<sup>n</sup>* ! given by,

$$\begin{cases} \mathbf{Q}\_0(\mathbf{z}) = \mathbf{1} \\ \frac{\mathbf{z}^n}{[n]!} = \sum\_{k=0}^n \frac{z\_k^{n-k}}{[n-k]!} \mathbf{Q}\_k(\mathbf{z} \, \mathbf{z}\_0, \dots, \mathbf{z}\_{k-1}); n \ge 1 \end{cases} \tag{33}$$

and the Goncarov series associated with the function *f z*ð Þ¼ <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>*anz<sup>n</sup>* is

$$f(z) \sim \sum\_{k=0}^{\infty} D\_q^k f\_k(z\_k) Q\_k(z; z\_0, \dots, z\_{k-1}). \tag{34}$$

Writing

$$R\_n(z\_0, \dots, z\_{n-1}) = Q\_n(0; z\_0, \dots, z\_{n-1}) \tag{35}$$

so that

$$R\_n(0; z\_1, \dots, z\_{n-1}) = 0, \ n \ge 1 \tag{36}$$

then we have from, (32) that

$$R\_n(z\_0, \dots, z\_{n-1}) = -\sum\_{k=0}^{n-1} \frac{z^{n-k}}{[n-k]!} R\_k(z\_0, \dots, z\_{k-1}).\tag{37}$$

Also, Nassif ([14], Lemma 4.1), proved that

$$Q\_{\mathfrak{n}}(\mathbf{z}; \mathbf{z}\_{0}, \dots, \mathbf{z}\_{n-1}) = R\_{\mathfrak{n}}(\mathbf{z}\_{0}, \dots, \mathbf{z}\_{n-1}) - R\_{\mathfrak{n}}(\mathbf{z}; \mathbf{z}\_{1}, \dots, \mathbf{z}\_{n-1}). \tag{38}$$

We can verify, with Buckholtz ([10], Lemma 1), from the formulae (33), the following:

$$Q\_n(\lambda z, \lambda z\_0, \dots, \lambda z\_{n-1}) = \lambda^n Q\_n(z, z\_0, \dots, z\_{n-1}); \ n \ge 1. \tag{39}$$

$$Q\_n(z\_0, z\_0, \dots, z\_{n-1}) = 0; \ n \ge 1. \tag{40}$$

$$D\_q Q\_n(z, z\_0, \dots, z\_{n-1}) = Q\_{n-1}(z, z\_1, \dots, z\_{n-1}); \ n \ge 1. \tag{41}$$

And hence, by repeated application of *Dq*, we obtain

$$D\_q^k Q\_n(\mathbf{z}; \mathbf{z}\_0, \dots, \mathbf{z}\_{n-1}) = Q\_{n-k}(\mathbf{z}; \mathbf{z}\_k, \dots, \mathbf{z}\_{n-1}); \ 1 \le k \le n-1. \tag{42}$$

Expressing *Qn*ð Þ *z*; *z*0, … , *zn*�<sup>1</sup> as a polynomial of degree *n* in *z*, then we have from (27), (29) and (42), that

*Recent Advances in Polynomials*

$$Q\_n(z; z\_0, \dots, z\_{n-1}) = \sum\_{k=0}^n \frac{z^k}{[n]!} R\_{n-k}(z\_k, \dots, z\_{n-1}) . \tag{43}$$

The identities (39) and (43) have been obtained, in their general form, in ([3]; formulae (2.5), (2.9)). Also, a combination of (38) and (42) yields

$$D\_q^k R\_n(0, z\_1, \dots, z\_{n-1}) = -R\_{n-k}(z\_k, \dots, z\_{n-1}),\tag{44}$$

for 1≤*k*≤*n* � 1, where the differentiation is with respect to the first argument. Expanding *Rn*ð Þ *z*0, *z*1, … , *zn*�<sup>1</sup> in powers of *z*0, then (36) and (44) imply that

$$R\_n(z\_0, z\_1, \dots, z\_{n-1}) = -\sum\_{k=1}^n \frac{z\_0^k}{[k]!} R\_{n-k}(z\_k, \dots, z\_{n-1}) . \tag{45}$$

Finally, if we put

$$h\_n = \max |R\_n(z\_0, z\_1, \dots, z\_{n-1})|,\tag{46}$$

where the maximum is taken over all sequences ð Þ *zk <sup>n</sup>*�<sup>1</sup> <sup>0</sup> and the terms lie in the unit disk *U*, then Buckholtz and Frank ([3], Corollary 5.2), proved that

$$\lim\_{n \to \infty} h\_n^{1\_{\tilde{\eta}}} = h = \sup\_{1 \le n < \infty} h\_n^{1\_{\tilde{\eta}}}.\tag{47}$$

Also, in view of the formulae (33), we can verify that, when *q*< 1,

$$h \ge h\_2^{\frac{1}{2}} = \left(\mathbf{1} + \frac{[1]}{[2]}\right)^{\frac{1}{2}} > \left(\frac{3}{2}\right)^{\frac{1}{2}} > \mathbf{1}.\tag{48}$$

#### **3. Results on the zeros of simple sets**

#### **3.1 Zeros of simple sets of polynomials and the conjecture of Nassif on the Whittaker constant are discussed here**

The following result is known for simple sets of polynomials whose zeros all lie in the unit disk.

Theorem A.([1], Theorem 1).

When the zeros of polynomials belonging to a simple set all lying within or on the unit circle the set will be of increase not exceeding order 1 type 1.378.

Using known contributions in the theory of Goncarov polynomials, we show that the alternative form of the above theorem is as follows:

#### **Theorem 3.1.1 ([Nassif and Adepoju [15], Theorem B)**

When the zeros of the polynomials belonging to a simple set all lying in the unit disk, the set will be of increase not exceeding order 1 type <sup>1</sup> *<sup>W</sup>*, where *W* is the Whittaker constant. It is shown also that the result in this theorem is bes t possible.

Indeed, applying the result of Buckholtz ([10], formula 2), the following theo-

rem which resolved the conjecture of Nassif ([8], p.138), is established.

#### **Theorem 3.1.2 ([15], Theorem B**)

Given a positive number *<sup>ε</sup>*, a simple set *pn*ð Þ*<sup>z</sup>* � � of polynomials, whose zeros all lie in *U* can be constructed such that the increase of the set is not less than order 1 type H–*ε*.

For completeness, we give the proof of Theorem 3.1.1 as a revised version of Theorem A.

#### **Proof of Theorem 3.1.1 (Proof of alternative form of Theorem A)**

Let f g *bn* <sup>∞</sup> <sup>1</sup> be a sequence of points lying in the unit disk and consider the set *qn*ð Þ*<sup>z</sup>* � � of polynomials given by

$$q\_0(\mathbf{z}) = \mathbf{1}; \ q\_n(\mathbf{z}) = (\mathbf{z} + b\_n)\_1^n \; ; \; n \ge \mathbf{1}. \tag{49}$$

Suppose that *z<sup>n</sup>* admits the representation

$$z^n = \sum\_{k=0}^n \tilde{w}\_{n,k} q\_{nk}(z). \tag{50}$$

Then multiplying the matrix of coefficients *<sup>n</sup> k* � �*b<sup>n</sup>*�*<sup>k</sup> n* � � with its inverse ð Þ *<sup>w</sup>*<sup>~</sup> *<sup>n</sup>*,*<sup>k</sup>* , we obtain

$$-\tilde{w}\_{n,0} = \sum\_{k=1}^{n} \binom{n}{k} b\_n^k \tilde{w}\_{n-k,0}; \quad n \ge 1.$$

Write.

$$
u\_n = \frac{\tilde{w}\_{n,0}}{n!};\ n \ge 0,\tag{51}$$

then the above relation will give

$$u\_n = -\sum\_{k=1}^n \frac{b\_n^k}{k!} u\_{n-k}; \quad n \ge 1.$$

And to show the dependence of *un* on the points ð Þ *bn* , this relation can be rewritten as

$$u\_n(b\_1, b\_2, \dots, b\_n) = -\sum\_{k=1}^n \frac{b\_n^k}{k!} u\_{n-k}(b\_1, b\_2, \dots, b\_{n-k}) \dots$$

Comparing this relation with the identify

$$F\_n(z\_0, z\_1, \dots, z\_{n-1}) = -\sum\_{k=1}^n \frac{z\_0^k}{k!} F\_n(z\_k, \dots, z\_{n-1}),$$

of Levinson [12], we infer that

$$
\mu\_n(b\_1, b\_2, \dots, b\_n) = F\_n(b\_n, b\_{n-1}, \dots, b\_1). \tag{52}
$$

Differentiating (50) *k* times, *k* ¼ 1, 2, … , *n* � 1, we obtain that

$$
\tilde{w}\_{n,k} = \binom{n}{k} \tilde{w}\_{n-k,0}(b\_{k+1}, b\_{k+2}, \dots, b\_n). \tag{53}
$$

Hence, a combination of (15), (16), (20), (51)-(53) leads to the inequality.

$$|\ddot{w}\_{n,k}| \le \frac{n!}{k!} H^{n-k};\ \ 0 \le k \le n. \tag{54}$$

Observing that *M qk*;*<sup>r</sup>* � �≤ð Þ <sup>1</sup> <sup>þ</sup> *<sup>r</sup> <sup>k</sup>* for any value of *r*≥ 0, then the Cannon sum of the set *qn*ð Þ*<sup>z</sup>* � � for j j *<sup>z</sup>* <sup>¼</sup> *<sup>r</sup>* will, in view of (54), be

$$
\omega\_n(r) = \sum\_{k=0}^n |\bar{w}\_{n,k}| \mathbf{M}(q\_k; r) \le n! \, \mathbf{H}'' \exp\left(\frac{\mathbf{1} + r}{\mathbf{H}}\right).
$$

It follows from (17) that the set *qn*ð Þ*<sup>z</sup>* � � is of increase not exceeding order 1 type 1 *<sup>W</sup>*. The proof is now completed by applying the results of Walsh and Lucas, cf. Marden ([16], pp. 15,46), with (54) and following exactly the same lines of argument as in ([1], pp.109–110), to arrive at the inequality.

$$|\pi\_{n,k}| \le \frac{n!}{k!} \left( H^{n-k} \right). \tag{55}$$

Since *pn*ð Þ*<sup>z</sup>* � � � �≤ð Þ <sup>1</sup> <sup>þ</sup> *<sup>r</sup> <sup>n</sup>* in j j *<sup>z</sup>* <sup>≤</sup>*r*, it follows that the set *pn*ð Þ*<sup>z</sup>* � � is of increase not exceeding order 1 type H = <sup>1</sup> *W*.

This completes the proof of the theorem.

#### **3.2 Background and the proof of the conjecture**

Before the proof of Theorem 3.2.1, we note that we can take, *ε*< *H* � 1 .(In fact, according to Macintyre ([13]; p. 241), we have H > <sup>1</sup> <sup>0</sup>*:*7378). Hence it follows from (16) that corresponding to *ε*, there exists an integer *m* such that

$$m > (\log \text{H}) / \log \left( 1 + \frac{\mathsf{E}}{2\,\mathrm{H}} \right) \tag{56}$$

such that

$$H\_m^{\frac{1}{m}} > H - \frac{\Xi}{2}.\tag{57}$$

Moreover, from (20), the definition (15) ensures the existence of the points ð Þ *ak <sup>m</sup>* <sup>1</sup> lying in j j *z* ≤1 such that

$$H\_m = |F\_m(a\_m, a\_{m-1}, \dots, a\_1)|.\tag{58}$$

Having fixed the integer *<sup>m</sup>* and the sequence ð Þ *ak <sup>m</sup>* <sup>1</sup> , the following Lemma is to be first established.

Lemma 3.2.1 ([15], Lemma 3.2). For any integer *j*≥1, write

$$f\_{\cdot j}(\mathbf{z}\_1, \mathbf{z}\_2, \dots, \mathbf{z}\_j) = F(\cdot j + 1)\_{m + j}(a\_m, \dots, a\_i; \mathbf{z}\_j; a\_m, \dots, a\_i; \mathbf{z}\_{j-1}; \dots, a\_m, \dots, a\_i; \mathbf{z}\_1; a\_m, \dots, a\_1) \tag{59}$$

Then, the complex numbers *<sup>ξ</sup><sup>k</sup>* ð Þ<sup>∞</sup> <sup>1</sup> can be chosen so that *Effectiveness of Basic Sets of Goncarov and Related Polynomials DOI: http://dx.doi.org/10.5772/intechopen.99411*

$$|\xi\_k| = 1; \ k \ge 1,\tag{60}$$

and

$$\left| \left| f \left( \xi\_1, \xi\_2, \dots, \xi\_j \right) \right| = H\_m^{j+1}; \ j \ge 1. \tag{61}$$

Proof.

The proof is by induction.

When *j* ¼ 1, we have from (59) that

$$f\_1(\mathbf{z}\_1) = F\_{2m+1}(a\_m, \dots, a\_1; \mathbf{z}\_1; a\_m, \dots, a\_1).$$

Then the value *ξ*<sup>1</sup> will be chosen so that

$$|\xi\_1| = \mathbf{1}; \left| \, f\_1(\xi\_1) \right| = \sup\_{|\mathbf{z}\_1| \le 1} \left| \, f\_1(\mathbf{z}\_1) \right|. \tag{62}$$

Applying the identify (25) of Macintyre to *F*2*m*þ<sup>1</sup>ð Þ *am*, … , *a*1; *z*1; *am*, … , *a*<sup>1</sup> , we obtain

$$\frac{d}{dz\_1}\left(f\_1(z\_1)\right) = -F\_m(a\_m, \dots, a\_1) \, G\_m(z\_1; a\_m, \dots, a\_1),$$

so that (20) and (58) imply that

$$\left|f\_{1}^{\prime\prime}(\mathbf{0})\right| = H\_{m}^{2},$$

where the prime denotes differentiation with respect to *z*1. Hence, in view of (62), Cauchy's inequality yields

$$\left| \left| f\_1(\xi\_1) \right| \geq H\_m^2, \right| $$

and the inequality (61) is satisfied for *j* ¼ 1. Suppose then that, for some value *j* ¼ *k*, the complex numbers *ξ*1, *ξ*2, … , *ξ<sup>k</sup>* have been chosen satisfying (60) and (61).

The numbers *ξ<sup>k</sup>*þ<sup>1</sup> will be fixed so that

$$\left\{ \left| F\_{k+1}(\xi\_1, \xi\_2, \dots, \xi\_{k+1}) \right| = \sup\_{|\pi\_{k+1}| \le 1} \left| F\_{k+1}(\xi\_1, \xi\_2, \dots, \xi\_k, z\_{k+1}) \right| \right. \tag{63}$$

Proceeding in a similar manner as for the Case *j* ¼ 1 and applying the identity (25) of Macintyre with (58), (59) and (61),we can obtain the inequality.

$$\left| \left| f\_{k+1}^{1}(\xi\_1, \xi\_2, \dots, \xi\_k, \mathbf{0}) \right| \geq H\_m^{k+2}, \tag{64}$$

where the prime denotes differentiation with respect to *zk*þ<sup>1</sup>*:*.

Applying Cauchy's inequality to the polynomial *Fk*þ<sup>1</sup> *ξ*1, *ξ*2, … , *ξ<sup>k</sup>* ð Þ , *zk*þ<sup>1</sup> ,we can deduce, using (63) and (64), that

$$|F\_{k+1}(\xi\_1, \xi\_2, \dots, \xi\_k, \mathfrak{s}\_{k+1})| \ge H\_m^{k+2}.$$

Hence, by induction, the inequality (61) of the Lemma is established.

We now prove theorem 3.1.2.

The required simple set *pn*ð Þ*<sup>z</sup>* � � of polynomials is constructed as follows:

$$\begin{cases} \begin{array}{c} P\_0(\mathbf{z}) = \mathbf{1}, \\ \end{array} \\ \begin{aligned} p\_{j(m+1)}(\mathbf{z}) = \left(\mathbf{z} + \boldsymbol{\xi}\_j\right)^{j(m+1)}; \ j \ge \mathbf{1}, \\ p\_{j(m+1)+i}(\mathbf{z}) = \left(\mathbf{z} + a\_j\right)^{j(m+1)+i}; \ \mathbf{1} \le i \le m; j \ge \mathbf{0}, \end{aligned} \end{cases} \tag{65}$$

where the points ð Þ *ak <sup>m</sup>* <sup>1</sup> are chosen to satisfy (63) and the numbers *<sup>ξ</sup><sup>k</sup>* ð Þ<sup>∞</sup> <sup>1</sup> are fixed as in the Lemma.

It follows that the zeros of the polynomials *pn*ð Þ*<sup>z</sup>* � � all lie in the unit disk *<sup>U</sup>:* Also, if *z<sup>n</sup>* admits the unique linear representation.

$$x^n = \sum\_{k=0}^n \pi\_{n,k} p\_k(x),\tag{66}$$

and if we write

$$
u\_n = \frac{\pi\_{n,0}}{n!} \; ; \quad n \ge 0,\tag{67}$$

then from the relation (52), we deduce from (59) and (65), that

$$u\_{(\
u\_{j+1})m+j} = f\_{\
j} \left( \xi\_1, \xi\_2, \dots, \xi\_j \right); \ j \ge 1. \tag{68}$$

Now, in view of (66), the Cannon sum of the set *pn*ð Þ*<sup>z</sup>* � � for j j *<sup>z</sup>* <sup>¼</sup> *<sup>r</sup>*, is *wn*ð Þ*r* >j j *π<sup>n</sup>*,0 .

Hence, combining (57), (61), (67) and (68) yields

$$
\omega\_{(\
u\_{j+1})m+j}(r) \ge \{(\
u\_j+1)m+j\}! \binom{H-\bigoplus}{2}^{(\
u\_j+1)m};\ j \ge 1.
$$

vIt follows from this inequality and Theorem 3.1 that the order of the set f g *Pn*ð Þ*z* is exactly 1 and since *<sup>H</sup>* � <sup>∈</sup> <sup>2</sup> >1, the type of the set will be

$$\gamma \ge \left( H - \frac{\mathsf{E}}{2} \right)^{\frac{m}{m+1}} \ge \left( H - \frac{\mathsf{E}}{2} \right)^{\frac{m+1}{m}}.\tag{69}$$

In view of the inequality (56), we deduce from (69) that

$$\gamma > H - \in$$

and Theorem 3.1.2 is established. This settles the conjecture.

#### **4. Generalization**

#### **4.1 As a generalization of the above problem, we consider the simple set** *pn*ð Þ *zn* � � **given by**

$$p\_0(\mathbf{z}) = \mathbf{1}; p\_n(\mathbf{z}) = p\_n(\mathbf{z}; a) = \sum\_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix} a\_n^{n-k} \mathbf{z}^k; n \ge \mathbf{1},\tag{70}$$

*Effectiveness of Basic Sets of Goncarov and Related Polynomials DOI: http://dx.doi.org/10.5772/intechopen.99411*

where *n k* � � is the *<sup>q</sup>*-analogue of the binomial coefficient *<sup>n</sup> k* � � and ð Þ *ak* <sup>∞</sup> <sup>1</sup> is a

sequence of given complex numbers. The set *pn*ð Þ*<sup>z</sup>* � � is in fact, the *<sup>q</sup>*-analogue of the set *qn*ð Þ*<sup>z</sup>* � � in (49). This study is motivated by the fact that this set is related to the generalized Goncarov polynomials belonging to the *Dq*-derivative operator. Our results show that effectiveness properties of the set.

*pn*ð Þ*<sup>z</sup>* � � depend on whether *<sup>q</sup>*<1 or *<sup>q</sup>*<sup>&</sup>gt; 1.

We establish the following:

**Theorem 4.1.1 ([17], Theorem 1.1)**

When the points ð Þ *ak* <sup>∞</sup> <sup>1</sup> all lie in the unit disk *<sup>U</sup>*, the corresponding set *pn*ð Þ*<sup>z</sup>* � � for *<sup>q</sup>*<1,will be effective in j j *<sup>z</sup>* <sup>≤</sup>*<sup>r</sup>* for *<sup>r</sup>*<sup>≥</sup> *<sup>h</sup>* <sup>1</sup>�*<sup>q</sup>*, where *<sup>h</sup>* is as in (47).

#### **Theorem 4.1.2. ([17], Theorem 3.1)**

Given <sup>∈</sup> <sup>&</sup>gt;0, the points ð Þ *ak* <sup>∞</sup> <sup>1</sup> lying in j j *z* ≤ 1 can the chosen so that the correspondence set *pn*ð Þ*<sup>z</sup>* � � of (70) with *<sup>q</sup>*<sup>&</sup>lt; 1 will not be effective in j j *<sup>z</sup>* <sup>&</sup>lt; *<sup>r</sup>* for *<sup>r</sup>*<sup>&</sup>lt; *<sup>h</sup>*� <sup>∈</sup> <sup>1</sup>�*<sup>q</sup> :*.

**Theorem 4.1.3 ([17], Theorem 1.2)**

When *q*> 1 and

$$|a\_k| \le q^{-k} \; ; \; q \ge 1 \tag{71}$$

the corresponding set *pn*ð Þ*<sup>z</sup>* � � of (70) will be effective in j j *<sup>z</sup>* <sup>≤</sup>*<sup>r</sup>* for *<sup>r</sup>*<sup>&</sup>gt; *<sup>q</sup><sup>γ</sup> q*�1 , where 1 *<sup>γ</sup>* is the least root of the equation.

$$\sum\_{n=0}^{\infty} q^{-n^2} \mathfrak{x}^n = \mathbf{2}.\tag{72}$$

Theorem 4.1.2 shows that the result in Theorem 4.1.1 is best possible. Also, the restriction (71) on the sequence ð Þ *ak* <sup>∞</sup> <sup>1</sup> when *q*> 1, is shown to be justified in the sense that if the restriction is not satisfied, the corresponding set *pn*ð Þ*<sup>z</sup>* � � may be of infinite order and not effective.

#### **Proof.**

Proof of Theorem 4.1.1 is similar to the first part of Theorem 3.1.1. Let *z<sup>n</sup>* admits the representation

$$z^n = \sum\_{k=0}^n \pi\_{n,k}(a\_1, a\_2, \dots, a\_n) p\_k(z),\tag{73}$$

then multiplying the matrix of coefficients *<sup>n</sup> k* � �*a<sup>n</sup> n*�*k* � � of the set *pn*ð Þ*<sup>z</sup>* � � with the inverse matrix ð Þ *π<sup>n</sup>*,*<sup>k</sup>* we obtain

$$\sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix} a^n\_{\ n-k} \pi\_{k,0}(a\_1, a\_2, \dots, a\_k) = 0; \ n \ge 1.$$

Putting

$$v\_0 = 1, v\_k = v\_k(a\_1, a\_2, \dots, a\_k) = \frac{1}{[k]!} \pi\_{k, \rho}(a\_1, \dots, a\_{n-k}),\tag{74}$$

the above relation yields

$$v\_n(a\_1, \dots, a\_n) = -\sum\_{k=1}^n \frac{a^k}{[k]!} v\_{n-k}(a\_1, \dots, a\_{n-k}) \tag{75}$$

Comparing the formulae (45) and (75) we infer that

$$v\_k(a\_1, \ldots, a\_k) = R\_k(a\_k, \ldots, a\_1). \tag{76}$$

Moreover, operating *Dq* on the polynomials *pn*ð Þ*<sup>z</sup>* � �, we can deduce, from (28) and (29), that

$$D\_q\left(p\_k(\mathbf{z}; a\_k)\right) = [K] p\_{k-1}(\mathbf{z}, a\_k);\ k \ge \mathbf{1}.\tag{77}$$

Hence, when the operator *Dq* acts on the representation (73), then (77) leads to the equality

$$
\pi\_{n,k}(a\_1, \ldots, a\_n) = \frac{[n]}{[k]} \pi\_{n-1,k-1}(a\_1, \ldots, a\_n),
$$

which, on reduction, yields

$$
\pi\_{n,k}(a\_1, \ldots, a\_n) = \begin{bmatrix} n \\ k \end{bmatrix} \pi\_{n-k,0}(a\_{K+1}, \ldots, a\_n); 0 \le k \le n. \tag{78}
$$

Applying (74), (76) and (78), we obtain

$$\pi\_{n,k}(a\_1, \ldots, a\_n) = \frac{[n]!}{[k]!} R\_{n-k}(a\_n, \ldots, a\_{K+1}); 0 \le k \le n. \tag{79}$$

Identify (79) is the bridge relation between the set *pn*ð Þ*<sup>z</sup>* � � and the Goncarov polynomials mentioned earlier.

Suppose *q* <1 and assume that

$$r \geq \frac{h}{1-q}.\tag{80}$$

Since *<sup>h</sup>*>1 as in (47), and restricting the points ð Þ *ak* <sup>∞</sup> <sup>1</sup> to lie in the unit disk *U* as in the theorem, it follows from (28) and (80) that

$$\mathbf{M}(p\_k; r) \le (k+1)r^k \; ; \; k \ge 0. \tag{81}$$

The Cannon sum of the set *pn*ð Þ*<sup>z</sup>* � � for j j *<sup>z</sup>* <sup>¼</sup> *<sup>r</sup>*, is evaluated from (46), (47), (79), (80) and (81) to obtain

$$w\_n(r) = \sum\_{k=0}^n |\pi\_{n,k}| \mathbf{M}(p\_k; r) \le (n+1)^2 r^n,\tag{82}$$

from which it follows that the set *pn*ð Þ*<sup>z</sup>* � � is effective in j j *<sup>z</sup>* <sup>≤</sup>*<sup>r</sup>* for *<sup>r</sup>*<sup>≥</sup> *<sup>h</sup>* <sup>1</sup>�*<sup>q</sup>* and the theorem is established.

*Effectiveness of Basic Sets of Goncarov and Related Polynomials DOI: http://dx.doi.org/10.5772/intechopen.99411*

#### **5. Proof**

#### **5.1 Proof of Theorem 4.1.2**

We argue as in the Proof of Theorem 3.1.2. We first obtain an identity similar to (25) of Macintyre using the following Lemma:

Lemma 5.1.1.

For *n* ≥1 and *k*≥0, the following identity holds.

$$\begin{cases} & D\_{q,x\_k} Q\_{k+n}(z; z\_0, \dots, z\_{k+n-1}) \\ & = -Q\_k(z; z\_0, \dots, z\_{k-1}) Q\_{n-1}(z\_k, z\_{k+1}, \dots, z\_{k+n-1}), \end{cases} \tag{83}$$

where *Dq*,*zk* denote the *Dq*-derivative with respect to *zk*.

Proof of Lemma

The proof is by induction.

For *n* ¼ 1, *k*≥0, we have from the construction formulae (33),

$$\begin{aligned} &Q\_{K+1}(z; z\_0, \dots, z\_k) = \frac{z^{k+1}}{[k+1]!} \\ &- \sum\_{j=0}^{k-1} \frac{z\_j^{k+1-j}}{[k+1-j]!} Q\_j(z; z\_0, \dots, z\_{j-1}) - z\_k Q\_k(z; z\_0, \dots, z\_{j-1}). \end{aligned}$$

Hence, operating *Dq*,*zk* on this equality, we have that

$$D\_{q,x\_k}Q\_{k+1}(z;z\_0,\ldots,z\_{k-1}) = -Q\_k(z;z\_0,\ldots,z\_{k-1}),$$

so that the identity (83) is satisfied for *n* ¼ 1, *k*≥0. Suppose that (83) is satisfied for*n* ¼ 1, 2, … , *m*; *k*≥ 0. The formulae (33) can be written for *k* þ *m* þ 1 in the form,

$$\begin{aligned} \mathbf{Q}\_{k+m+1}(z; z\_0, \dots, z\_{k+m}) &= \frac{z^{k+m+1}}{[k+m+1]!} - \sum\_{j=0}^{k-1} \frac{z\_j^{k+m+1-j}}{[k+m+1-j]!} \mathbf{Q}\_j \left( z; z\_{0,\dots,x\_{j-1}} \right), \\\ &- \frac{z^{m+1}}{[m+1]!} (\mathbf{Q}\_k(z; z\_0, \dots, z\_{k-1})) - \sum\_{j=1}^m \frac{z\_{k+j}^{k+j+1-j}}{[m+1-j]!} \mathbf{Q}\_{k+j} \left( z; z\_0, \dots, z\_{k+j-1} \right). \end{aligned}$$

Hence, the derivative *Dq*,*zk* operating on this equation gives, in view of (83),

$$\begin{aligned} \mathbf{D}\_{q,x\_k} \mathbf{Q}\_{k+m+1}(\mathbf{z}; \mathbf{z}\_0, \dots, \mathbf{z}\_{k+m}) &= -\frac{\mathbf{z}^m}{[m]!} \mathbf{Q}\_k(\mathbf{z}; \mathbf{z}\_0, \dots, \mathbf{z}\_{k-1}) \\ + \sum\_{j=1}^m \frac{\mathbf{z}\_{k+j}^{m+1-j}}{[m+\mathbf{1}-j]!} \mathbf{Q}\_k(\mathbf{z}; \mathbf{z}\_0, \dots, \mathbf{z}\_{K-1}) &\times \mathbf{Q}\_{j-1}(\mathbf{z}\_k; \mathbf{z}\_{k+1}, \dots, \mathbf{z}\_{K+j-1}). \end{aligned}$$

Or equivalently,

$$\begin{split} D\_{q, \boldsymbol{z}k} \mathbf{Q}\_{k+m+1}(\boldsymbol{z}; \boldsymbol{z}\_{0}, \dots, \boldsymbol{z}\_{k+m}) &= -\mathbf{Q}\_{k}(\boldsymbol{z}; \boldsymbol{z}\_{0}, \dots, \boldsymbol{z}\_{k-1}) \\ &\times \left[ \frac{\boldsymbol{z}\_{k}^{m}}{[m]!} - \sum\_{j=0}^{m-1} \frac{\boldsymbol{z}\_{k+j+1}^{m+j}}{[m-j]!} \mathbf{Q}\_{j}(\boldsymbol{z}\_{k}; \boldsymbol{z}\_{k+1}, \dots, \boldsymbol{z}\_{K+j}) \right]. \end{split}$$

Hence, formulae (33) imply that

$$D\_{q,x\_k}Q\_{k+m+1}(z;z\_0,\ldots,z\_{k+m}) = -Q\_k(z;z\_0,\ldots,z\_{k-1})Q\_m(z\_k;z\_{k+1},\ldots,z\_{K+m}),$$

and the relation (83) is also valid for *n* ¼ *m* þ 1

The Lemma is thus proved by induction. Now, following similar lines paralleling those of the proof of Theorem 3.1.2, we need to establish a Lemma similar to that used for Theorem 3.1.2.

Indeed, observing that *h*>1 as in (39), the ∈ >0 of Theorem 4.1.2 can always be picked less than *h* � 1. Also, from (39) it follows that, corresponding to the number ∈, there exists an integer *m* for which

$$m > (\log h) / \log \left( 1 + \frac{\epsilon}{2h} \right),\tag{84}$$

such that

$$h\_m^\ddagger > h - \frac{\in}{2}.\tag{85}$$

Also, from the definition (46) of *hm*, the points ð Þ *<sup>α</sup><sup>i</sup> <sup>m</sup>* <sup>1</sup> lying in *U* can be chosen so that

$$h\_m = |R\_m(a\_m, \dots, a\_1)|.\tag{86}$$

With this choice of the integer *<sup>m</sup>* and the points ð Þ *<sup>α</sup><sup>i</sup> <sup>m</sup>* <sup>1</sup> , the Lemma to be established is the following:

Lemma 5.1.2.

With the notation

$$u\_j(z\_1, z\_2, \dots, z\_j) = R\_{(-j+1)m+j}(a\_m, \dots, a\_1; z\_j; a\_m, \dots, a\_1; z\_{j-1}; \dots; a\_m, \dots, a\_1; a\_m, \dots, a\_1),\tag{87}$$

we can choose a sequence *ξ <sup>j</sup>* � �*<sup>m</sup>* <sup>1</sup> of points on j j *<sup>z</sup>* <sup>¼</sup> 1 such that

$$\left| u\_j \left( \xi\_1, \xi\_2, \dots, \xi\_j \right) \right| \ge m^{j+1}; j \ge 1. \tag{88}$$

Proof.

We first observe, from a repeated application of (30), that an analytic function *f z*ð Þ regular at the origin, can be expanded in a certain disk j j *z* ≤1 in a series of the form

$$f(\mathbf{z}) = \sum\_{n=0}^{\infty} \frac{z^n}{[n]!} D\_q^n f(\mathbf{0}).$$

Hence, by Cauchy's inequality, we have

$$\mathbf{M}(f, r) \ge r \big| D\_{\mathfrak{g}} f(\mathbf{0}) \big|. \tag{89}$$

Applying the usual induction process, we obtain, from (87) for the case *j* ¼ 1, that

$$\mu\_1(z\_1) = R\_{2m+1}(a\_m, \dots, \dots, a\_1; z\_i; a\_m, \dots, a\_1)$$

Hence the identity (83) yields

$$\begin{array}{c} D\_{q}u\_{1}(\mathbf{z}) = D\_{q,\mathbf{z}\_{1}}Q\_{2m+1}(\mathbf{0};a\_{m},\ldots,\mathbf{0},\mathbf{z};a\_{i};a\_{m},\ldots,\mathbf{0},a\_{1})\\ = -R\_{m}(a\_{m},\ldots,\mathbf{0},a\_{1})Q\_{m}(z\_{i},a\_{m},\ldots,a\_{1}). \end{array}$$

Therefore, we obtain

$$D\_q u\_1(\mathbf{0}) = \mathcal{R}\_m^2(a\_m, \dots, \dots, a\_1),\tag{90}$$

where the *Dq* is operating with respect to *z*1. Pick the number *ξ*1, with *ξ*<sup>1</sup> j j ¼ 1, such that

$$|\mu\_1(\xi\_1)| = \sup\{ |\mu\_1(z\_1)| : |z\_1| = 1 \};$$

hence, a combination of (86), (89) and (90) yields

$$|u\_1(\xi\_1)| \ge h\_m^2,$$

and the inequality (88) is satisfied for *j* ¼ 1. The similarity with the proof of Lemma 3.2.1 shows that the proof of this Lemma can be completed in the same manner as that for ealier Lemma.

We can now prove Theorem 5.1.4.

We note that the points ð Þ *ak* <sup>∞</sup> <sup>1</sup> lying in *<sup>U</sup>* which define the required set *pn*ð Þ*<sup>z</sup>* � � of polynomials (70), are chosen as follows:

$$\begin{cases} \mathcal{a}\_{\,j(m+1)} = \xi\_j\\ \mathcal{a}\_{\,j(m+1)+i} = \mathcal{a}\_i \,; \mathbf{1} \le i \le m; j \ge \mathbf{0} \end{cases} \tag{91}$$

where the points ð Þ *<sup>α</sup><sup>i</sup> <sup>m</sup>* <sup>1</sup> are fixed as in (86) and the sequence *ξ <sup>j</sup>* � �<sup>∞</sup> <sup>0</sup> of points is determined as in Lemma 5.1.2; and the integer *m* is chosen as in (84) and (85).

If *z<sup>n</sup>* admits the representation (86), then applying (79), (87) and (91) we have that

$$
\pi\_{(\
u\_{j+1})m+j,o} = [(\
u\_j+\mathbf{1})m+i]! u\_j \left(\xi\_1, \xi\_2, \dots, \xi\_j\right), j \ge \mathbf{1},\tag{92}
$$

so that, for the Cannon sum of the set *pn*ð Þ*<sup>z</sup>* � � for j j *<sup>z</sup>* <sup>¼</sup> *<sup>r</sup>*, we obtain, from (85), (88) and (92),

$$w\_{(\
u\_{(j+1)m+j})}(r) > [(\
u\_{j}+1)m+j]! \left(h - \frac{\mathsf{E}}{2}\right)^{(\
u\_{(j+1)m})};\ r>0..\tag{93}$$

Since *q* <1, we have that

$$\lim\_{n \to \infty} ([n]!)^{\vee\_n} = \frac{1}{1 - q} \,. \tag{94}$$

Hence, (93) and (94) yield, for the Cannon function,

$$\begin{aligned} \lambda(r) &= \limsup\_{n \to \infty} \{w\_n(r)\}^{\mathbb{1}\_n} \\ &\ge \limsup\_{j \to \infty} \{w\_{(\cdot, j+1)m + j}(r)\}^{\mathbb{1}\_{\left(\cdot, j+1\right)m + j}} \\ &\ge \frac{1}{1 - q} \left(h - \frac{\mathbf{e}}{2}\right)^{\frac{m}{m+1}}; \ r > 0. \end{aligned}$$

Noting that *<sup>h</sup>* � <sup>∈</sup> <sup>2</sup> >1, we conclude, from (84), as in the proof of Theorem (50), that

$$
\lambda(r) \ge \frac{h - \in}{1 - q} \; ; \; r > 0,
$$

and *pn*ð Þ*<sup>z</sup>* � � will not be effective in j j *<sup>z</sup>* <sup>≤</sup>*<sup>r</sup>* for *<sup>r</sup>*<sup>&</sup>lt; *<sup>h</sup>*� <sup>∈</sup> <sup>1</sup>�*<sup>q</sup>* . This completes the proof.

#### **5.2 Proof of Theorem 4.1.3**

Let *pn*ð Þ*<sup>z</sup>* � � be the basic set in (70) with *<sup>q</sup>*>1. We first justify the statement that if the restriction (71) is not satisfied the corresponding set *pn*ð Þ*<sup>z</sup>* � � may be of infinite order.

For this, we put

$$a\_k = t^k \quad ; \ k \ge 1,\tag{95}$$

and let *t* be such that

$$|t| = \beta, \frac{1}{q} < \beta < q \tag{96}$$

We claim that, in this case, the corresponding set *pn*ð Þ*<sup>z</sup>* � � will be of infinite order and hence the effectiveness properties of the set will be violated.

Now, in the identity (37), we let

$$z\_k = a\_{n-k} = t^{n-k}; \ 0 \le k \le n-1,$$

to obtain

$$\sum\_{k=0}^{n} \frac{t^{nk}}{[k]!} R\_{n-k} (t^{n-k}, \dots, t) = 0; n > 0. \tag{97}$$

Put

$$R\_j(t^j, \dots, t) = t^{\sharp(\cdot \mid j-1)} u\_j; j \ge 1,\tag{98}$$

so that (97) yields

$$\sum\_{k=0}^{n} \frac{t^{k(k+1)}}{[k]!} u\_{n-k} = 0 \quad \text{;} \text{ } n > 0. \tag{99}$$

Hence, if we put

*Effectiveness of Basic Sets of Goncarov and Related Polynomials DOI: http://dx.doi.org/10.5772/intechopen.99411*

$$u(z) = \sum\_{n=0}^{\infty} u\_n z^n,\tag{100}$$

then (97) implies that

$$
\mu(\mathbf{z}) = \frac{1}{\varrho(\mathbf{z})},
\tag{101}
$$

where

$$\phi(z,t) = \sum\_{n=0}^{\infty} \frac{t^{-\frac{1}{2^n}(n-1)}}{[n]!} z^n.$$

Since j j *t* ¼ *β* < *q*, the function *ϕ*ð Þ *z*, *t* is entire of zero order and hence it will have zeros in the finite part of the plane.

Let

$$\sigma = \inf \left\{ |z|; \rho(z) = 0 \right\} < \infty,\tag{102}$$

then from (100) and (101), we have lim sup*<sup>n</sup>*!<sup>∞</sup>j j *un* 1 *<sup>n</sup>* <sup>¼</sup> <sup>1</sup> *<sup>σ</sup>* >0*:*

Thus, for the Cannon sum of the set *pn*ð Þ*<sup>z</sup>* � �, we have, from (79), (96) and (98), that

$$|\omega\_n(r) > |\pi\_{n,0}| = [n]! \beta^{\frac{1}{2}n(n-1)} |u\_n|. \tag{103}$$

Since *q* >1 and *β* > <sup>1</sup> *<sup>q</sup>* then, in view of (102), we deduce from (103) that the set *pn*ð Þ*<sup>z</sup>* � � is of infinite order; as claimed.

To prove Theorem 4.1.3 we first note, from (72), that if we put

$$
\sigma = \frac{q\chi}{q-1},
\tag{104}
$$

then

$$c > \frac{1}{q - 1}.\tag{105}$$

We then multiply the matrix *n k* � �*a<sup>n</sup>*�*<sup>k</sup> <sup>n</sup>* � � with the inverseð Þ *<sup>π</sup><sup>n</sup>*,*<sup>k</sup>* to get

$$
\pi\_{n,k} = -\sum\_{j=k}^{n-1} \begin{bmatrix} n \\ k \end{bmatrix} a\_n^{n-j} \pi\_{j,k} \; : \; n > k; \; \pi\_{k,k} = 1. \tag{106}
$$

Now, imposing the restriction (71) on the points ð Þ *ak* <sup>∞</sup> <sup>1</sup> , we have from (105) and (106) that

$$|\mathfrak{a}\_{k+1,k}| \le c.$$

Thus, the inequality

$$|\pi\_{m\,k}| \le c^{m-k} \; ; \; m \ge k,\tag{107}$$

is true for *m* ¼ *k*, *k* þ 1.

To prove (107), in general, we observe that, since *q*>1,

$$
\binom{n}{j} \le q^{j(n-j)} \left\{ \frac{q}{q-1} \right\}^{n-j}; \ 1 \le j \le n. \tag{108}
$$

Assume that (107) is satisfied for *m* ¼ *k*, *k* þ 1, … , *n* � 1; then a combination of (71), (72), (104), (106), (107) and (108) leads to the inequality.

$$|\pi\_{n,k}| \le c^{n-k} \sum\_{j=1}^{\infty} \left(\frac{q}{c(q-1)}\right)^j q^{-j^2} = c^{n-k}.$$

Hence, it follows by induction, that the inequality (107) is true for *m* ≥*k:* Noting that

$$
\begin{bmatrix} k \\ j \end{bmatrix} = q^{j(k-j)} \begin{Bmatrix} k \\ j \end{Bmatrix} \quad q > 1,
$$

where *k j* � � is the *<sup>q</sup>*–analogue of *<sup>k</sup> j* � �, *<sup>q</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> *<sup>q</sup>* <1, we then deduce from (70) and (71), that

$$\begin{aligned} \mathbf{M}(p\_k; r) &\leq r^k \sum\_{j=0}^k \begin{Bmatrix} k \\ j \end{Bmatrix} q^{-j^2} r^{-j} \\ &\leq r^k \sum\_{j=0}^k \begin{Bmatrix} k \\ j \end{Bmatrix} q^{-\frac{1}{2}(\
u^{-1})} (qr)^{-j} \ \vdots \ \mathbf{q} > \mathbf{1}. \end{aligned}$$

Appealing to a result of Al-Salam ([18]; formula 2.5), we deduce that

$$\mathbf{M}(p\_k; r) \le r^k \prod\_{j=1}^k \left( 1 + \frac{1}{q^j r} \right); k \ge 1, r > 0. \tag{109}$$

The Cannon sum of the set *pn*ð Þ*<sup>z</sup>* � � for j j *<sup>z</sup>* <sup>¼</sup> *<sup>r</sup>* can be evaluated from (107) and (109) in the form

$$w\_n(r) \le \left\{ \prod\_{j=1}^n \left( 1 + \frac{1}{q^j r} \right) \right\} \sum\_{k=0}^n c^{n-k} r^k. \tag{110}$$

Hence, when *r*≥*c* we should have

$$w\_n(r) \le (n+1) \left\{ \prod\_{j=1}^n \left( 1 + \frac{1}{q^j r} \right) \right\} r^n,$$

from which it follows that the set *pn*ð Þ*<sup>z</sup>* � � is effective in j j *<sup>z</sup>* <sup>≤</sup> *<sup>r</sup>* and Theorem 4.1.3 is proved.

#### **6. Other related results**

The Goncarov polynomials belonging to the *Dq–*derivative operator have other properties of interest and worth recording. Hence, we present, in this section, more results regarding the Goncarov polynomials *Qn* f g ð Þ*z* ; *z*0, … , *zn*�<sup>1</sup> as defined in (84) which belong to the derivative operator *Dq* and whose points ð Þ *zn* <sup>∞</sup> <sup>0</sup> lie in the unit disk *U* for which *q*<1 or *q* >1.

When *q*< 1, the result of Buckoltz and Frank ([3]; Theorem 1.2) applied to the derivative operator *Dq* leads, in the language of basic sets, to the following theorem:

#### **Theorem 6.1 ([19], Theorem 1).** The set of Gancarov polynomials *Qn* f g ð Þ*z* ; *z*0, … , *zn*�<sup>1</sup> belonging to the *Dq* oper-

ator, with *<sup>q</sup>*<1 and associated with the sequence of points ð Þ *zn* <sup>∞</sup> <sup>0</sup> in *U*, is effective in j j *<sup>z</sup>* <sup>≤</sup>*<sup>r</sup>* for *<sup>r</sup>*<sup>≥</sup> *<sup>h</sup>* 1�*q*.

Theorem 1.5 of Buckholtz and Frank [3] shows that the result of Theorem 6.1 above is best possible. They also showed that when *q*>1 the Goncarov polynomials fail to be effective and also, that if j j *<sup>z</sup>* <sup>≤</sup>*q*�*<sup>n</sup>*, no favorable effectiveness results will occur, thus justifying the restriction j j *<sup>z</sup>* <sup>≤</sup>*q*�*<sup>n</sup>* on the points ð Þ *zn* <sup>∞</sup> 0 .

We also state and prove the following theorem.

#### **Theorem 6.2 ([19], Theorem 2).**

Suppose that *<sup>q</sup>*>1 and that the points ð Þ *zn* <sup>∞</sup> <sup>0</sup> satisfy the restriction (111). Then the Goncarov set *Qn* f g ð Þ *z*; *z*0, … , *zn*�<sup>1</sup> belonging to the *Dq*–derivative operator, will be effective in j j *<sup>z</sup>* <sup>≤</sup> *<sup>r</sup>* for *<sup>r</sup>*<sup>≥</sup> *hq <sup>q</sup>*�<sup>1</sup> and this result is best possible.

To prove this theorem we put, as in the proof of Theorem (72),

$$q^1 = \frac{1}{q},\tag{111}$$

so that *q*<sup>1</sup> <1 and we differentiate between the Goncarov polynomials belonging to the operations *Dq* and *Dq*<sup>1</sup> by adopting the notation.

$$Q\_n(z; z\_0, \dots, z\_{n-1}) \text{ and } p\_n(z; z\_0, \dots, z\_{n-1}),$$

for these respective polynomials. Thus, the constructive formulae (33) for these polynomials will be

$$\mathbf{Q}\_{n}(\mathbf{z};\mathbf{z}\_{0},\ldots,\mathbf{z}\_{n-1}) = \frac{\mathbf{z}^{n}}{[n]!} - \sum\_{k=0}^{n-1} \frac{z\_{k}^{n-k}}{[n-k]!} \mathbf{Q}\_{k}(\mathbf{z};\mathbf{z}\_{0},\ldots,\mathbf{z}\_{k-1}),\tag{112}$$

and

$$P\_n(z; z\_0, \ldots, z\_{n-1}) = \frac{z^n}{\{n\}!} - \sum\_{k=0}^{n-1} \frac{z\_k^{n-k}}{\{n-k\}!} P\_k(z; z\_0, \ldots, z\_{k-1}),\tag{113}$$

where ½ � *<sup>k</sup>* ! and f g*<sup>k</sup>* ! are the respective *<sup>q</sup>* and *<sup>q</sup>*<sup>1</sup> analogues of the factorial *<sup>k</sup>*. With this notation, the following Lemma is to be proved.

Lemma 6.1.

The following identity is true for *n*≥1 and *q*> 1 :

$$q^{-\frac{1}{2}n(n+1)}Q\_n(q^nz; q^nz\_0, \dots, qz\_{n-1}) = p\_n(z; z\_0, \dots, z\_{n-1}).\tag{114}$$

Proof.

We finish note, from the definition of the analogue ½ � *k* ! and f g*k* !, that

$$\frac{q^{n^2}}{\{n\}!} = \frac{q^{\frac{1}{2}n(n+1)}}{\{n\}!}; \ n \ge 1,\tag{115}$$

and

$$\frac{q^{(n-k)^2+(n-k)k}}{\{n-k\}!} = \frac{1}{\{n-k\}!} q^{\frac{1}{2}n(n+1) - \frac{1}{2}k(k+1)} \; ; \; 0 \le k \le n. \tag{116}$$

Hence, applying the relations (37) and (112) to *QN qnz*; *qnz*0,…, *qzn* � � � � , we get

$$Q\_n(qz, q^n z\_0, \dots, qz\_{n-1}) = \frac{q^{n^2}}{\{n\}!} z^n - \sum\_{k=0}^{n-1} \frac{q^{(n-k)^2 + (n-k)k}}{\{n-k\}!} z\_k^{n-1} Q\_k(q^k z, q\_k z\_0, \dots, qz\_{k-1}) \dots$$

Hence, the relations (115) and (116) can be introduced to yield

$$\begin{aligned} q^{-\frac{1}{2}n(n+1)} \mathbf{Q}\_n(q^n z, q^n z\_0, \dots, q z\_{n-1}) &= \frac{z^n}{\{n\}!} \\ &- \sum\_{k=0}^{n-1} \frac{z\_k^{n-k}}{\{n-k\}!} q^{-\frac{1}{2}k(k+1)} \mathbf{Q}\_k(q^k z, q^k z\_0, \dots, q z\_{k-1}). \end{aligned} \tag{117}$$

Now, since

$$q^{-1}Q\_1(qz;qz\_0) = z - z\_0 = p\_1(z;z\_0),$$

the identity (114) is satisfied for *n* ¼ 1.

Moreover, if (114) is valid for *k* ¼ 1, 2, … , *n* � 1, the relations (113) and (117) will give

$$\begin{aligned} q^{-\frac{1}{2}n(n+1)}Q\_n\left(q^n z, q^n z\_0, \dots, qz\_{n-1}\right) &= \frac{z^n}{\{n\}!} - \sum\_{k=0}^{n-1} \frac{z\_k^{n-k}}{\{n-k\}!} P\_k(z; z\_0, \dots, z\_{k-1}) \\ &= P\_n(z; z\_0, \dots, z\_{n-1}), \end{aligned}$$

and hence the Lemma is established. Proof of Theorem 6.2. Write

> *zk* <sup>¼</sup> *<sup>q</sup>*�*<sup>k</sup> ak*; *k*≥ 0, (118)

so that the restriction (111) implies that

$$|a\_k| \le 1 \; ; \; k \ge 0 \tag{119}$$

Therefore, a combination of (37), (114), (118) yields

$$Q\_n(z; z\_0, \dots, z\_{k-1}) = q^{-\frac{1}{2}k(k+1)} P\_k(z, z\_0, \dots, z\_{k-1}).\tag{120}$$

Also, by actual calculation we have that

*Effectiveness of Basic Sets of Goncarov and Related Polynomials DOI: http://dx.doi.org/10.5772/intechopen.99411*

$$\frac{[n]!}{[n-k]!}q^{-k(n-k)-\frac{1}{2}k(k+1)} = \frac{\{n\}!}{\{n-k\}!} \; ; \; 0 \le k \le n \tag{121}$$

Inserting (118), (120) and (121) into (33), we obtain

$$z^n = \sum\_{k=0}^n \frac{[n]!}{[n-k]!} z\_k^{n-k} Q\_k(z; z\_0, \dots, z\_{k-1})$$

$$= \sum\_{k=0}^n \frac{\{n\}!}{\{n-k\}!} a\_k^{n-k} P\_k(\mathfrak{s}, a\_0, \dots, a\_{k-1}),$$

in the sense that each term in the sum on the left hand side of this relation is equal to the corresponding term in the sum on the right hand side.

Hence, if

$$\mathbf{M}\_k(r) = \sup\_{|\mathbf{z}| = r} |Q\_k(\mathbf{z}; \mathbf{0}, \dots, \mathbf{z}\_{k-1})|$$

$$m\_k(r) = \sup\_{|\pi|=r} |P\_k(z; a\_0, \dots, a\_{k-1})|$$

and Ω*n*ð Þ*r* and *wn*ð Þ*r* are the respective Cannon sums of the sets *Qn* f g ð Þ *z*; *z*0, … , *zn*�<sup>1</sup> and f g *Pn*ð Þ *z*; *a*0, … , *an*�<sup>1</sup> , it follows that

$$\mathfrak{Q}\_n(r) = \sum\_{k=0}^n \frac{[n]!}{[n-k]!} |x\_k|^{n-k} \mathfrak{M}\_k(r) \tag{122}$$

$$= \sum\_{k=0}^n \frac{\{n\}!}{\{n-k\}!} |a\_k|^{n-k} m\_k(r) = w\_n(r).$$

Since the points ð Þ *ak* <sup>∞</sup> <sup>0</sup> lie in *U*, from (119), then applying Theorem 6.1 we deduce from (122) that the set *Qn* f g ð Þ *z*; *z*0, … , *zn* will be effective in j j *z* ≤*r* for *r*≥ *<sup>h</sup>* <sup>1</sup>�*<sup>q</sup>* <sup>¼</sup> *qh <sup>q</sup>*�<sup>1</sup> as to be proved.

To show that the result of the Theorem is best possible we appeal to Theorem 1.5 of Buckholtz and Frank [3] to deduce that the set f g *Pn*ð Þ *z*; *a*0, … , *an*�<sup>1</sup> may not be effective in j j *<sup>z</sup>* <sup>≤</sup> *<sup>r</sup>* for *<sup>r</sup>*<sup>&</sup>lt; *qh q*�1 .

In view of the relation (122), we may conclude that the set *Qn* f g ð Þ *z*; *z*0, … , *zn* will not be effective in j j *<sup>z</sup>* <sup>≤</sup> *<sup>r</sup>* for *<sup>r</sup>*<sup>&</sup>lt; *qh <sup>q</sup>*�<sup>1</sup> and Theorem 6.2 is fully established.

### **6.1 The case of Goncarov polynomials with** *Zk* <sup>¼</sup> *at<sup>k</sup>***,** *<sup>k</sup>* **<sup>≥</sup> <sup>0</sup>**

Nassif [14] studied the convergence properties of the class of Goncarov polynomials *Qn* f g ð Þ *<sup>z</sup>*; *<sup>z</sup>*0, … , *zn*�<sup>1</sup> generated through the *<sup>q</sup>*th derivative described in (33) where now, *zk* <sup>¼</sup> *atk*, *<sup>k</sup>*<sup>≥</sup> 0 and *<sup>a</sup>* and *<sup>t</sup>* are any complex numbers. By considering possible variations of t and q, it was shown that except for the cases j j*t* ≥ 1, *q*<1 and j j*<sup>t</sup>* <sup>&</sup>gt; <sup>1</sup> *<sup>q</sup>* ; *<sup>q</sup>*<sup>&</sup>gt; 1, all other cases lead to the effectiveness of the set *Qn <sup>z</sup>*; *<sup>a</sup>*, *at*, … *atn*�<sup>1</sup> ð Þ in finite circles ([14]; Theorems 1.1, 1.2, 1.3, 3.2, 3.3).

#### **6.2 Quasipower basis (QP-basis)**

Kazmin [20] announced results on some systems of polynomials that form a quasipower basis, (QP-basis), in specified spaces. These include the systems of Goncarov polynomials and of polynomials of the form:

$$\{\left(\mathbf{z} + a\_n\right)^n\}, n = \mathbf{0}, i, 2\dots; a\_n \in [-1, 1]. \tag{123}$$

For full details of QP-basis and some of the results announced, cf. ([20]; Corollaries 3, 4).

Of interest is his results that the system in (123), for arbitrary sequence f g*<sup>a</sup>* <sup>∞</sup> <sup>0</sup> of complex numbers with j j *an* ≤1, forms a QP- basis in the space 1, ½ � *σ* , for 0 <*σ* <*W* and in the space 1, ½ Þ *σ* , for 0< *σ* ≤*W*, where W = 0.7377 is the Whittaker constant. This value of W = 0.7377 is attributed to Varga [21]. He also added that Corollaries 3 and 4 contain known results in [5, 9, 15, 22, 23].

#### **7. Conclusions**

The chapter presents a compendium of diverse but related results on the convergence properties of the Goncarov and Related polynomials of a single complex variable. Most of the results of the author (or joint), have appeared in print but are here presented in considerable details in the proofs and in their development, for easy reading and assimilation. The results of other authors are summarized with related and relevant ones mentioned to complement the thesis of the chapter. Some recent works related to the Goncarov and related polynomials, cf. [24–29], which provides further applications are included in the references.

The comprehensiveness of the presentation is for the needs of those who may be interested in the subject of the Goncarov polynomials in general and also in their application to the problem of the determination of the exact value of the Whittaker constant, a problem that is still topical and challenging.

#### **Acknowledgements**

I acknowledge the mentorship of Professor M. Nassif, (1916-1986), who taught me all I know about Basic Sets. I thank Dr. A. A. Mogbademu and his team for typesetting the manuscript at short notice and also the Reviewer for helpful comments which greatly improved the presentation.

#### **No conflict of interest**

The author declares no conflict of interest.

*Effectiveness of Basic Sets of Goncarov and Related Polynomials DOI: http://dx.doi.org/10.5772/intechopen.99411*

### **Author details**

Jerome A. Adepoju Formerly of the Department of Mathematics, University of Lagos, Akoka-Yaba, Lagos, Nigeria

\*Address all correspondence to: jadi1011@yahoo.com

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

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[5] Whittaker JM. Sur les series des polynomes quelcuonques, Gauthier-Villars, Paris. 1949.

[6] Cannon TW. On the convergence of integral functions by general basic series. Math. Zeit. 1949; 45:185-205

[7] Boas Jr. R P. Basic sets of polynomials I. Duke Math. J. 1948; 15: 717-724.

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[10] Buckholtz JD. The Whittaker constant and successive derivatives of entire functions. J. Approx. Theory, 1970; 3: 194-212.

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[16] Marden M. The Geometry of the Zeros of a polynomial in a complex variable. Amer. Math. Soc. Math. Surv. 3; 1958 :NewYork.

[17] Nassif M, Adepoju J A. A polynomial set Related to Goncarov polynomials. Indian J. Pure Appl. Math 1980; 1 (12): 1665-1672.

[18] AL- Salam, W. A., q- analogue of Cauchy formula. Proc. Amer. Math. Soc. 1966; 17: 616-621.

[19] Adepoju JA. On the convergence of a class of Generalized Goncarov polynomials. Soochow J. Math. 1995; 21 (1): 71-80.

[20] Kaz'min Yu A. Perturbed Appell polynomials and systems of functions Associated with them. Soviet. Math. Dokl. 1985; 31(3): 452-455.

[21] Varga RS. Topics in polynomials and rational interpolation and Approximation. Sem. Math. Sup. 1982. Presses Univ. Montreal, Montreal.

[22] Eweida, M; A note on Abel's polynomials. Proc. Math. Phys. Soc. Egypt. 1959; 5 (3): 63-66

[23] Kaz'min Yu. Vestnic Moscow Univ. Ser. I. Math. Mekh. 1978; 2: 88-90, English Transl. in Moscow Univ. Math. Bull. 1978; 33: 1978.

[24] Eweida MA. Proc. Math. Phys. Soc. Egypt. 1959; 22: 83-88.

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[25] Adeniran A, Yan C. Goncarov polynomials in partition lattice and and exponentials families. Advanced in Appl. Math. Elsevier. May 2021.

[26] Lorentz R, et al. Generalized Goncarov polynomials. In Butler J. Cooper and G. Hierlbert (eds). Cambridge Univ. P. Press . 2018.

[27] Wu R. Abel- Goncarov Type multiquadric Quasi- interpolation operators with higher approximation order. Hindawi J. Math. July 2021.

[28] Anjorin A. Akinbode S. Chebychev polynomials of the first kind and the Whittaker Constant. Global J. Sci. Frontier Research: For Math. and Decision Sci. 2015 (2):Version 10

[29] Yacubovich S. On some properties of Abel- Goncerov polynomials and the Casas- Alvero problem. Integral Transforms and special functions. 2016; 27 (8).

#### **Chapter 4**

## Irreducible Polynomials: Non-Binary Fields

*Mani Shankar Prasad and Shivani Verma*

#### **Abstract**

Irreducible polynomials play an important role in design of Forward Error Correction (FEC) codes for data transmission with integrity and automatic correction of data, as for example, *Low-Density Parity Check codes*. The usage of irreducible polynomials enables construction of non-prime-order finite fields. Most of the irreducible polynomials belong to binary Galois field. The important analytical concept is optimisation of irreducible polynomials for use in FECs in nonbinary Galois (NBG) field, leading to the development of an algorithm for LDPC that can work with nonbinary Galois fields. According to studies, the Tanner graph for 'nonbinary Low Density Parity Check' codes might get sparser as the field's dimensions rise, ensuring that they do much better than their binary counterparts. A detailed discussion of representation of nonbinary irreducible polynomial and the computations involved have been illustrated. The concept has been tried for NB-LDPC codes. To prove the notion, computational complexity is found from different parameters such as performance of error correction capability, complexity cost and simulation time taken. Such detail study makes the NBG fields-based FEC very suitable for high-speed data transmission with self-error correction.

**Keywords:** irreducible polynomial, nonbinary Galois field, FEC, LDPC, Galois field

#### **1. Introduction**

Polynomials are algebraic expressions that consist of variables and coefficients. Indeterminates are another word for variables. Polynomial expressions can undergo a variety of mathematical operations, but they cannot be split by the variable.

The Greek phrases 'poly' and 'nomial', which imply 'many' and 'period', respectively, from the word Polynomial. A polynomial is a mathematical equation that is formed by multiplying the sum of terms in one or more variables by coefficients.

For example

$$5\mathbf{x}^3 + \frac{7}{4}\mathbf{x}^2 - \frac{2}{3}\mathbf{x} + \mathbf{1} \tag{1}$$

is a polynomial in single variable.

$$
\sqrt{2x^3y^2 + 6x^2y + 5xy + \sqrt{3}} \tag{2}
$$

is a polynomial in two variables.

For the polynomial, *anx<sup>n</sup>* <sup>þ</sup> *an*�<sup>1</sup>*xn*�<sup>1</sup> <sup>þ</sup> … … *::a*0, the degree is defined to be n if *an* 6¼ 0. The degree of the polynomial <sup>P</sup>*aijxi y <sup>j</sup>* in two variables x, and y is given by max *<sup>i</sup>* <sup>þ</sup> *<sup>j</sup>* <sup>j</sup> *aij* 6¼ <sup>0</sup> � �*:*

Given two polynomials *f x*ð Þ and *g x*ð Þ, there exist unique quotient and remainder polynomials *q x*ð Þ and *r x*ð Þ, such that

$$f(\mathbf{x}) = q(\mathbf{x})\mathbf{g}(\mathbf{x}) + r(\mathbf{x}),\\
\text{where } \text{degree of } r(\mathbf{x}) < \mathbf{g}(\mathbf{x})\tag{3}$$

If *gcd f x* ½ �¼ ð Þ, *g x*ð Þ *d*, then there exists two polynomials *p x*ð Þ, *q x*ð Þ such as

$$d = p(\mathfrak{x})f(\mathfrak{x}) + q(\mathfrak{x})\mathfrak{g}(\mathfrak{x})\tag{4}$$

#### **1.1 Irreducible polynomials**

Irreducible polynomials are considered as the basic constituents of all polynomials.

A polynomial of degree n ≥ 1 with coefficients in a field *F* is defined as irreducible over *F* in case it cannot be expressed as a product of two non-constant polynomials over *F* of degree less than *n*.

Example 1:

Consider the *<sup>x</sup>*<sup>2</sup>–2 polynomial. There are no zeroes in *<sup>x</sup>*<sup>2</sup> � 2 over Q. This is the same as asserting that √ 2 is not rational [1].

In case*x*<sup>2</sup> � 2 is reducible, we may write *<sup>x</sup>*<sup>2</sup>–2 = g(x)h(x), where g(x) and h(x) are both fewer than two degrees. Because the LHS has a degree of two, the sole option is that both g(x) and h(x) have a degree of one. *x*<sup>2</sup>–2 has a zero in Q in this example, which is a contradiction. As a result, *<sup>x</sup>*<sup>2</sup>–2 is irreducible over Q. *<sup>x</sup>*<sup>2</sup>–<sup>2</sup> <sup>¼</sup> ð Þ <sup>x</sup> � <sup>2</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>2</sup> , and on the contrary, is reducible over R, *<sup>x</sup>*<sup>2</sup> � <sup>2</sup> <sup>¼</sup> *<sup>x</sup>* � <sup>√</sup><sup>2</sup> � � *<sup>x</sup>* <sup>þ</sup> <sup>√</sup><sup>2</sup> � �*:*

Example 2: *<sup>x</sup>*<sup>4</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> 1 is an irreducible polynomial having degree 4 over *GF* ð Þ<sup>2</sup> but *<sup>x</sup>*<sup>4</sup> <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> 1 is not irreducible because *<sup>x</sup>*<sup>4</sup> <sup>þ</sup> *<sup>x</sup>*<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> <sup>¼</sup> *<sup>x</sup>*ð Þ <sup>3</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> .

Every polynomial of degree one is irreducible. The polynomial *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> 1 is irreducible over R but reducible over C.

Gauss's lemma.

A polynomial *f* ∈*Z x*½ �⊆ *Q x*½ � of the form

$$f(\mathbf{x}) = \mathbf{x}^n + a\_{n-1}\mathbf{x} + n - 1 + \dots + a\_1\mathbf{x} + a\_0\mathbf{x}$$

is irreducible in Q x½ � iff it is irreducible in Z x½ �. More precisely, if f xð Þ∈Z x½ �, then f xð Þ can be factorised and represented as multiplication of two polynomials of lesser degrees r and s in Q x½ � iff it has such a factorisation with polynomials of similar degrees r and s in Z x½ �*:*

Eisenstein irreducibility criterion (1850).

Suppose that *f x*ð Þis the polynomial with coefficients in the ring Z of integers, given by *f x*ð Þ¼ *cnxn* <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *<sup>c</sup>*1*<sup>x</sup>* <sup>þ</sup> *<sup>c</sup>*0and p is a prime which satisfies

1.*p* does not divide *cn*;

2.*p* divides *cn*�1, … , *p*1, *p*0;

3.*p*<sup>2</sup> does not divide *c*0;

Then, *f x*ð Þ is irreducible over field Q of rational numbers. [2]

*Irreducible Polynomials: Non-Binary Fields DOI: http://dx.doi.org/10.5772/intechopen.101897*

Dumas criterion (1906).

Let F xð Þ¼ *anx<sup>n</sup>* <sup>þ</sup> *an*�<sup>1</sup>*xn*�<sup>1</sup> <sup>þ</sup> … … *::a*<sup>0</sup> be a polynomial with coefficients in Z [2]. Suppose there exists a prime p whose exact power *<sup>p</sup>ri* dividing *ai* (where *ri* <sup>¼</sup> <sup>∞</sup> if *ai* ¼ 0), 0 ≤ i ≤ n, satisfy.

$$\begin{array}{c} \bullet \quad r\_n = 0, \\ \bullet \quad \bullet \quad \bullet \end{array}$$


Then, F(x) is irreducible over Q. Note that Eisenstein's criterion is a special case of Dumas criterion with *r*<sup>0</sup> ¼ 1*:*

#### **1.2 Monic polynomials**

A polynomial *xn* <sup>þ</sup> *an*�<sup>1</sup>*x<sup>n</sup>*�<sup>1</sup> <sup>þ</sup> … … *:* <sup>þ</sup> *<sup>a</sup>*1*<sup>x</sup>* <sup>þ</sup> *<sup>a</sup>*<sup>0</sup> in which the coefficient of the highest-order term is 1 is called a monic polynomial. For example, *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> 3 is monic but, 7*x*<sup>2</sup> <sup>þ</sup> 3 is not monic (the highest power of *<sup>x</sup>*<sup>2</sup> has a coefficient of 7, not 1).

Let *Fq* stand for the finite field of q elements, where q is the prime number. Gauss' formula, in general, gives the number of monic irreducible polynomials *Mn*ð Þ *p* of degree *n* over the finite field *Fq*. [3]

$$M\_n(p) = \frac{1}{n} \sum\_{d/n} \mu(d/n) q^d \tag{5}$$

where *μ*ð Þ*r* denotes the Mobius function and *d* includes all positive divisors of *n*. *d* also includes 1 and *n.* Also, *μ* ð Þ¼ 1 1. The value of *μ*ð Þ *d* calculated at a product of distinct primes is 1 if number of factors is even and it is equal to �1 if the number of factors is odd. For all other natural numbers *μ*ð Þ¼ *d* 0, that is,

*μ*ð Þ¼ *d* 1, *d* is squarefree positive integer with an even number of prime factors �1, *d* is square freepositive integer with an odd number of prime factors 0, *d* has a squared prime factor 8 >< >:

In particular, *μ*ð Þ¼� *d* 1 for all primes *p.* Example: *GF*ð Þ¼ <sup>256</sup> *GF* 28 � �*:* That is, *<sup>p</sup>* <sup>¼</sup> <sup>2</sup> *and n* <sup>¼</sup> <sup>8</sup>*:*

$$Mn(p) = \frac{1}{8} \sum\_{d/8} \mu(8/n) 2^d = \frac{1}{8} \sum\_{d \epsilon(1,2,4,\aleph)} \mu\left(\frac{8}{n}\right) 2^d$$

#### **1.3 Primitive polynomials**

<sup>A</sup> 'primitive polynomial' has its roots as primitive elements in the field *GF p<sup>n</sup>* ð Þ. It is an irreducible polynomial of degree d. It can be proved that there are ∅ *<sup>p</sup>d*�<sup>1</sup>*=d* � � number of primitive polynomials, where ∅ is Euler phi-function. For example, if p = 2, d = 4, ∅ <sup>2</sup>4�<sup>1</sup>*=*4 � � is 2, so there exist exactly two primitive polynomials of degree 4 over *GF* ð Þ2 *:* The MATLAB function pol = gfprimfd(m,opt,p) searches for one or more primitive polynomials for *GF pn* ð Þ, where p represents a number that is prime and n is greater than 0. The MATLAB code below seeks primitive polynomials for *GF* ð Þ 81 having various other properties.

```
p = 3; n = 4;
pol1 = gfprimfd(m,'min',p)
pol2 = gfprimfd(m,3,p)
pol3 = gfprimfd(m,4,p)
The output is as shown below:
pol1 =
  21001
pol2 =
  21001
  22001
  20011
  20021
pol3 = [ ]
```
Because no primitive polynomial for *GF*ð Þ 81 has exactly four nonzero terms, pol3 is empty. Also, pol1 represents a single three-term polynomial, whereas pol2 represents all of the three-term primitive polynomials for *GF* ð Þ 81 .

Further, *Primpoly(m)* gives the primitive polynomial for *GF* <sup>2</sup>*<sup>m</sup>* ð Þ, where m is an integer between 2 and 16. The output is an integer, whose binary representation represents the polynomial coefficients.

For *GF* <sup>2</sup>*<sup>m</sup>* ð Þ, *primpoly(m,opt)* returns one or more primitive polynomials. As seen in **Table 1**, the output depends on the argument opt. The output argument (an integer represented in binary format) represents the coefficients of the relevant polynomial. There is no element in the output if no primitive polynomial satisfies the conditions [4].

```
m = 4;
defaultprimpoly = primpoly(m)
allprimpolys = primpoly(m,'all')
i1 = isprimitive(25)
i2 = isprimitive(21)
The output is as shown below:
Primitive polynomial(s) =
D^4+D^1+1
defaultprimpoly =
19
Primitive polynomial(s) =
D^4+D^1+1
D^4+D^3+1
allprimpolys =
19
25
i1 = logical
```


#### **Table 1.**

*Different options for argument 'opt' in primpoly (m,opt).*

*Irreducible Polynomials: Non-Binary Fields DOI: http://dx.doi.org/10.5772/intechopen.101897*

```
=1
i2 = logical
=0
```
The MATLAB function *isprimitive(a)* gives the output as 1 if *a* represents a primitive polynomial for the Galois field *GF* <sup>2</sup>*<sup>m</sup>* ð Þ, and 0 otherwise [4].

### **2. Role of primitive polynomials in nonbinary LDPC codes [12]**

Error control codes find applications in the transmission and storage of vast amounts of error-prone data. Mostly, binary and nonbinary channels use ECC codes such as BCH codes, Reed Solomon codes [5], and Low-Density Parity Check codes. Berlekamp and Massey, after Peterson, created powerful algorithms that proved possible with the use of latest digital techniques. In addition to this, the usage of primitive polynomials and the Galois field offered these routines a structured and systematic approach.

LDPC codes got an initial explanation from Gallager in 1963. A parity check matrix (PCM) having sparse characteristics with a minimal number of nonzero components feature LDPC codes. These codes were overlooked until the mid-1990s despite their superior performance due to their decoding complexity, which exceeded the capacity of then electronic systems. When Mackay et al. reviewed the LDPC codes in 1996 [6], they discovered that when decoded using probabilistic soft choice decoding methods, they obtain performance close to the Shannon limit. Gallager later proposed nonbinary LDPC (NB-LDPC) codes by extending the concept of LDPC codes to nonbinary alphabets.

The NB-LDPC codes are defined for Galois fields of order, strictly higher than 2. They are considered as a good alternative to LDPC codes (Refer **Figure 1**) because:


#### **2.1 Galois fields**

A Galois field is a finite field with a finite order, which is either a prime number or the power of a prime number. A field of order *np* <sup>¼</sup> *<sup>q</sup>* is represented as *GF np* ð Þ. A

**Figure 1.** *Binary vs. nonbinary LDPC, Nb = 3008 bits and R = 1/2 [12].*

specific type called as characteristic-2 fields are the fields when *n* ¼ 2*:* All the elements of a characteristic-2 field can be shown in a polynomial format [10].

In coding applications, for*p*≤ 32, it is normal to represent an entire polynomial in *GF* <sup>2</sup>*<sup>p</sup>* ð Þ as a single integer value in which individual bits of the integer represent the coefficients of the polynomial. The least significant bit of the integer represents the a0 coefficient. For example, the polynomial form of the Galois field with 16 elements (known as *GF (16)*, so that *p=4*), is:

$$\mathfrak{a}\_3 \mathfrak{x}^3 + \mathfrak{a}\_2 \mathfrak{x}^2 + \mathfrak{a}\_1 \mathfrak{x}^1 + \mathfrak{a}\_0 \mathfrak{x}^0$$

with *a*3*a*2*a*1*a*<sup>0</sup> corresponding to the binary numbers *0000* to *1111*.

For any finite field *GF* <sup>2</sup>*<sup>p</sup>* ð Þ, there exists a primitive polynomial of degree *<sup>p</sup>* over *GF q*ð Þ [11].

**Table 2** lists the primitive polynomials.

The following MATLAB functions provide default primitive polynomials for Galois field:

*The row vector that supplies the coefficients of the default primitive polynomial for GF(pm), given by gfprimdf(m,p), is shown in polynomial format by the gfpretty function*

For binary field, *p* ¼ 1, while for *p*≥2, it represents a nonbinary field. Hence, nonbinary LDPC codes can be visualised as a direct generalisation of binary LDPC codes. **Table 3** shows the example for *p* ¼ 3 while considering the primitive polynomial1 <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>2</sup>*:*

The binary coefficients of the polynomial representation can be used to represent the Parity Check Matrix of a NB-LDPC code in a binary matrix form (Refer **Figures 2** and **3**).

The field's nonbinary elements can be represented as a polynomial [11] when the field order of an NB-LDPC code is a power of 2, giving the field elements a binary representation.

Consider a normal ð Þ *M* � *N* NB-LDPC code written in a Galois field *GF q*ð Þ, where *<sup>q</sup>* <sup>¼</sup> <sup>2</sup>*<sup>p</sup>* is the field's order. ð Þ *dv*, *dc* are the degrees of connection of the variable and check nodes, respectively. The nonzero elements of the NB parity check matrix H associated with the code correspond to the Galois field *GF* <sup>2</sup>*<sup>p</sup>* ð Þ <sup>¼</sup> *<sup>q</sup>* .

#### *Irreducible Polynomials: Non-Binary Fields DOI: http://dx.doi.org/10.5772/intechopen.101897*


#### **Table 2.**

*Primitive polynomials.*


#### **Table 3.**

*For* p *= 3 while considering the primitive polynomial 1 +* x *+* x*<sup>2</sup> :*


**Figure 2.** *Nonbinary* LDPC *parity check matrix [12].*

**Figure 3.** *Tanner graph for a nonbinary parity check matrix [12].*

We define a primitive polynomial of degree *p* for the field:

$$p(\mathbf{x}) = a\_0 + \ \boldsymbol{a}\_1 \mathbf{x} + \ \boldsymbol{a}\_2 \mathbf{x}^2 \dots + \mathbf{x}^p$$

A matrix A of size p � p is also associated to the primitive polynomial [10].


where [*a*0, *a*<sup>1</sup> … , *ap*�1] are primitive polynomial p(x)'s coefficients.

Because p(x) represents the field's primitive polynomial, A can be called as its primitive element in matrix representation. As a result, the binary matrix representations of all the other elements of the field are generated by the powers of the matrix A. Thus, the nonzero elements *hij* of the parity check matrix can be written in the form of a ð Þ *p* � *p* binary matrices *Hij*, where *Hij* is the result of the tranpose of a power of the primitive matrix of the Galois field. Subsequently, the ð Þ *M* � *N* nonbinary parity check matrix can be written in the form of a *MbXNb* binary parity check matrix, where *Mb* ¼ *pM* and *Nb* ¼ *pN* as shown in **Figure 4**. The zero elements of the parity check matrix are represented with all-zero matrices of size ð Þ *p* � *p :*

Modulo-2 addition and multiplication of polynomial representations are used to do arithmetic on the GF(q) elements. Modulo-2 arithmetic over the matrix representations can also be used to do arithmetic on the matrix representations. As a result, in the vectorial domain, the parity check equation may be represented as:

$$\sum\_{j:H\_{\vec{\imath}} \neq \mathbf{0}} H\_{\vec{\imath}\vec{\jmath}} X\_j^T = \mathbf{0}^T \tag{6}$$

where *Hij* is the matrix representation of the Galois field element *hij*, *X <sup>j</sup>* is the *p*-bits binary mapping of the symbol *cj*, and *0* is the all zero-component vector.

The methods for decoding binary LDPC codes may be generalised to nonbinary LDPC codes defined over finite fields by performing modifications in

*Irreducible Polynomials: Non-Binary Fields DOI: http://dx.doi.org/10.5772/intechopen.101897*


#### **Figure 4.**

*Nonbinary parity check matrix [12].*

correspondence to finite fields. MacKay et al. [13] expanded the belief propagation technique to nonbinary LDPC codes constructed over finite fields. The main obstacle in the development of the hardware realisation of the BP decoding algorithm is its computational complexity, the major factor being the check nodes processing, which is composed of a high number of additions and multiplications. A combination of iterative and list decoding algorithms [12] can be used to design low complexity nonbinary LDPC decoders.

#### **3. Conclusion**

In both binary (Galois field) and nonbinary fields, this chapter introduces monic, irreducible, primal polynomials. With examples, the MATLAB functions related to primitive polynomials were also discussed. The relevance of polynomials and the Galois field in creating the nonbinary LDPC code's parity check matrix for error detection and correction has also been described.

#### **Author details**

Mani Shankar Prasad and Shivani Verma\* Amity Institute of Space Science and Technology, Amity University Uttar Pradesh, India

\*Address all correspondence to: sverma2@amity.edu

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] http://www.math.ucsd.edu/ jmckerna/Teaching/15-16/Spring/ 103B/l\_17.pdf

[2] http://maths.du.ac.in/Events/IWM/ talks/S.K.Khanduja.pdf

[3] Chebolu SK, Miná J. Counting irreducible polynomials over finite fields using the inclusion-exclusion principle. Mathematics Magazine. 2011; **84**:369-371. DOI: 10.4169/math. mag.84.5.369 c Mathematical Association of America

[4] https://in.mathworks.com/help/

[5] Hanzlík P. Steganography in Reed-Solomon Codes. Sweden: Luleå University of Technology; 2011

[6] MacKay DJC, Neal RM. Near shannon limit performance of lowdensity parity check codes. Electronics Letters. 1997;**33**(6):457-458. DOI: 10.1049/el:19970362

[7] MacKay DJC, Davey MC. Evaluation of gallager codes for short block length and high-rate applications. In: Marcus B, Rosenthal J, editors. Codes, Systems, and Graphical Models. The IMA Volumes in Mathematics and its Applications, Vol. 123. New York, NY: Springer; 2001. DOI: 10.1007/978-1- 4613-0165-3\_6

[8] Ma L, Wang L, Zhang J. Performance Advantage of Non-binary LDPC Codes At High Code Rate under AWGN Channel. 2006 International Conference on Communication Technology. 2006. pp. 1-4. DOI: 10.1109/ ICCT.2006.341827

[9] Li G, Fair IJ, Krzymien W. Density evolution for nonbinary LDPC codes under Gaussian approximation. IEEE Transactions on Information Theory. 2009;**55**(3):997-1015

[10] Poulliat C, Fossorier M, Declercq D. Design of non binary ldpc codes using their binary image: algebraic properties. Seattle, USA: In Proceedings of IEEE ISIT; 2006

[11] Poulliat C, Fossorier M, Declercq D. Opitmization of non-binary ldpc codes using their binary images. Munich, Germnay: In Proceedings of IEEE Int. Symp. on Turbo codes; 2006

[12] Shams B. Next generation nonbinary LDPC codes. Information Theory [cs.IT]. French: University of Cergy Pontoise; 2010

[13] Davey M, Mackay DJC. Low Density Parity Check Codes Over gf(q). IEEE Communication Letter. 1998;**2**:165-167

#### **Chapter 5**

## On the Use of Homogeneous Polynomial Yield Functions in Sheet Metal Forming Analysis

*Mehmet Firat, Bora Şener,Toros Arda Akşen and Emre Esener*

#### **Abstract**

Sheet metal forming techniques are a major class of stamping and manufacturing processes of numerous parts such as doors, hoods, and fenders in the automotive and related supplier industries. Due to series of rolling processes employed in the sheet production phase, automotive sheet metals, typically, exhibit a significant variation in the mechanical properties especially in strength and an accurate description of their so-called plastic anisotropy and deformation behaviors are essential in the stamping process and methods engineering studies. One key gradient of any engineering plasticity modeling is to use an anisotropic yield criterion to be employed in an industrial content. In literature, several orthotropic yield functions have been proposed for these objectives and usually contain complex and nonlinear formulations leading to several difficulties in obtaining positive and convex functions. In recent years, homogenous polynomial type yield functions have taken a special attention due to their simple, flexible, and generalizable structure. Furthermore, the calculation of their first and second derivatives are quite straightforward, and this provides an important advantage in the implementation of these models into a finite element (FE) software. Therefore, this study focuses on the plasticity descriptions of homogeneous second, fourth and sixth order polynomials and the FE implementation of these yield functions. Finally, their performance in FE simulation of sheet metal cup drawing processes are presented in detail.

**Keywords:** Homogeneous polynomials, yield criteria, finite element, plastic anisotropy, cup drawing

#### **1. Introduction**

Sheet materials represent significant anisotropic behavior due to their thermomechanical process history. Anisotropy states the variation of the mechanical properties with direction. This material property is determined from tensile test and it is calculated by dividing width plastic strain increments to thickness plastic strain increments. From this definition, it is seen that anisotropy indicates the resistance to the thinning. Therefore, it can be said that increasing anisotropy values improves the deep drawability of the material. Two approaches are applied in the description of the anisotropy. The first approach is the phenomenological approach in which global material behavior is determined according to the average behavior of all grains. The second approach is crystal plasticity which investigates the behavior of one grain to determine the material behavior.

In the phenomenological plasticity approach, the transition from the elastic deformation to plastic deformation is defined with yield functions [1]. A yield function establishes the relationship between principal stresses and yield stress of the material. Plastic flow occurs when the yield function reaches a critical value which is the yield stress of the material. Therefore, yield condition actually indicates a state of equilibrium and it can be defined by the following equation:

$$\mathbf{F} = \mathbf{f}\_{\mathbf{y}}(\overline{\mathbf{c}}) - \mathbf{c}\_{\mathbf{y}} = \mathbf{0} \tag{1}$$

where σ and *σ<sup>y</sup>* denote the equivalent and yield stresses, respectively. Eq. (1) defines a surface in three dimensional stress space and it is called as yield surface. According to Drucker's postulate [2], this surface must be closed, convex, and smooth in order to establish a relationship between plastic strain increments and stresses. In the literature, Tresca and von Mises are well known and have been most commonly used yield criteria. However, these yield criteria are isotropic and they could not give satisfactory results for sheet metal forming processes. Therefore, the usage of anisotropic yield functions is required for representation of sheet metal behavior and several anisotropic yield functions have been proposed by researchers. The first phenomenological anisotropic yield function was proposed by Hill in 1948 [3]. Hill added some coefficients to von Mises criterion to transform isotropic von Mises criterion into an anisotropic form. Hill's quadratic criterion could be used for both plane stress (2D) and general stress (3D) states. The criterion has four coefficients for 2D stress state, and it has six coefficients for 3D stress state. These coefficients could be obtained analytically according to stress or plastic strain ratios. Hill48 quadratic criterion has a simple form and useful coefficient identification procedure. However, this criterion could not simultaneously predict the variations of the stress and strain ratios within the sheet plane. Therefore, it could not successfully define the plastic behavior of highly anisotropic materials such as Al-Mg alloys, Ti alloys, etc. Different type yield criteria have been applied to accurately describe the anisotropic behavior of these materials. The most popular approach used to derive an anisotropic yield criterion is the linear transformation method. In this method, Cauchy stress tensor or the deviatoric stress tensor is transformed linearly, and an anisotropic yield function is obtained by substitution of this transformed tensor in an isotropic yield function [4]. Yld89 is one of the functions developed by this approach. Barlat and Lian [5] applied linear transformation method to Hosford 1972 [6] isotropic yield criterion and developed this anisotropic material model. The criterion has four coefficients and it could be used for only 2D stress state. Then, Barlat et al. [7] extended this yield criterion for 3D stress state and developed a criterion has six coefficients in 1991. However, these yield criteria could not accurately describe the anisotropic behavior of especially Al-Mg alloys. Another yield criterion based on linear transformation approach was developed by Karafillis and Boyce [8] in 1993. Karafillis and Boyce generalized Hosford's yield function and proposed an isotropic yield function. Then researchers applied to linear transformation approach and developed an anisotropic yield criterion. They applied their developed yield criterion for modeling of AA2008- T4 alloy and could successfully define the angular variations of both stress and plastic strain ratios of the material. Barlat et al. have inspired by this method and developed Yld2000 and Yld2004 yield criteria, respectively [9, 10]. From these models, Yld2000 could only be used for plane stress condition, whereas the other could be used for both plane stress and general stress states. Yld2004 criterion has 18 coefficients and it could successfully describe in-plane variations of plastic properties of highly anisotropic aluminum alloys. These models are effective in the representation of the anisotropic behavior. However, their parameter identification procedures consist of complex nonlinear formulas and computation of the derivatives is difficult.

*On the Use of Homogeneous Polynomial Yield Functions in Sheet Metal Forming Analysis DOI: http://dx.doi.org/10.5772/intechopen.99412*

Another method which is applied to derive anisotropic yield function is the polynomial approach. Due to inability of quadratic Hill48 criterion, Hill suggested that the usage of general homogeneous polynomials as yield functions in 1950 [11]. In the literature, firstly Gotoh [12, 13] applied this method and modeled the anisotropic behavior of commercial Al-killed steel and Cu-(1/4)H sheets with fourthorder polynomial yield function. Gotoh determined explicitly the coefficients of the polynomial function for these materials and successfully predicted the angular variations of the plastic properties. However, Gotoh did not take into account the convexity of the yield surface in the parameter identification. This deficiency was noticed by Soare et al. [14] and they proposed changes to Gotoh's identification procedure. This modification has contributed to the applicability of the polynomial criteria and important results have been obtained.

In the present work, polynomial yield criteria, their modeling capability and applications on the sheet metal forming simulations have been investigated. Article consists of four sections. In Section 2, the theoretical background of the developed polynomial yield functions are briefly explained. Then, applications of polynomial criteria and results are presented. In Section 4, the main conclusions and findings are summarized.

#### **2. Homogeneous polynomial yield functions**

It is seen from the literature that the second, the fourth, and the sixth-order homogeneous polynomials have been used as yield functions. Therefore, the general formulation of these functions are explained in this section.

#### **2.1 Second-order polynomial yield function**

Conventional quadratic Hill48 yield criterion can be defined as second-order polynomial yield function (P2). The form of the criterion for plane stress state could be written as follows:

$$\mathbf{P\_2 = \mathbf{a\_1}\sigma\_x\,^2 + \mathbf{a\_2}\sigma\_y\,^2 - 2\mathbf{a\_3}\sigma\_x\sigma\_y + 2\mathbf{a\_4}\sigma\_{xy}\,^2\tag{2}$$

a1, a2, a3, and a4 are function parameters and they can be determined based on stress or plastic strain ratios. The equations related to stress and plastic strain ratios are given in below. The coefficients determined with stress or strain based definition are distinguished by subscripts σ and R, respectively.

$$\begin{split} a\_{1\\_\sigma} = 1; a\_{2\\_\sigma} &= \left(\frac{1}{\overline{\sigma}\_{90}}\right)^2; a\_{3\\_\sigma} = \frac{1}{2} \left(1 + \left(\frac{1}{\overline{\sigma}\_{90}}\right)^2 - \left(\frac{1}{\overline{\sigma}\_b}\right)^2\right); a\_{4\\_\sigma} \\ &= 2 \left(\frac{1}{\overline{\sigma}\_{45}}\right)^2 - \frac{1}{2} \left(\frac{1}{\overline{\sigma}\_b}\right)^2 \end{split} \tag{3}$$

$$a\_{1\cdot R} = \mathbf{1}; a\_{2\cdot R} = \frac{r\_0(\mathbf{1} + r\_{90})}{r\_{90}(\mathbf{1} + r\_0)}; a\_{3\cdot R} = \frac{r\_0}{\mathbf{1} + r\_0}; a\_{4\cdot R} = \frac{(r\_0 + r\_{90})(\mathbf{1} + 2r\_{45})}{2r\_{90}(\mathbf{1} + r\_0)}\tag{4}$$

#### **2.2 Fourth-order polynomial yield function**

For plane stress state, the fourth-order polynomial yield function (P4) is expressed as following:

$$\begin{aligned} P\_4 &= \mathbf{a}\_1 \mathbf{\sigma}\_{\mathbf{x}}^4 + \mathbf{a}\_2 \mathbf{\sigma}\_{\mathbf{x}}^3 \mathbf{\sigma}\_{\mathbf{y}} + \mathbf{a}\_3 \mathbf{\sigma}\_{\mathbf{x}}^2 \mathbf{\sigma}\_{\mathbf{y}}^2 + \mathbf{a}\_4 \mathbf{\sigma}\_{\mathbf{x}} \mathbf{\sigma}\_{\mathbf{y}}^3 + \mathbf{a}\_5 \mathbf{\sigma}\_{\mathbf{y}}^4 + \left( \mathbf{a}\_6 \mathbf{\sigma}\_{\mathbf{x}}^2 + \mathbf{a}\_7 \mathbf{\sigma}\_{\mathbf{x}} \mathbf{\sigma}\_{\mathbf{y}} + \mathbf{a}\_8 \mathbf{\sigma}\_{\mathbf{y}}^2 \right) \mathbf{\sigma}\_{\mathbf{xy}}^2 \\ &+ \mathbf{a}\_9 \mathbf{\sigma}\_{\mathbf{xy}}^4 \end{aligned} \tag{5}$$

where a1, a2, a3 … .a9 are the material coefficients. In order to determine these nine coefficients, nine experimental data are required. Direct approach for coefficient determination can lead to oscillations in the predictions of the plastic strain or yield stress ratios. Therefore, Soare et al. [14] proposed a different coefficient identification procedure and derived upper and lower bounds on coefficients to obtain a convex and smooth yield surface. In this section, the coefficient identification procedure developed by Soare is explained:

(i) Firstly, the first five coefficients are determined with explicit formulas are given below:

$$\mathbf{a}\_1 = \mathbf{1}, \mathbf{a}\_2 = -4\mathbf{r}\_0/(\mathbf{1} + \mathbf{r}\_0), \mathbf{a}\_5 = \mathbf{1}/(\overline{\mathbf{a}}\_{90})^4, \mathbf{a}\_4 = -4\mathbf{a}\_5\mathbf{r}\_{90}/(\mathbf{1} + \mathbf{r}\_{90})\tag{6}$$

where r0 and r90 indicate plastic strain ratios (r-values) along rolling and transverse directions, whereas *σ*<sup>90</sup> denotes yield stress ratio along transverse direction.

(ii) The coefficient a3 is determined according to the Eq. (7).

$$\mathbf{a}\_3 = \left(\mathbf{1}\langle\overline{\sigma}\_\mathbf{b}\,^4\right) - \left(\mathbf{a}\_1 + \mathbf{a}\_2 + \mathbf{a}\_4 + \mathbf{a}\_5\right) \tag{7}$$

where *σ<sup>b</sup>* indicates the biaxial yield stress ratio.

(iii) The coefficient a9 is determined according to Eq. (8)

$$\mathbf{a}\_{\mathfrak{d}} = \frac{(2/\overline{\sigma}\_{45})^4 \mathbf{r}\_{45}}{\mathbf{1} + \mathbf{r}\_{45}} + \left(\mathbf{1}/\overline{\sigma}\_{\mathfrak{b}}\mathbf{^4}\right) \tag{8}$$

where σ<sup>45</sup> and r45 indicate the yield stress and plastic strain ratios along the diagonal direction.

(iv) The coefficients a6 and a8 are determined with the minimization of the error (distance) function given in Eq. (9).

$$\mathbf{E} = \mathbf{w}\_1 \sum\_{i=1}^{2} \left[ \frac{(\overline{\sigma}\_{\theta})\_{\text{pred}} - (\overline{\sigma}\_{\theta})\_{\text{exp}}}{(\overline{\sigma}\_{\theta})\_{\text{exp}}} \right]^2 + \mathbf{w}\_2 \sum\_{i=1}^{2} \left[ \frac{(\mathbf{r}\_{\theta})\_{\text{pred}} - (\mathbf{r}\_{\theta})\_{\text{exp}}}{(\mathbf{r}\_{\theta})\_{\text{exp}}} \right]^2 \tag{9}$$

where w1 and w2 are the weight coefficients for stress and plastic strain ratios at the interval angles. In this minimization problem, interval angles could be 150 -750 , 300 -600 or 22.50 -67.50 . After determination of the coefficients a6 and a8, these coefficients are checked for positivity and convexity of the yield surface. In order to obtain convex and smooth yield surface, a6 and a8 must satisfy the following inequalities:

$$0 \le a\_6 \le 6\sqrt{a\_1 a\_9}, \ 0 \le a\_8 \le 6\sqrt{a\_5 a\_9} \tag{10}$$

v) The coefficient a7 is determined with Eq. (11)

$$\mathbf{a}\_{\mathsf{V}} = \frac{\left(2/\overline{\sigma}\_{45}\right)^{4}}{\mathbf{1} + \mathbf{r}\_{45}} - 2\left(\mathbf{1}/\overline{\sigma}\_{\mathsf{b}}\,^{4}\right) \tag{11}$$

Inequalities related to convexity and positivity conditions are given detailed in [14].

*On the Use of Homogeneous Polynomial Yield Functions in Sheet Metal Forming Analysis DOI: http://dx.doi.org/10.5772/intechopen.99412*

#### **2.3 The sixth-order polynomial yield function**

The sixth-order polynomial yield function (P6) has 16 coefficients for plane stress state and the form of the criterion is given below:

$$\begin{array}{l} \mathbf{P}\_{6} = \mathbf{a}\_{1}\mathbf{\sigma}\_{\mathbf{x}}\,^{6} + \mathbf{a}\_{2}\mathbf{\sigma}\_{\mathbf{x}}\,^{5}\mathbf{\sigma}\_{\mathbf{y}} + \mathbf{a}\_{3}\mathbf{\sigma}\_{\mathbf{x}}\,^{4}\mathbf{\sigma}\_{\mathbf{y}}\,^{2} + \mathbf{a}\_{4}\mathbf{\sigma}\_{\mathbf{x}}\,^{3}\mathbf{\sigma}\_{\mathbf{y}}\,^{3} + \mathbf{a}\_{5}\mathbf{\sigma}\_{\mathbf{x}}\,^{2}\mathbf{\sigma}\_{\mathbf{y}}\,^{4} + \mathbf{a}\_{6}\mathbf{\sigma}\_{\mathbf{x}}\mathbf{\sigma}\_{\mathbf{y}}\,^{5} + \mathbf{a}\_{7}\mathbf{\sigma}\_{\mathbf{y}}\,^{6} \\ \quad + \left(\mathbf{a}\_{8}\mathbf{\sigma}\_{\mathbf{x}}\,^{4} + \mathbf{a}\_{9}\mathbf{\sigma}\_{\mathbf{x}}\,^{3}\mathbf{\sigma}\_{\mathbf{y}} + \mathbf{a}\_{10}\mathbf{\sigma}\_{\mathbf{x}}\,^{2}\mathbf{\sigma}\_{\mathbf{y}}\,^{2} + \mathbf{a}\_{11}\mathbf{\sigma}\_{\mathbf{x}}\mathbf{\sigma}\_{\mathbf{y}}\,^{3} + \mathbf{a}\_{12}\mathbf{\sigma}\_{\mathbf{y}}\,^{4}\right)\mathbf{\sigma}\_{\mathbf{xy}}\,^{2} \\ \quad + \left(\mathbf{a}\_{13}\mathbf{\sigma}\_{\mathbf{x}}\,^{2} + a\_{14}\sigma\_{\mathbf{x}}\sigma\_{\mathbf{y}} + \mathbf{a}\_{15}\mathbf{\sigma}\_{\mathbf{y}}\,^{2}\right)\mathbf{\sigma}\_{\mathbf{xy}}\,^{4} + a\_{16}\sigma\_{\mathbf{x}}\mathbf{\sigma}\_{\mathbf{y}}\,^{6} \end{array} \tag{12}$$

The coefficients a1, a2, a6, and a7 are calculated explicitly and the equations are given below:

$$\mathbf{a}\_1 = \mathbf{1}, \mathbf{a}\_2 = -\frac{6\mathbf{r}\_0}{\left(\mathbf{1} + \mathbf{r}\_0\right)}, \mathbf{a}\_7 = \left(\mathbf{1}\middle|\overline{\mathbf{a}}\_{90}\right)^6, \ \mathbf{a}\_6 = -6\mathbf{r}\_{90}\mathbf{a}\_7/\left(\mathbf{1} + \mathbf{r}\_{90}\right) \tag{13}$$

The remained coefficients are determined by minimization of the error function given in Eq. (8).

#### **3. Applications of polynomial yield functions**

Three validation studies are generally performed in the literature in order to evaluate the prediction capability of orthotropic yield criteria: These are the description of the planar variations of plastic properties, the prediction of the earing profile and number of ears in cup drawing test, and prediction of the thickness strain distributions along the different directions in a drawn part, respectively. Obtained results with polynomial yield functions are presented in below.

#### **3.1 Description of the directional properties**

Soare et al. [14] investigated the prediction capability of the polynomial yield functions. They described the anisotropic behavior of AA2090-T3 with P4 and P6 yield criteria. **Figures 1** and **2** show the P4 and P6 predictions of the angular variation of plastic properties for AA2090-T3 alloy, respectively.

It is seen from **Figures 1** and **2** that both criteria could simultaneously predict the angular variations of stress and plastic strain ratio. In addition to that the predictions of P6 criterion were more successful than P4 criterion especially at interval

**Figure 1.** *Comparison of the predicted results from P4 criterion with experiment (a) stress ratio, (b) r-value.*

#### *Recent Advances in Polynomials*

angles. Sener et al. [15] investigated the evolution of anisotropic behavior of Al5754 with P2 and P4 yield criteria. They determined the coefficients of the yield functions at four different plastic strain levels and predicted the angular variations of yield stress and plastic strain ratios. Then, researchers compared the predicted results from yield criteria with experimental data for each plastic strain level.

**Figures 3** and **4** show the comparison results for P2 and P4 criteria, respectively. It is seen from **Figures 3** and **4** that P2 criterion could only accurately predict the variation of r-values in the sheet plane, while P4 criterion could predict both the angular variations of stress and strain ratios. This result is related to the identification procedures of the yield criteria. As it is declared in Section 2 that, P2 criterion takes as input either stress or strain ratios. However, the coefficients of P4 criterion

**Figure 2.** *Comparison of the predicted results from P6 criterion with experiment (a) stress ratio, (b) r-value.*

**Figure 4.** *Comparison of the predicted results from P4 criterion with experiment (a) stress ratio, (b) r-values.*

*On the Use of Homogeneous Polynomial Yield Functions in Sheet Metal Forming Analysis DOI: http://dx.doi.org/10.5772/intechopen.99412*

are calibrated with both stress and strain ratios. In addition to description of the planar anisotropy, researchers investigated the variation of the yield locus shape with plastic strain. **Figure 5a** and **b** show the variation of yield locus contours with plastic strain for P2 and P4 yield criteria, respectively.

It is seen from **Figure 5** that the contours of the yield locus are changed with plastic strain and this evolution is more pronounced in P4 criterion.

#### **3.2 Prediction of the earing profile**

Cup drawing is a test which is used for validation of an anisotropic yield criterion. If material has a strong anisotropy, the height of the formed cup is not uniform and a series of crests and valleys are observed around the cup perimeter. This waviness in the top edge of a cup is called as earing and four, six or eight ears could be occurred in a drawn cup depend on the degree of the anisotropy [16, 17]. Soare et al. [14] investigated the prediction capability of polynomial yield functions on the cup drawing test. They implemented P4 and P6 yield criteria into FE code ABAQUS and performed FE analyses of the test. Researchers also studied the effect of element type on the predictions and they carried out simulations with shell and solid elements. After FE analyses, they predicted the number of ears, cup height, and compared the numerical results with the Yld96 criterion and experiment. Yld96 criterion was selected as reference by the researchers due to involving the same number of material coefficients of both criteria. **Figures 6** and **7** show the geometry of the drawn cup and the comparison of the predicted cup profiles from P4 and Yld96 yield criteria with experiment for AA2090-T3 alloy.

It is seen from **Figure 7** that P4 and Yld96 criteria could successfully predict cup heights, however the predictions of P4 were closer to the experiment in the rolling direction. Both criteria predicted two extra ears along the transverse direction (90° and 270°). It was also observed that there are no significant differences between the predictions of P4-2D, and P4-3D models. Researchers also investigated the capability of P6 criterion on earing prediction and compared the predictions with Yld2004 and experiment. These comparisons are shown in **Figure 8**.

From the comparisons, it is observed that P6 criterion could accurately predict both the number of ears and cup height. Another observation in this study is related to Yld2004 and P6 predictions. Both criteria gave similar results and this shows that P6 has higher capability in the modeling of the anisotropy.

#### **3.3 Prediction of thickness strains in rectangular cup drawing**

Another study related to polynomial yield functions was carried out by Sener et al. [18]. They investigated the anisotropic behavior of AISI 304 stainless steel

**Figure 5.** *Variation of the yield locus contours with plastic strain (a) P2, (b) P4.*

**Figure 6.** *Drawn cup [14].*

**Figure 7.** *Experimental and predicted cup profiles from the fourth-order polynomial and Yld96 criteria for AA2090-T3 [14].*

with P4 yield criterion. Investigation was conducted on the uniaxial tensile test and a rectangular cup drawing process. Criterion could successfully describe stress anisotropy and r-value variations. Researchers implemented the criterion into explicit FE code Ls-Dyna by using user defined material subroutines and performed FE simulation of rectangular cup drawing process. They investigated the thickness distributions and flange geometry. **Figures 9** and **10** show the comparisons of the numerical and experimental results in terms of the thickness distributions and flange geometry of the cup.

It is seen from the **Figures 9** and **10** that the predicted thickness distributions and flange geometry matches well with the experimental results. Then, Sener et al. [19] expanded the study [18] and studied the variation of anisotropy during plastic *On the Use of Homogeneous Polynomial Yield Functions in Sheet Metal Forming Analysis DOI: http://dx.doi.org/10.5772/intechopen.99412*

**Figure 8.**

*Experimental and predicted cup profiles from the sixth-eight order polynomial and Yld2004 criteria for AA2090-T3 [14].*

**Figure 9.**

*Numerical and experimental thickness distributions (a) rolling (RD) (b) diagonal (DD), (c) transverse directions (TD).*

deformation experimentally and numerically. They carried out FE simulations of same industrial part at different plastic strain levels (0.2%, 2%, 5%, and 18%) and compared P4 predictions with experimental data. **Figure 11** shows the comparison of the predicted thickness distributions along the three directions with experiment.

It is seen from **Figure 11** that different thickness predictions were obtained at different plastic strain levels. After the comparison of the predicted thickness results with experiment, researchers eliminated two strain levels and then they investigated the flange geometry results (**Figure 12**).

**Figure 10.** *Numerical and experimental flange geometry.*

**Figure 11.**

*Comparison of the predicted thickness distributions with experiment (a) RD, (b) DD, (c) TD.*

**Figure 12.** *Comparison of the numerical and experimental flange geometry.*

*On the Use of Homogeneous Polynomial Yield Functions in Sheet Metal Forming Analysis DOI: http://dx.doi.org/10.5772/intechopen.99412*

From the comparison of the predicted and experimental flange geometry results, it is seen that numerical results were matched well with the experiment.

#### **4. Conclusions**

In the present study, homogeneous anisotropic polynomial yield functions, their types, and application areas in the metal forming process were investigated. In the literature, generally anisotropic yield functions derived from linear transformation approach are used. These functions have high modeling capability and they could be used for different materials. However, yield functions based on linear transformation approach have some disadvantages. They have complex coefficient identification procedure and nonlinear formulas. Therefore, calculations of the first and second order gradients of these models are difficult and it causes to difficulties in the implementation of the models into FE codes. On the other hand, polynomial yield functions have a generalized, simple structure and derivatives of these functions could easily calculated.

It is seen from the studies carried out in the literature that researchers generally use the fourth and the sixth order polynomial functions to model of the anisotropic behavior of the materials. Based on the results obtained from the studies performed in the literature, the following conclusions could be drawn:


*Recent Advances in Polynomials*

#### **Author details**

Mehmet Firat<sup>1</sup> \*, Bora Şener<sup>2</sup> , Toros Arda Akşen<sup>1</sup> and Emre Esener<sup>3</sup>


\*Address all correspondence to: firat@sakarya.edu.tr

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*On the Use of Homogeneous Polynomial Yield Functions in Sheet Metal Forming Analysis DOI: http://dx.doi.org/10.5772/intechopen.99412*

#### **References**

[1] Banabic D. Sheet Metal Forming Processes. Springer-Verlag: Berlin Heidelberg; 2010. 30 p. DOI: 10.1007/ 978-3-540-88113-1\_1

[2] Drucker DC. A more fundamental approach to plastic stress-strain relations. In: 1 st U.S. Congress of Applied Mechanics (ASME), New York, 1952, p.116-126

[3] Hill R. A theory of the yielding and plastic flow of anisotropic metals. Proceedings of the Royal Society of London Series. 1948; 281-297. DOI: 10.1098/rspa.1948.0045

[4] Barlat F, Yoon JW, Cazacu O. On linear transformations of stress tensors for the description of plastic anisotropy. International Journal of Plasticity. 2007; 23: 876-896

[5] Barlat F, Lian J. Plastic behavior and stretchability of sheet metals. Part I: a yield function for orthotropic sheets under plane stress conditions. International Journal of Plasticity. 1989; 5: 51-66

[6] Hosford WF. A generalized isotropic yield criterion. Journal of Applied Mechanics. 1972; 39: 607-609

[7] Barlat F, Lege DJ, Brem JC. A sixcomponent yield function for anisotropic materials. International Journal of Plasticity. 1991; 7: 693-712

[8] Karafillis AP, Boyce MC. A general anisotropic yield criterion using bounds and a transformation weighting tensor. Journal of the Mechanics and Physics of Solids. 1993; 41: 1859-1886

[9] Barlat F, Brem JC, Yoon JW, Chung K, Dick RE, Lege DJ, Pourboghrat F, Choi S-H, Chu E. Plane stress yield function for aluminum alloy sheets-part I:theory. International Journal of Plasticity. 2003; 19: 1297-1319 [10] Barlat F, Aretz H, Yoon JW, Karabin ME, Brem JC, Dick RE. Linear transformation-based anisotropic yield functions. International Journal of Plasticity. 2005; 21: 1009-1039

[11] Hill R. The Mathematical Theory of Plasticity. 1st ed. Oxford University Press: New York; 1950. 330p.

[12] Gotoh M. A theory of plastic anisotropy based on a yield function of fourth order (plane stress state)-I. International Journal of Mechanical Sciences. 1977; 19: 505-512.

[13] Gotoh M. A theory of plastic anisotropy based on a yield function of fourth order (plane stress state)-II. International Journal of Mechanical Sciences. 1977; 19: 513-520.

[14] Soare S, Yoon JW, Cazacu O. On the use of homogeneous polynomials to develop anisotropic yield functions with applications to sheet forming. International Journal of Plasticity. 2008; 24: 915-944.

[15] Sener B, Esener E, Firat M. Determining the effect of yield criterias on plasticity modeling in different plastic strain levels. In: Proceedings of the International Automotive Technologies Congress (OTEKON 2018); 07-08 May 2018; Bursa: 2018, p.754-760.

[16] Hosford W, Caddell RM. Metal Forming Mechanics and Metallurgy: Cambridge University Press; 2007.228 p.

[17] Yoon JW, Barlat F, Dick RE, Karabin ME. Prediction of six or eight ears in a drawn cup based on a new anisotropic yield function. International Journal of Plasticity. 2006; 22: 174-193.

[18] Sener B, Kilicarslan ES, Firat M. Modelling anisotropic behavior of AISI 304 stainless steel sheet using a fourthorder polynomial yield function. Procedia Manufacturing. 2020; 47: 1456-1461.

[19] Sener B, Esener E, Firat M. Modeling plastic anisotropy evolution of AISI 304 steel sheets by a polynomial yield function. SN Applied Sciences. 2021; 3: 1-12.

#### **Chapter 6**

## On the Irreducible Factors of a Polynomial and Applications to Extensions of Absolute Values

*Lhoussain El Fadil and Mohamed Faris*

#### **Abstract**

Polynomial factorization over a field is very useful in algebraic number theory, in extensions of valuations, etc. For valued field extensions, the determination of irreducible polynomials was the focus of interest of many authors. In 1850, Eisenstein gave one of the most popular criterion to decide on irreducibility of a polynomial over . A criterion which was generalized in 1906 by Dumas. In 2008, R. Brown gave what is known to be the most general version of Eisenstein-Schönemann irreducibility criterion. Thanks to MacLane theory, key polynomials play a key role to extend absolute values. In this chapter, we give a sufficient condition on any monic plynomial to be a key polynomial of an absolute value, an irreducibly criterion will be given, and for any simple algebraic extension *L* ¼ *K*ð Þ *α* , we give a method to describe all absolute values of *L* extending ∣∣, where ð Þ *K*, jj is a discrete rank one valued field.

**Keywords:** Irreducibly criterion, irreducible factors, Extensions of absolute values, Newton polygon's techniques

#### **1. Introduction**

Polynomial factorization over a field is very useful in algebraic number theory, for prime ideal factorization. It is also important in extensions of valuations, etc. For valued field extensions, the determination of irreducible polynomials was the focus of interest of many authors (cf. [1–7]). In 1850, Eisenstein gave one of the most popular criterion to decide on irreducibility of a polynomial over [1]. A criterion which was generalized in 1906 by Dumas in [8], who showed that for a polynomial *f x*ð Þ¼ *anxn* <sup>þ</sup> *an*�<sup>1</sup>*x<sup>n</sup>*�<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> *<sup>a</sup>*<sup>0</sup> <sup>∈</sup> ½ � *<sup>x</sup>* (*a*<sup>0</sup> 6¼ 0), if *<sup>ν</sup>p*ð Þ¼ *an* 0, *<sup>n</sup>νp*ð Þ *ai* <sup>≥</sup>ð Þ *<sup>n</sup>* � *<sup>i</sup> <sup>ν</sup>p*ð Þ *<sup>a</sup>*<sup>0</sup> <sup>&</sup>gt;<sup>0</sup> for every 0 <sup>¼</sup> *<sup>i</sup>*, … , *<sup>n</sup>* � 1, and *gcd <sup>ν</sup>p*ð Þ *<sup>a</sup>*<sup>0</sup> , *<sup>n</sup>* <sup>¼</sup> 1 for some prime integer *<sup>p</sup>*, then *f x*ð Þ is irreducible over . In 2008, R. Brown gave what is known to be the most general version of Eisenstein-Schönemann irreducibility criterion [9]. He showed for a valued field ð Þ *<sup>K</sup>*, *<sup>ν</sup>* and for a monic polynomial *f x*ð Þ¼ *<sup>ϕ</sup><sup>n</sup>*ð Þþ *<sup>x</sup> an*�<sup>1</sup>ð Þ *<sup>x</sup> <sup>ϕ</sup><sup>n</sup>*�<sup>1</sup> ð Þþ *x* … þ *a*0ð Þ *x* ∈ *Rν*½ � *x* , where *R<sup>ν</sup>* is a valuation ring of a discrete rank one valuation and *ϕ* being a monic polynomial in *Rν*½ � *x* whose reduction *ϕ* is irreducible over *ν*, *ai*ð Þ *x* ∈*Rν*½ � *x* , degð Þ *ai* < degð Þ *ϕ* for every *i* ¼ 0, … , *n* � 1, if *ν*ð Þ *ai* ≤ð Þ 1 � *i=n ν*ð Þ *a*<sup>0</sup> for every *i* ¼ 0, … , *n* � 1 and gcdð Þ¼ *ν*ð Þ *a*<sup>0</sup> , *n* 1, then *f x*ð Þ is irreducible over the field *K*. In this paper, based on absolute value, we give an irreduciblity criterion of monic polynomials. More precisely, let ð Þ *K*, jj be a discrete rank one valued field, *R*∣ ∣ its valuation ring, ∣ ∣, its residue field, and <sup>Γ</sup> <sup>¼</sup> <sup>∣</sup>*<sup>K</sup>* <sup>∗</sup> <sup>∣</sup> its value group, we show that for a monic

polynomial *f x*ð Þ¼ *<sup>ϕ</sup>n*ð Þþ *<sup>x</sup> an*�1ð Þ *<sup>x</sup> <sup>ϕ</sup>n*�<sup>1</sup> ð Þþ *x* … þ *a*0ð Þ *x* ∈*R*∣ ∣½ � *x* , where *ϕ* being a monic polynomial in *R*∣ ∣½ � *x* whose reduction *ϕ* is irreducible over ∣ ∣, *ai*ð Þ *x* ∈ *R*∣ ∣½ � *x* , degð Þ *ai* <sup>&</sup>lt; degð Þ *<sup>ϕ</sup>* for every *<sup>i</sup>* <sup>¼</sup> 0, … , *<sup>n</sup>* � 1, if *an*�*<sup>i</sup>* j j<sup>∞</sup> <sup>≥</sup>*γ<sup>i</sup>* for every *<sup>i</sup>* <sup>¼</sup> 0, … , *<sup>n</sup>* � <sup>1</sup> and *<sup>n</sup>* is the smallest integer satisfying *<sup>γ</sup><sup>n</sup>* <sup>∈</sup> <sup>Γ</sup>, where *<sup>γ</sup>* <sup>¼</sup> j j *<sup>a</sup>*<sup>0</sup> <sup>∞</sup> � �1*=<sup>n</sup>* , then *f x*ð Þ is irreducible over *K*. Similarly for the results of extensions of valuations given in [10, 11], for any simple algebraic extension *L* ¼ *K*ð Þ *α* , we give a method to describe all absolute values of *L* extending ∣∣, where ð Þ *K*, jj is a discrete rank one valued field. Our results are illustrated by some examples.

#### **2. Preliminaries**

#### **2.1 Newton polygons**

Let *L* ¼ ð Þ *α* be a number field generated by a complex root *α* of a monic irreducible polynomial *f x*ð Þ∈½ � *x* and *<sup>L</sup>* the ring of integers of *L*. In 1894, K. Hensel developed a powerful approach by showing that the prime ideals of *<sup>L</sup>* lying above a prime *p* are in one–one correspondence with monic irreducible factors of *f x*ð Þ in *p*½ � *x* . For every prime ideal corresponding to any irreducible factor in *p*½ � *x* , the ramification index and the residue degree together are the same as those of the local field defined by the irreducible factors [6]. These results were generalized in ([12], Proposition 8.2). Namely, for a rank one valued field ð Þ *K*, *ν* , *R<sup>ν</sup>* its valuation ring, and *L* ¼ *K*ð Þ *α* a simple extension generated by *α*∈ *K* a root of a monic irreducible polynomial *f x*ð Þ∈*Rν*½ � *x* , the valuations of *L* extending *ν* are in one–one correspondence with monic irreducible factors of *f x*ð Þ in *<sup>K</sup><sup>h</sup>*½ � *<sup>x</sup>* , where *<sup>K</sup><sup>h</sup>* is the henselization of ð Þ *K*, jj will be defined later. So, in order to describe all valuations of *L* extending *ν*, one needs to factorize the polynomial *f x*ð Þ into monic irreducible factors over *<sup>K</sup><sup>h</sup>* . The first step of the factorization was based on Hensel's lemma. Unfortunately, the factors provided by Hensel's lemma are not necessarily irreducible over *K<sup>h</sup>*. The Newton polygon techniques could refine the factorization. Namely, theorem of the product, theorem of the polygon, and theorem of residual polynomial say that we can factorize any factor provided by Hensel's lemma, with as many sides of the polygon and with as many of irreducible factors of the residual polynomial. For more details, we refer to [7, 13] for Newton polygons over *p*-adic numbers and [14, 15] for Newton polygons over rank one discrete valued fields. As our proofs are based on Newton polygon techniques, we recall some fundamental notations and techniques on Newton polygons. Let ð Þ *<sup>K</sup>*, *<sup>ν</sup>* be a rank one discrete valued field (*<sup>ν</sup> <sup>K</sup>*<sup>∗</sup> ð Þ¼ ), *<sup>R</sup><sup>ν</sup>* its valuation ring, *M<sup>ν</sup>* its maximal ideal, *<sup>ν</sup>* its residue field, and *K<sup>h</sup>*, *ν<sup>h</sup>* � � its henselization; the separable closure of *<sup>K</sup>* in *<sup>K</sup>*^, where *<sup>K</sup>*^ is the completion of ð Þ *<sup>K</sup>*, jj , and ∣∣ is an associated absolute value of *<sup>ν</sup>*. By normalization, we can assume that *<sup>ν</sup> <sup>K</sup>*<sup>∗</sup> ð Þ¼ , and so *<sup>M</sup><sup>ν</sup>* is a principal ideal of *R<sup>ν</sup>* generated by an element *π* ∈*K* satisfying *ν π*ð Þ¼ 1. Let also *ν* be the Gauss's extension of *<sup>ν</sup>* to *<sup>K</sup><sup>h</sup>*ð Þ *<sup>x</sup>* . For any monic polynomial *<sup>ϕ</sup>* <sup>∈</sup>*Rν*½ � *<sup>x</sup>* whose reduction modulo *<sup>M</sup><sup>ν</sup>* is irreducible in *ν*½ � *<sup>x</sup>* , let *<sup>ϕ</sup>* be the field *ν*½ � *<sup>x</sup>* ð Þ*ϕ* .

Let *f x*ð Þ∈*Rν*½ � *x* be a monic polynomial and assume that *f x*ð Þ is a power of *ϕ* in *ν*½ � *x* , with *ϕ*∈*Rν*½ � *x* a monic polynomial, whose reduction is irreducible in *ν*½ � *x* . Upon the Euclidean division by successive powers of *ϕ*, we can expand *f x*ð Þ as follows *f x*ð Þ¼ <sup>P</sup>*<sup>l</sup> <sup>i</sup>*¼<sup>0</sup>*ai*ð Þ *<sup>x</sup> <sup>ϕ</sup>*ð Þ *<sup>x</sup> i* , where degð Þ *ai* <degð Þ *ϕ* for every *i* ¼ 0, … , *l*. Such a *ϕ*-expansion is unique and called the *ϕ*-expansion of *f x*ð Þ. The *ϕ*-Newton polygon of *f*, denoted by *Nϕ*ð Þ*f* is the lower boundary of the convex envelope of the set of points ð Þ *i*, *ν*ð Þ *ai* f g , *i* ¼ 0, … , *l* in the Euclidean plane. For every edge *S <sup>j</sup>*, of the

polygon *Nϕ*ð Þ*f* , let *l <sup>j</sup>* be the length of the projection of *S <sup>j</sup>* to the *x*-axis and *H <sup>j</sup>* the length of its projection to the *y*-axis. *l <sup>j</sup>* is called the length of *S <sup>j</sup>* and *H <sup>j</sup>* is its height. Let *d <sup>j</sup>* ¼ gcd *l <sup>j</sup>*, *H <sup>j</sup>* � � be the degree of *<sup>S</sup> <sup>j</sup>*, *ej* <sup>¼</sup> *<sup>l</sup> <sup>j</sup> d j* the ramification degree of *S <sup>j</sup>*, and �*<sup>λ</sup> <sup>j</sup>* ¼ � *<sup>H</sup> <sup>j</sup> l j* ∈ the slope of *S <sup>j</sup>*. Geometrically, we can remark that *Nϕ*ð Þ*f* is the process of joining the obtained edges *S*1, … , *Sr* ordered by increasing slopes, which can be expressed by *Nϕ*ð Þ¼ *f S*<sup>1</sup> þ … þ *Sr*. The segments *S*1, … , and *Sr* are called the sides of *Nϕ*ð Þ*f* . The principal *ϕ*-Newton polygon of *f x*ð Þ, denoted by *N*<sup>þ</sup> *<sup>ϕ</sup>* ð Þ*f* , is the part of the polygon *Nϕ*ð Þ*f* , which is determined by joining all sides of negative slopes. For every side *S* of the polygon *N*<sup>þ</sup> *<sup>ϕ</sup>* ð Þ*f* of slope �*λ* and initial point ð Þ *s*, *us* , let *l* be its length, *H* its height and *e* the smallest positive integer satisfying *eλ*∈. Since *lλ* ¼ *H* ∈ , we conclude that *e* divides *l*, and so *d* ¼ *l=e*∈ called the degree of *S*. Remark that *d* ¼ gcdð Þ *l*, *H* . For every *i* ¼ 0, … , *l*, we attach the following residue coefficient *ci* ∈*ϕ*:

$$x\_i = \begin{cases} 0, & \text{if } \ (s+i, u\_{s+i}) \text{ lies strictly above } S\\ \left(\frac{a\_{s+i}(\mathbf{x})}{\pi^{u\_{s+i}}}\right) (\text{mod } (\pi, \phi)), & \text{if } (s+i, u\_{s+i}) \text{ lies on } S. \end{cases} \tag{1}$$

where ð Þ *π*, *ϕ* is the maximal ideal of *Rν*½ � *x* generated by *π* and *ϕ*.

Let *λ* ¼ �*h=e* be the slope of *S*, where *h* ¼ *H=d* and *d* ¼ *l=e*. Notice that, the points with integer coordinates lying in *S* are exactly ð Þ *s*, *us* ,ð Þ *s* þ *e*, *us* � *h* , … ,ð Þ *s* þ *de*, *us* � *dh* . Thus, if*i* is not a multiple of*e*, then ð Þ *s* þ *i*, *us*þ*<sup>i</sup>* does not lie on *S*, and so *ci* ¼ 0. It follows that the candidate abscissas which yield nonzero residue coefficient are *s*, *s* þ *e*, … , and *<sup>s</sup>* <sup>þ</sup> *de*. Let *<sup>R</sup>λ*ð Þ*<sup>f</sup>* ð Þ¼ *<sup>y</sup> tdyd* <sup>þ</sup> *td*�<sup>1</sup>*y<sup>d</sup>*�<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> *<sup>t</sup>*1*<sup>y</sup>* <sup>þ</sup> *<sup>t</sup>*<sup>0</sup> <sup>∈</sup> *ϕ*½ � *<sup>y</sup>* be the residual polynomial of*f x*ð Þ associated to the side *S*, where for every *i* ¼ 0, … , *d*, *ti* ¼ *cie*. For every *λ*∈ þ, the *λ*-component of *Nϕ*ð Þ*f* is the largest segment of *Nϕ*ð Þ*f* of slope �*λ*. If *Nϕ*ð Þ*f* has a side *S* of slope �*λ*, then *T* ¼ *S*. Otherwise, *T* is reduced to a single point; the end point of a side *Si*, which is also the initial point of *Si*þ<sup>1</sup> if *λ<sup>i</sup>*þ<sup>1</sup> < *λ*<*λ<sup>i</sup>* or the initial point of *Nϕ*ð Þ*f* if *λ<sup>i</sup>* <*λ* for every side *Si* of *Nϕ*ð Þ*f* or the end point of *Nϕ*ð Þ*f* if *λ<sup>i</sup>* <*λ* for every side *Si* of *Nϕ*ð Þ*f* . In the sequel, we denote by *Rλ*ð Þ*f* ð Þ*y* , the residual polynomial of *f x*ð Þ associated to the *λ*-component of *Nϕ*ð Þ*f* .

The following are the relevant theorems from Newton polygon. Namely, theorem of the product and theorem of the polygon. For more details, we refer to [15].

**Theorem 2.1.** (theorem of the product) *Let f x*ð Þ¼ *f* <sup>1</sup>ð Þ *x f* <sup>2</sup>ð Þ *x in Rν*½ � *x be monic polynomials such that f x*ð Þ *is a positive power of ϕ. Then for every λ*∈ þ*, if Ti is the λ-componenet of N<sup>ϕ</sup> fi* � �*, then T* <sup>¼</sup> *<sup>T</sup>*<sup>1</sup> <sup>þ</sup> *<sup>T</sup>*<sup>2</sup> *is the <sup>λ</sup>-componenet of Nϕ*ð Þ*<sup>f</sup> and*

$$R\_{\vec{\lambda}}(f)(\mathfrak{y}) = R\_{\vec{\lambda}}\left(f\_1\right)(\mathfrak{y})R\_{\vec{\lambda}}\left(f\_2\right)(\mathfrak{y})$$

*up to multiplication by a nonzero element of ϕ.*

**Theorem 2.2.** (theorem of the polygon) *Let f* ∈*Rν*½ � *x be a monic polynomial such that f x*ð Þ *is a positive power of ϕ. If Nϕ*ð Þ¼ *f S*<sup>1</sup> þ … þ *Sg has g sides of slope* �*λ*1, … , � *<sup>λ</sup><sup>g</sup> respectively, then we can split f x*ð Þ¼ *<sup>f</sup>* <sup>1</sup> � … � *<sup>f</sup> <sup>g</sup>* ð Þ *<sup>x</sup> in K<sup>h</sup>*½ � *<sup>x</sup> , such that N<sup>ϕ</sup> fi* � � <sup>¼</sup> *Si and R<sup>λ</sup><sup>i</sup> fi* � �ð Þ¼ *<sup>y</sup> <sup>R</sup><sup>λ</sup><sup>i</sup>* ð Þ*f* ð Þ*y up to multiplication by a nonzero.*

**Theorem 2.3.** (theorem of the residual polynomial) *Let f* ∈ *Rν*½ � *x be a monic polynomial such that Nϕ*ð Þ¼ *<sup>f</sup> S has a single side of finite slope* �*λ. If Rλ*ð Þ*<sup>f</sup>* ð Þ¼ *<sup>y</sup>* <sup>Q</sup>*<sup>t</sup> <sup>i</sup>*¼<sup>1</sup>*ψi*ð Þ*<sup>y</sup> ai is the factorization in ϕ*½ � *y , then f x*ð Þ *splits as f x*ð Þ¼ *f* <sup>1</sup>ð Þ� *x* ⋯ � *ft* ð Þ *<sup>x</sup> in K<sup>h</sup>*½ � *<sup>x</sup> such that N<sup>ϕ</sup> fi* � � <sup>¼</sup> *Si has a single side of slope* �*<sup>λ</sup> and R<sup>α</sup> fi* � �ð Þ¼ *<sup>y</sup> <sup>ψ</sup>i*ð Þ*<sup>y</sup> ai up to multiplication by a nonzero element of <sup>ϕ</sup> for every i* ¼ 1, ⋯, *t*.

#### **2.2 Absolute values**

Let ∣∣ be an absolute value of *K*; a map ∣∣ : *K* ! þ, which satisfies the following three axioms:

1.∣*a*∣ ¼ 0 if and only if *a* 6¼ 0,

2.∣*ab*∣ ¼ ∣*a*k*b*∣, and

3.∣*a* þ *b*∣≤∣*a*∣ þ ∣*b*∣. ðtriangular inequalityÞ

for every ð Þ *<sup>a</sup>*, *<sup>b</sup>* <sup>∈</sup>*K*<sup>2</sup> .

If the triangular inequality is replaced by an ultra-inequality, namely ∣*a* þ *<sup>b</sup>*∣ ≤ max f g <sup>j</sup>*a*j, <sup>j</sup>*b*<sup>j</sup> for every ð Þ *<sup>a</sup>*, *<sup>b</sup>* <sup>∈</sup>*K*<sup>2</sup> , then the absolute value ∣∣ is called a non archimidean absolute value and we say that ð Þ *K*, jj is a non archimidean valued field.

**Lemma 2.4.** *Let K*ð Þ , jj *be a valued field. Then* ∣∣ *is a non archimidean absolute value if and only if the set* f g j*n*1*K*j, *n* ∈ *is bounded in .*

*Proof.* By induction if ∣∣ is a non archimidean absolute value, then the set f g j*n*1*K*j, *n*∈ is bounded by 1.

Conversely, assume that there exists *M* ∈ <sup>þ</sup> such that ∣*n*1*K*∣ ≤ *M* for every *n* ∈ .

$$\text{Let } (a, b) \in K^2, n \in \mathbb{N}, \text{ and set } m = \sup \left( |a|, |b| \right). \text{ Then } |a + b|^n = |\sum\_{k=0}^n \binom{n}{k} a^k b^{n-k}|,$$

where *n k* � � is the binomial coefficient. As ∣∣ is a non archimidean absolute value, *n*

j j *<sup>a</sup>* <sup>þ</sup> *<sup>b</sup> <sup>n</sup>* <sup>≤</sup>supf<sup>∣</sup> *k* � �1*K*<sup>∣</sup> � <sup>∣</sup>*akbn*�*<sup>k</sup>* ∣ ≤ *Mm<sup>n</sup>*. Thus <sup>∣</sup>*<sup>a</sup>* <sup>þ</sup> *<sup>b</sup>*∣ ≤ *<sup>M</sup>*<sup>1</sup>*<sup>=</sup>nm*. Go over the limit, we obtain <sup>∣</sup>*<sup>a</sup>* <sup>þ</sup> *<sup>b</sup>*∣ ≤ *<sup>m</sup>* <sup>¼</sup> *sup*ð Þ <sup>j</sup>*a*j, <sup>j</sup>*b*<sup>j</sup> as desired. □

**Exercices 1.** Let ð Þ *K*, jj be a valued field.

1.Show that if *K* is a finite field, then ∣∣ is a non archimidean absolute value.


#### **2.3 Characteristic elements of an absolute value**

Let ð Þ *K*, jj be a non archimedian valued field.

*On the Irreducible Factors of a Polynomial and Applications to Extensions of Absolute… DOI: http://dx.doi.org/10.5772/intechopen.100021*

Let *R*∣ ∣ ¼ f g *a*∈*K*, j*a*j≤1 and *M*∣ ∣ ¼ f g *a*∈*K*, j*a*j<1 . Then *R*∣ ∣ is a valuation ring, called the valuation ring of ∣∣, *M*∣ ∣ its maximal ideal, and so ∣ ∣ ¼ *R*∣ ∣*=M*∣ ∣ is a field, called the residue field of ∣∣.

**Exercices 2.** Let *p* be a prime integer and ∣∣ the *p*-adic absolute value of , defined by <sup>∣</sup>*a*<sup>∣</sup> <sup>¼</sup> *<sup>p</sup>*�*νp*ð Þ *<sup>a</sup>* for every *<sup>a</sup>*∈, where *<sup>ν</sup>p*ð Þ *<sup>a</sup>* is the greatest integer satisfying *<sup>p</sup><sup>ν</sup>p*ð Þ *<sup>a</sup>* divides *<sup>a</sup>* for *<sup>a</sup>* 6¼ 0 and *<sup>ν</sup>p*ð Þ¼ <sup>0</sup> <sup>∞</sup>.

1.Show that ð Þ , jj is a non archimidean valued field.

2.Determine the characteristic elements of ∣∣.

**Exercices 3.** Let ð Þ *K*, jj be a non archimidean valued field.


#### **2.4 Completion and henselization**

Let ð Þ *K*, jj be a valued field and consider the map *d* : *K* � *K* ! ≥0, defined by *d a*ð Þ¼ , *b* ∣*a* � *b*∣. Then *d* is a metric on *K*.

**Definition 1.** A sequence ð Þ *un* <sup>∈</sup>*K* is said to be a Cauchy sequence if for every positive real number *ε*, there exists an integer *N* such that for every natural numbers *m*, *n* ≥ *N*, we have ∣*un* � *um*∣ ≤*ε*.

**Example 1.**

Any convergente sequence of ð Þ *K*, jj is a Cauchy sequence.

The converse is false, indeed, it suffices to consider the valued field , jj<sup>0</sup> � � with jj<sup>0</sup> is the usual absolute value of and *un* ¼ 1 þ 1*=*1! þ … þ 1*=n*! for every natural integer *n*. Then ð Þ *un* is a Cauchy sequence, which is not convergente.

**Definition 2.** A valued field ð Þ *K*, jj is said to be complete if every Cauchy sequence of ð Þ *K*, jj is convergente.

**Example 2.**

1. , jj<sup>0</sup> � � is a complete valued field.

2. , jj<sup>0</sup> � � is not a complete valued field.

**Definition 3.** Let ð Þ *K*, jj be a valued field, *L=K* an extension of fields, and jj*<sup>L</sup>* an absolute value of *L*.

1.We say that jj*<sup>L</sup>* extends ∣∣ if jj*<sup>L</sup>* and ∣∣ coincide on *K*. In this case *L*, jj*<sup>L</sup>* � �*=*ð Þ *<sup>K</sup>*, jj is called a valued field extension.

2.Let *L*, jj*<sup>L</sup> <sup>=</sup>*ð Þ *<sup>K</sup>*, jj be a valued field extension and *<sup>Δ</sup>* <sup>¼</sup> *<sup>L</sup>*<sup>∗</sup> j j*L*. Then *<sup>e</sup>* ¼ j*Δ=*Γ<sup>j</sup> the cardinal order of *Δ=*Γ, is called the ramification index of the extension and *<sup>f</sup>* <sup>¼</sup> j j*<sup>L</sup>* : *<sup>F</sup>*∣ ∣ is called its residue degree.

**Definition 4.** Let *K*1, jj<sup>1</sup> and *<sup>K</sup>*2, jj<sup>2</sup> be two valued fields and *<sup>f</sup>* : *<sup>K</sup>*<sup>1</sup> ! *<sup>K</sup>*<sup>2</sup> be an isomorphism of fields. *f* is said to be an isomorphism of valued fields if it preserves the absolute values.

**Exercices 4.** Let *L*, jj*<sup>L</sup> <sup>=</sup>*ð Þ *<sup>K</sup>*, jj be a valued field extension.


**Theorem 2.5.** ð[16], Theorem 1.1.4Þ

*There exists a complete valued field L*, jj*<sup>L</sup> , which extends K*ð Þ , jj *.*

**Definition 5.** The smallest complete valued field extending ð Þ *K*, jj is called the completion of ð Þ *<sup>K</sup>*, jj and denoted by *<sup>K</sup>*^.

Furtheremore, the completion is unique up to a valued fields isomorphism. Now we come to an important property of complete fields. This theorem is

widely known as Hensel's Lemma. For the proof, we refer to ([16], Lemma 4.1.3). **Theorem 2.6.** ð*Hensel's lemma*Þ

*Let f* ∈ *R*∣ ∣½ � *x be a monic polynomial such that f x*ð Þ¼ *g*1ð Þ *x g*2ð Þ *x in* ∣ ∣½ � *x and g*1ð Þ *x and g*2ð Þ *x are coprime in* ∣ ∣½ � *x . If K*ð Þ , jj *is a complete valued field valued field, then there exists two monic polynomials f* <sup>1</sup>ð Þ *x and f* <sup>2</sup>ð Þ *x in R*∣ ∣½ � *x such that f* <sup>1</sup>ð Þ¼ *x g*1ð Þ *x and f* <sup>2</sup>ð Þ¼ *x g*2ð Þ *x* .

The following example shows that for any prime integer *p*, Hensel's lemma is not applicable in ð Þ , jj , with ∣∣ is the *<sup>p</sup>*-adic absolute value defined by <sup>∣</sup>*a*<sup>∣</sup> <sup>¼</sup> *<sup>p</sup>*�*νp*ð Þ *<sup>a</sup>* . Indeed, let *q* be a prime integer which is coprime to *p*, *n*≥2 an integer, and *f x*ð Þ¼ *xn* <sup>þ</sup> *qx* <sup>þ</sup> *pq*∈*Z x*½ �. First *f x*ð Þ¼ *x x*ð Þ *<sup>n</sup>*�<sup>1</sup> <sup>þ</sup> *<sup>q</sup>* in ∣ ∣½ � *<sup>x</sup>* . As *f x*ð Þ is *<sup>q</sup>*-Eisenstein *f x*ð Þ is irreducible over . Thus, we conclude that Hensel's lemma is not applicable in ð Þ , jj .

**Definition 6.** A valued field ð Þ *K*, jj is said to be Henselian if Hensel's lemma is applicable in ð Þ *K*, jj . The smallest Henselian field extending ð Þ *K*, jj is called the henselization of <sup>ð</sup>*K*, ∣∣ and denoted by *<sup>K</sup><sup>h</sup>*.

**Exercices 5.** Let ð Þ *K*, jj be a valued field ð Þ *K*, jj .

Show that *K* ⊂ *K<sup>h</sup>* ⊂*K*^. Furthermore, these three fields have the same value group and same residue fields.

We have the following apparently easier characterization of Henselian fields. For the proof, we refer to ([16], Lemma 4.1.1).

**Theorem 2.7.** *The valued field K*ð Þ , jj *is Henselian if and only if it extends uniquely to K<sup>s</sup>* , where *K<sup>s</sup>* is the separable closure of *K*.

In particular, we conclude the following characterization of the henselization *K<sup>h</sup>* of ð Þ *K*, jj .

**Theorem 2.8.** *Let K*ð Þ , jj *be a valued field. Then K<sup>h</sup> is the separable closure of K in K<sup>s</sup> .*

*On the Irreducible Factors of a Polynomial and Applications to Extensions of Absolute… DOI: http://dx.doi.org/10.5772/intechopen.100021*

#### **3. Main results**

Let ð Þ *K*, jj be a non archimidean valued field, *ν* the associated valuation to ∣∣ defined by *<sup>ν</sup>*ð Þ¼� *<sup>a</sup> Ln*∣*a*<sup>∣</sup> for every *<sup>a</sup>*∈*<sup>K</sup>* <sup>∗</sup> , *<sup>R</sup>*∣ ∣ its valuation ring, *<sup>M</sup>*∣ ∣ its maximal ideal, ∣ ∣ its residue field, and *Kh*, *ν<sup>h</sup>* � � its henselization.

#### **3.1 Key polynomials**

The notion of key polynomials was introduced in 1936, by MacLane [17], in the case of discrete rank one absolute values and developed in [18] by Vaquié to any arbitrary rank valuation. The motivation of introducing key polynomials was the problem of describing all extensions of ∣∣ to any finite simple extension *K*ð Þ *α* . For any simple algebraic extension of *K*, MacLane introduced the notions of key polynomials and augmented absolute with respect to the gievn key.

**Definition 7.** Two nonzero polynomials *f* and *g* in *R*∣ ∣½ � *x* ,


**Definition 8.** A polynomial *ϕ*∈*R*∣ ∣½ � *x* is said to be a MacLane-Vaquié key polynomial of ∣∣ if it satisfies the following three conditions:

1.*ϕ* is monic,


It is easy to prove the following lemma:

**Lemma 3.1.** *Let ϕ*∈*R*∣ ∣½ � *x be a monic polynomial. If ϕ is irreducible over* ∣ ∣*, then ϕ is a MacLane-Vaquié key polynomial of* ∣∣*.*

#### **3.2 Augmented absolute values**

Let *ϕ*∈*R*∣ ∣½ � *x* be a MacLane-Vaquié key polynomial of ∣∣ and *γ* ∈ <sup>þ</sup> with *γ* ≤j j *ϕ* <sup>∞</sup>. Let *ω* : *K x*ð Þ! ≥0, defined by *ω*ð Þ¼ *P* max *pi* � � � � ∞*γi* , *i* ¼ 0, … , *l* n o for every *<sup>P</sup>*∈*K x*½ �, with *<sup>P</sup>* <sup>¼</sup> <sup>P</sup>*<sup>l</sup> <sup>i</sup>*¼<sup>0</sup>*pi ϕ<sup>i</sup>* and deg *pi* � �<sup>&</sup>lt; degð Þ *<sup>ϕ</sup>* for every *<sup>i</sup>* <sup>¼</sup> 0, … , *<sup>l</sup>* and extended by *ω*ð Þ¼ *A=B ω*ð Þ� *A ω*ð Þ *B* for every nozero *A* and *B* of *K x*ð Þ.

**Lemma 3.2** *Let P* <sup>¼</sup> <sup>P</sup>*<sup>n</sup> <sup>i</sup>*¼<sup>0</sup>*biϕ<sup>i</sup> be a <sup>ϕ</sup>-expansion of P, where the condition deg b*ð Þ*<sup>i</sup>* <*deg*ð Þ *ϕ for every i* ¼ 0, … , *n is omitted. If ϕ does not divide bi*ð Þ *x =b for every <sup>i</sup>* <sup>¼</sup> 0, … , *n, then <sup>ω</sup>*ð Þ¼ *<sup>P</sup> max bi* j j∞*γ<sup>i</sup>* , *<sup>i</sup>* <sup>¼</sup> 0, … , *<sup>n</sup>* � �*, where b*∈*R<sup>ν</sup> such that bi* j j<sup>∞</sup> <sup>¼</sup> ∣*b*∣*. Such an expansion is called an admissible expansion.*

**Theorem 3.3.** *Let ϕ*∈ *R*∣ ∣½ � *x be MacLane-Vaquié key polynomial of* ∣∣ *and γ* ∈ <sup>þ</sup> *with γ* ≤j j *ϕ* <sup>∞</sup>*. The map ω* : *K x*ð Þ! ≥<sup>0</sup>*, defined by ω*ð Þ¼ *P max pi* � � � � ∞*γi* , *i* ¼ 0, … , *l* n o *for every P*∈*K x*½ �*, with P* <sup>¼</sup> <sup>P</sup>*<sup>l</sup> <sup>i</sup>*¼<sup>0</sup>*pi ϕ<sup>i</sup> and deg pi* � �<*deg*ð Þ *<sup>ϕ</sup> for every i* <sup>¼</sup> 0, … , *l, and extended by <sup>ω</sup>*ð Þ¼ *<sup>A</sup>=<sup>B</sup> <sup>ω</sup>*ð Þ *<sup>A</sup> <sup>=</sup>ω*ð Þ *<sup>B</sup> for every nonzero polynomials A*ð Þ , *<sup>B</sup>* <sup>∈</sup> *K x*½ �<sup>2</sup> *, is an absolute value of K x*ð Þ*.*

*Proof.* It suffices to check that *ω* satisfies the three proprieties of an absolute value in *K x*½ �. Let ð Þ *<sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup>*K x*½ � be tow polynomials, *<sup>A</sup>* <sup>¼</sup> <sup>P</sup>*<sup>k</sup> <sup>i</sup>*¼<sup>0</sup>*aiϕ<sup>i</sup>* , and *<sup>B</sup>* <sup>¼</sup> <sup>P</sup>*<sup>s</sup> <sup>i</sup>*¼<sup>0</sup>*biϕ<sup>i</sup>* the *<sup>ϕ</sup>*-expansions.


**Definition 9.** The absolute value *ω* defined in Theorem 3.3 is denoted by ½ � jj*ϕ*, *γ*, and called the augmented absolue value of ∣∣ associated to *ϕ* and *γ*.

□

**Example 3.** Let ∣∥ be the 2-adic absolute value defined on by <sup>∣</sup>*a*<sup>∣</sup> <sup>¼</sup> *<sup>e</sup>*�*ν*2ð Þ *<sup>a</sup>* , where for every integer *<sup>b</sup>*, *<sup>ν</sup>*2ð Þ *<sup>b</sup>* is the largest integer satisfying 2*<sup>k</sup>* divides *<sup>b</sup>* in . Let *<sup>ϕ</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> <sup>1</sup><sup>∈</sup> ½ � *<sup>x</sup>* . By Lemma 3.1, *<sup>ϕ</sup>* is a MacLane-Vaquié key polynomial of ∣∣. Since j j *ϕ* <sup>∞</sup> ¼ 1, for every real *γ*, 0 <*γ* ≤1, the map *ω* : ð Þ! *x* ≥0, defined by *ω*ð Þ¼ *P*ð Þ *α* max *pi* � � � � ∞*γi* , *i* ¼ 0, … , *l* n o for every *<sup>P</sup>*∈*K x*½ �, with *<sup>P</sup>* <sup>¼</sup> <sup>P</sup>*<sup>l</sup> <sup>i</sup>*¼<sup>0</sup>*pi ϕ<sup>i</sup>* and deg *pi* � �<2.

#### **3.3 Extensions of absolute values**

The following Lemma makes a one–one correspondence between the absolute value of *<sup>L</sup>* and monic irreducible factors of *f x*ð Þ in *<sup>K</sup><sup>h</sup>*½ � *<sup>x</sup>* for any simple finite extension *L* ¼ *K*ð Þ *α* of *K* generated by a root *α* ∈*K* of a monic irreducible polynomial *f x*ð Þ∈ *K x*½ �.

*On the Irreducible Factors of a Polynomial and Applications to Extensions of Absolute… DOI: http://dx.doi.org/10.5772/intechopen.100021*

### **Lemma 3.4.** ð[19], Theorem 2.1Þ

*Let L* ¼ *K*ð Þ *α generated by a root α* ∈*K of a monic irreducible polynomial f x*ð Þ∈*K x*½ � *and f x*ð Þ¼ <sup>Q</sup>*<sup>t</sup> <sup>i</sup>*¼<sup>1</sup> *<sup>f</sup> ei <sup>i</sup>* ð Þ *x be the factorization into powers of monic irreducible factors of <sup>K</sup>h*½ � *<sup>x</sup> . Then ei* <sup>¼</sup> <sup>1</sup> *for every i* <sup>¼</sup> 1, … , *t and there are exactly t distinct valuations* jj1, … , *and* jj*<sup>t</sup> of L extending* ∣∣*. Furthermore for every absolute value* jj*<sup>i</sup> of L associated to the irreducible factor fi , P*j j ð Þ *<sup>α</sup> <sup>i</sup>* <sup>¼</sup> <sup>∣</sup>*P*ð Þ *<sup>α</sup><sup>i</sup>* <sup>∣</sup>*, where* ∣∣ *is the unique absolute value of <sup>K</sup><sup>h</sup> extending* ∣∣ *and α<sup>i</sup>* ∈ *K is a root of fi* ð Þ *x .*

**Lemma 3.5.** ð[16], Corollary 3.1.4Þ

*Let L=K be a finite extension and RL the integral closure of R*∣ ∣ *in L. Then*

$$\mathcal{R}\_L = \cap\_{\parallel\_L} \mathcal{R}\_{\parallel\_L};$$

*for any elemnt α*∈*L, α*∈*RL if and only if* j j *α <sup>L</sup>* ≤ 1 *for every absolute value* jj*<sup>L</sup> of L extending* ∣∣.

**Lemma 3.6.** *Let f x*ð Þ∈*R*∣ ∣½ � *x be a monic irreducible polynomial such that f x*ð Þ *is a power of ϕ in* ∣ ∣½ � *x for some monic polynomial ϕ*∈*R*∣ ∣½ � *x , whose reduction is irreducible over* ∣ ∣*. Let L* ¼ *K*ð Þ *α with α*∈ *K a root of f x*ð Þ*. Then for every absolute value* jj*<sup>L</sup> of L extending* ∣∣*, for every nonzero polynomial P*∈*K x*½ �*, P*j j ð Þ *α <sup>L</sup>* ≤j j *P* <sup>∞</sup>*.*

*The equality holds if and only if <sup>ϕ</sup> does not divide <sup>P</sup>*0*, where P*<sup>0</sup> <sup>¼</sup> *<sup>P</sup> <sup>a</sup>, with a*∈ *K such that P*j j<sup>∞</sup> ¼ ∣*a*∣*.*

*In particular,* j j *ϕ α*ð Þ *<sup>L</sup>* <1 *and P*j j ð Þ *α <sup>L</sup>* ¼ j j *P* <sup>∞</sup> *for every polynomial P*∈ *K x*½ � *such deg P*ð Þ<*deg*ð Þ *ϕ .*

*Proof.* Let jj*<sup>L</sup>* be an absolute value of *L* extending ∣∣, *P*∈*K x*½ � a nonzero polynomial, and *a*∈*K* with ∣*a*∣ ¼ j j *P* <sup>∞</sup>. Then j j *P*<sup>0</sup> <sup>∞</sup> ¼ 1. Since *α* is integral over *R*∣ ∣, we conclude that j j *P*0ð Þ *α <sup>L</sup>* ≤1. Thus, j j *P*ð Þ *α <sup>L</sup>* ≤ ∣*a*∣ ¼ j j *P* <sup>∞</sup>.

Moreover, the inequality j j *P*ð Þ *α <sup>L</sup>* < j j *P* <sup>∞</sup> means that *P*0ð Þ *α* ∈ *M*j j*<sup>L</sup>* , which means that *P*0ð Þ� *α* 0 *mod M*j j*<sup>L</sup>* � �. Consider the ring homomorphism *<sup>φ</sup>* : ∣ ∣½ �!*<sup>x</sup> <sup>M</sup>*j j*<sup>L</sup>* , defined by *φ P* � � <sup>¼</sup> *<sup>P</sup>*ð Þþ *<sup>α</sup> <sup>M</sup>*j j*<sup>L</sup>* . Then *<sup>P</sup>*0ð Þ *<sup>α</sup>* 6� <sup>0</sup> *mod M*j j*<sup>L</sup>* � � is equivalent to *ϕ* does not divide *P*0.

In particular, since *ϕ*∈*R*∣ ∣½ � *x* , j j *ϕ* <sup>∞</sup> ≤1. Furthermore as *ϕ* divide *ϕ*, we conclude that j j *ϕ* <sup>∞</sup> <1.

Let *P*∈*K x*½ � be a nonzero polynomial of degree less than degree of *ϕ*. Then *P*0ð Þ *x* ∈ *R*∣ ∣½ � *x* is a primitive polynomial; j j *P*<sup>0</sup> <sup>∞</sup> ¼ 1. As degree *P*<sup>0</sup> is less than degree of *<sup>ϕ</sup>*, *<sup>ϕ</sup>* does not divide *<sup>P</sup>*0. Thus j j *<sup>P</sup>*ð Þ *<sup>α</sup> <sup>L</sup>* <sup>¼</sup> j j *<sup>P</sup>* <sup>∞</sup>. □

**Theorem 3.7.** *Let f x*ð Þ∈*R*∣ ∣½ � *<sup>x</sup> be a monic polynomial. If f x*ð Þ *is irreducible over Kh, then f x*ð Þ *is a power of ϕ in* ∣ ∣½ � *x for some monic polynomial ϕ*∈*R*∣ ∣½ � *x , whose reduction is irreducible over* ∣ ∣*. Moreover if we set f x*ð Þ¼ <sup>P</sup>*<sup>n</sup> <sup>i</sup>*¼<sup>0</sup>*ai*ð Þ *<sup>x</sup> <sup>ϕ</sup><sup>i</sup>* ð Þ *x the ϕ-expansion of f x*ð Þ*, then an*�*<sup>i</sup>* j j<sup>∞</sup> <sup>≤</sup>*γ<sup>i</sup> for every i* <sup>¼</sup> 0, … , *n, where <sup>γ</sup>* <sup>¼</sup> j j *<sup>a</sup>*<sup>0</sup> 1*=n* <sup>∞</sup> *.*

*Proof.* The first point of the theorem is an immediate consequence of Theorem 2.6. For the second point, let *m* ¼ degð Þ *ϕ* .

1.For *<sup>m</sup>* <sup>¼</sup> 1, let *f x*ð Þ¼ <sup>Q</sup>*<sup>k</sup> <sup>i</sup>*¼<sup>1</sup>ð Þ *<sup>x</sup>* � *<sup>α</sup><sup>i</sup>* , where *<sup>α</sup>*1, … , *<sup>α</sup><sup>k</sup>* be the roots of *f x*ð Þ in *K*, the algebraic closure of *K*. Then the formula linking roots and coefficients of *f x*ð Þ, we conclude that *f x*ð Þ¼ <sup>P</sup>*<sup>k</sup> <sup>i</sup>*¼<sup>0</sup>*six<sup>i</sup>* , where *sk* ¼ 1, *si* ¼ P Q *j* <sup>1</sup> < … < *j i α j* 1 ⋯*α <sup>j</sup> i* . Keep the notation ∣∣ for the valuation of *K<sup>h</sup>* extending ∣∣ and let ∣∣ be the unique extension of ∣∣ to *<sup>K</sup><sup>h</sup>* <sup>¼</sup> *<sup>K</sup>*. Then <sup>∣</sup>*α*1<sup>∣</sup> <sup>¼</sup> … <sup>¼</sup> <sup>∣</sup>*αk*<sup>∣</sup> <sup>¼</sup> *<sup>τ</sup>*, <sup>∣</sup>*sk*�*<sup>i</sup>*∣ ≤*τ<sup>i</sup>* , and *τ* ¼ *γ*.

2.For *<sup>m</sup>* <sup>≥</sup> 2, let <sup>Ł</sup> <sup>¼</sup> *<sup>K</sup>h*ð Þ *<sup>α</sup>* , where *<sup>α</sup>* <sup>∈</sup>*<sup>K</sup>* is a root of *f x*ð Þ, *g x*ð Þ¼ *<sup>x</sup><sup>t</sup>* <sup>þ</sup> *bt*�<sup>1</sup>*xt*�<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> *<sup>b</sup>*<sup>0</sup> the minimal polynomial of *ϕ α*ð Þ over *<sup>K</sup>h*, and *F x*ð Þ¼ *<sup>g</sup>*ð Þ¼ *<sup>ϕ</sup>*ð Þ *<sup>x</sup> <sup>ϕ</sup>*ð Þ *<sup>x</sup> <sup>t</sup>* <sup>þ</sup> *bt*�1*ϕ*ð Þ *<sup>x</sup> <sup>t</sup>*�<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> *<sup>b</sup>*0. By the previous case, we conclude that <sup>∣</sup>*bt*�*<sup>i</sup>*∣ ≤*τ<sup>i</sup>* for every *i* ¼ 0, … , *t* with *τ* ¼ j j *b*<sup>0</sup> 1*=t* , which means that *Nϕ*ð Þ¼ *F S* has a single side of slope �*<sup>λ</sup>* ¼ � *<sup>ν</sup>*ð Þ *<sup>b</sup>*<sup>0</sup> *<sup>t</sup>* . Since *F*ð Þ¼ *α* 0, we conclude that *f x*ð Þ divides *F x*ð Þ, and so *<sup>N</sup>ϕ*ð Þ*<sup>f</sup>* has a single side of the same slope �*λ*. Therefore, *an*�*<sup>i</sup>* j j<sup>∞</sup> <sup>≤</sup> *<sup>γ</sup><sup>i</sup>* for every *i* ¼ 0, … , *n*, where *γ* ¼ j j *a*<sup>0</sup> 1*=n* <sup>∞</sup> .

□

**Exercices 6.** Let ð Þ *<sup>K</sup>*, jj be a non archimidean valued field and *f x*ð Þ∈*Kh*½ � *<sup>x</sup>* . Set *f x*ð Þ¼ <sup>P</sup>*<sup>n</sup> <sup>i</sup>*¼<sup>0</sup>*ai*ð Þ *<sup>x</sup> <sup>ϕ</sup><sup>i</sup>* ð Þ *x* the *ϕ*-expansion of *f x*ð Þ.

Show that j j *f x*ð Þ <sup>∞</sup> ¼ max j j *an* <sup>∞</sup>, j j *a*<sup>0</sup> <sup>∞</sup> � �.

Based on absolute value, the following theorem gives an hyper bound of the number of monic irreducible factors of monic polynomials. In particular, Corollary 3.9 gives a criterion to test the irreducibility of monic polynomials.

**Theorem 3.8.** *Let K*ð Þ , jj *be a non archimidean valued field,* <sup>Γ</sup> <sup>¼</sup> <sup>∣</sup>*<sup>K</sup>* <sup>∗</sup> <sup>∣</sup> *its value group, and f x*ð Þ∈*K x*½ � *a monic polynomial such that f x*ð Þ *is a power of ϕ in* ∣ ∣½ � *x . Let f x*ð Þ¼ <sup>P</sup>*<sup>n</sup> <sup>i</sup>*¼<sup>0</sup>*ai*ð Þ *<sup>x</sup> <sup>ϕ</sup><sup>i</sup>* ð Þ *<sup>x</sup> be the <sup>ϕ</sup>-expansion of f x*ð Þ *and assume that an*�*<sup>i</sup>* j j<sup>∞</sup> <sup>≤</sup>*γ<sup>i</sup> for every i* ¼ 0, … , *n, where γ* ¼ j j *a*<sup>0</sup> 1*=n* <sup>∞</sup> *. Let e be the smallest positive integer satisfying γ<sup>e</sup>* ∈Γ*. Then f x*ð Þ *has at most d irreducible monic factors in K<sup>h</sup>*½ � *<sup>x</sup> , where d* <sup>¼</sup> *<sup>n</sup>=e with degree at least em each, and m* ¼ *deg*ð Þ *ϕ .*

*Proof.* By applying the map �*Ln*, the hypothesis *an*�*<sup>i</sup>* j j<sup>∞</sup> <sup>≤</sup> *<sup>γ</sup><sup>i</sup>* for every *<sup>i</sup>* <sup>¼</sup> 0, … , *<sup>n</sup>* means that *<sup>ν</sup>*ð Þ *an*�*<sup>i</sup>* <sup>≥</sup>*iλ*, where *<sup>λ</sup>* <sup>¼</sup> *<sup>ν</sup>*ð Þ *<sup>a</sup>*<sup>0</sup> *<sup>n</sup>* , which means that *Nϕ*ð Þ¼ *f S* has a single side of slope �*<sup>λ</sup>* with respect to *<sup>ν</sup>*. Let *f x*ð Þ¼ <sup>Q</sup>*<sup>t</sup> <sup>i</sup>*¼<sup>1</sup> *fi* ð Þ *x* be a non trivial factorization of monic polynomials in *<sup>K</sup><sup>h</sup>*½ � *<sup>x</sup>* . Then by Theorem 2.2, *<sup>N</sup><sup>ϕ</sup> fi* � � <sup>¼</sup> *Si* has a single side of slope �*λ*. Fix *i* ¼ 1, … , *t* and let *fi* ð Þ¼ *<sup>x</sup>* <sup>P</sup>*li <sup>j</sup>*¼<sup>0</sup>*aij*ð Þ *<sup>x</sup> <sup>ϕ</sup><sup>j</sup>* be the *<sup>ϕ</sup>*-expansion of *fi* . Then deg *fi* � � <sup>¼</sup> *lim* and �*Ln*ð Þ¼� *<sup>γ</sup> <sup>λ</sup>* is the slope of *Si*. Since *<sup>e</sup>* is the smallest positive integer satisfying *γ<sup>e</sup>* ∈Γ, we conclude that *e* is the smallest positive integer satisfying *<sup>e</sup>λ*∈*<sup>ν</sup> <sup>K</sup>* <sup>∗</sup> ð Þ. On the other hand, since *<sup>λ</sup>* <sup>¼</sup> *ai*<sup>0</sup> *li* is the slope of *Si*, where *li* is the length of the side *Si*, we conclude that *e* divides *li*. Thus deg *fi* � � <sup>¼</sup> *diem*, where *di* <sup>¼</sup> *li e* . It follows that every non trivial factor *fi* ð Þ *x* has degree at least *em*. Since degð Þ¼ *<sup>f</sup>* <sup>P</sup>*<sup>t</sup> <sup>i</sup>*¼<sup>1</sup>deg *fi* � �≥ *tem*, we conclude that *t*≤ *<sup>n</sup> <sup>e</sup>* <sup>¼</sup> *<sup>d</sup>*. □

**Corollary 3.9.** *Under the hypothesis and notations of Theorem 3.8, if e* ¼ *n, then f x*ð Þ *is irreducible over Kh.*

*Proof.* If *n* ¼ *e*, then *d* ¼ 1, and so there is a unique monique polynomial of *K x*½ � which divides *f x*ð Þ and this factor has the degree at least *mn*. As degð Þ¼ *f nm*, we conclude that *f x*ð Þ is this unique monic factor. □

**Theorem 3.10.** *Let L* ¼ *K*ð Þ *α be a simple extension generated by α*∈ *K a root of a monic irreducible polynomial f x*ð Þ<sup>∈</sup> *<sup>R</sup>*∣ ∣½ � *<sup>x</sup> such that f x*ð Þ¼ *<sup>ϕ</sup><sup>n</sup> in* ∣ ∣½ � *<sup>x</sup> . Let f x*ð Þ¼ P*<sup>n</sup> <sup>i</sup>*¼<sup>0</sup>*ai*ð Þ *<sup>x</sup> <sup>ϕ</sup><sup>i</sup>* ð Þ *<sup>x</sup> be the <sup>ϕ</sup>-expansion of f x*ð Þ*. Assume that an*�*<sup>i</sup>* j j<sup>∞</sup> <sup>≤</sup>*γ<sup>i</sup> for every i* <sup>¼</sup> 0, … , *n, where γ* ¼ j j *a*<sup>0</sup> 1*=n* <sup>∞</sup> *. Then for every absolute value* jj*<sup>L</sup> of L extending* ∣∣*,* j j *P*ð Þ *α <sup>L</sup>* ≤ *max pi* � � � � ∞*γi* , *i* ¼ 0, … , *l* n o *for every P*<sup>∈</sup> *K x*½ �*, with P* <sup>¼</sup> <sup>P</sup>*<sup>l</sup> <sup>i</sup>*¼<sup>0</sup>*pi ϕi and l*< *n.*

*Proof.* Let jj*<sup>L</sup>* be an absolute value of *L* extending ∣∣ and let us show that j j *ϕ α*ð Þ *<sup>L</sup>* ¼ *γ*. For this reason, let *τ* ¼ j j *ϕ α*ð Þ *<sup>L</sup>*. By Lemma 3.6, 0 <*τ* < 1. By hypotheses *On the Irreducible Factors of a Polynomial and Applications to Extensions of Absolute… DOI: http://dx.doi.org/10.5772/intechopen.100021*

and Lemma 3.6, *ai*ð Þ *<sup>α</sup> ϕ α*ð Þ*<sup>i</sup>* � � � � � � *L* ≤*γn*�*<sup>i</sup> <sup>τ</sup><sup>i</sup>* for every *<sup>i</sup>* <sup>¼</sup> 0, … , *<sup>n</sup>*. Thus, if *<sup>τ</sup>* 6¼ *<sup>γ</sup>*, max *ai*ð Þ *<sup>α</sup> ϕ α*ð Þ*<sup>i</sup>* � � � � � � *L* , *i* ¼ 0, … , *n* n o <sup>¼</sup> max *<sup>τ</sup>n*, *<sup>γ</sup><sup>n</sup>* ð Þ. Since ∣∣ is a non archimidean absolute value, we conclude that jj*<sup>L</sup>* is a non archimidean absolute value, and so by the ultra-metric propriety, j j *<sup>f</sup>*ð Þ *<sup>α</sup> <sup>L</sup>* <sup>¼</sup> max *<sup>τ</sup>n*, *<sup>γ</sup><sup>n</sup>* ð Þ>0, which is impossible because *f*ð Þ¼ *α* 0. Therefore j j *ϕ α*ð Þ *<sup>L</sup>* ¼ *γ*.

Now, let *<sup>P</sup>* <sup>¼</sup> <sup>P</sup>*<sup>l</sup> <sup>i</sup>*¼<sup>0</sup>*pi <sup>ϕ</sup><sup>i</sup>* be a polynomial in *K x*½ �. By the ultra-metric propriety, j j *P*ð Þ *α <sup>L</sup>* ≤ *pi* ð Þ *<sup>α</sup>* � � � � *Lγi* . □

**Theorem 3.11.** *Let L* ¼ *K*ð Þ *α be a simple extension generated by α* ∈*K a root of a monic irreducible polynomial f x*ð Þ∈*R*∣ ∣½ � *<sup>x</sup> such that f x*ð Þ¼ *<sup>ϕ</sup><sup>n</sup> in* ∣ ∣½ � *<sup>x</sup> . Let f x*ð Þ¼ P*<sup>n</sup> <sup>i</sup>*¼<sup>0</sup>*ai*ð Þ *<sup>x</sup> <sup>ϕ</sup><sup>i</sup>* ð Þ *<sup>x</sup> be the <sup>ϕ</sup>-expansion of f x*ð Þ*. Assume that an*�*<sup>i</sup>* j j<sup>∞</sup> <sup>≤</sup>*γ<sup>i</sup> for every i* <sup>¼</sup> 0, … , *n, where γ* ¼ j j *a*<sup>0</sup> 1*=n* <sup>∞</sup> *. If n is the smallest positive integer satisfying γ<sup>e</sup>* ∈Γ*, then there is a unique absolute value* jj*<sup>L</sup> of L extending* ∣∣. *Moreover this absolute value is defined by P*j j ð Þ *α <sup>L</sup>* ¼ *max pi* � � � � ∞*γi* , *i* ¼ 0, … , *l* n o *for every P*<sup>∈</sup> *K x*½ �*, with P* <sup>¼</sup> <sup>P</sup>*<sup>l</sup> <sup>i</sup>*¼<sup>0</sup>*pi ϕi and l*< *n.*

*Furthermore, its ramification index is n and its residue degree is m* ¼ *deg*ð Þ *ϕ .*

*Proof.* By Corollary 3.9, if *<sup>n</sup>* <sup>¼</sup> *<sup>e</sup>*, then *f x*ð Þ is irreducible over *<sup>K</sup><sup>h</sup>*. Thus by Hensel's Lemma, there is a unique absolute value jj*<sup>L</sup>* of *L* extending ∣∣. By Theorem 3.10, we conclude that j j *ϕ α*ð Þ *<sup>L</sup>* ¼ *γ* and j j *P*ð Þ *α <sup>L</sup>* ≤ *pi* ð Þ *<sup>α</sup>* � � � � *<sup>L</sup>γ<sup>i</sup>* <sup>¼</sup> *pi* � � � � <sup>∞</sup> for every polynomial *<sup>P</sup>* <sup>¼</sup> <sup>P</sup>*<sup>l</sup> <sup>i</sup>*¼<sup>0</sup>*pi <sup>ϕ</sup><sup>i</sup>* in *K x*½ �. Let us show the equality. Let *<sup>s</sup>* be the smallest integer which satisfies *ω*ð Þ¼ *P ps* � � � � ∞*γs* . Let *i* be an integer satisfying *ω*ð Þ¼ *P pi* � � � � ∞*γi* . Then *<sup>γ</sup><sup>s</sup>*�*<sup>i</sup>* <sup>¼</sup> *pi* � � � � <sup>∞</sup>*= ps* � � � � <sup>∞</sup> ∈Γ. Thus *n* divides *i* � *s* because *n* is the smallest positive integer satisfying *<sup>γ</sup><sup>e</sup>* <sup>∈</sup> <sup>Γ</sup>. Since *<sup>l</sup>* <sup>&</sup>lt;*n*, then *<sup>i</sup>* � *<sup>s</sup>* <sup>¼</sup> 0. Therefore, j j *<sup>P</sup>*ð Þ *<sup>α</sup> <sup>L</sup>* <sup>¼</sup> *ps* � � � � <sup>∞</sup>*γ<sup>s</sup>* <sup>¼</sup> *<sup>ω</sup>*ð Þ *<sup>P</sup>* .

For the residue degree and ramification index, since j j *ϕ α*ð Þ *<sup>L</sup>* ¼ *γ* and *n* ¼ ð Þ Γð Þ*γ* : Γ , we conclude that *n* divides the ramification index *e* of jj*L*. On the other hand, since ∣ ∣ <sup>⊂</sup>*<sup>ϕ</sup>* <sup>⊂</sup>j j*<sup>L</sup>* , with *<sup>ϕ</sup>* <sup>¼</sup> ∣ ∣½ � *<sup>x</sup>* ð Þ*<sup>ϕ</sup>* , we have *<sup>m</sup>* <sup>¼</sup> *<sup>ϕ</sup>* : ∣ ∣ � � divides j j*<sup>L</sup>* : ∣ ∣ � �. As *<sup>m</sup>* � *<sup>n</sup>* <sup>¼</sup> degð Þ*<sup>f</sup>* , we conclude the equality. □

**Exercices 7.** For every positive integer *n*≥2 and *p* a positive prime integer, let *f x*ð Þ¼ *xn* � *<sup>p</sup>*.


Combining Lemma 3.4 and Theorem 3.8, we conclude the following result: **Corollary 3.12.** *Let L* ¼ *K*ð Þ *α be a simple extension generated by α* ∈*K a root of a monic irreducible polynomial f x*ð Þ∈*R*∣ ∣½ � *<sup>x</sup> . Let f x*ð Þ¼ <sup>Q</sup>*<sup>r</sup> <sup>i</sup>*¼<sup>1</sup>*ϕ<sup>i</sup> ni* ð Þ *x be the factorization of f x*ð Þ *in* ∣ ∣½ � *x , with every ϕ<sup>i</sup>* ∈ *R*∣ ∣½ � *x is a monic polynomial. For every i* ¼ 1, … ,*r, let N*<sup>þ</sup> *ϕi* ð Þ¼ *f Si*<sup>1</sup> þ … þ *Sigi be the principal ϕi-Newton polygon of f x*ð Þ*. Then L has t absolue value extending* ∣∣ *with r*≤*t* ≤P*<sup>r</sup> i*¼1 P*gi <sup>j</sup>*¼<sup>1</sup>*dij, where dij* <sup>¼</sup> *lij eij is the degree of Sij, lij is the length of Sij, and eij* <sup>¼</sup> *lij dij for every i* ¼ 1, … ,*r and j* ¼ 1, … , *gi .*

### **4. Applications**

1.Let <sup>∥</sup> be the *<sup>p</sup>*-adic absolute value defined on by <sup>∣</sup>*a*<sup>∣</sup> <sup>¼</sup> *<sup>p</sup>*�*νp*ð Þ *<sup>a</sup>* and *f x*ð Þ¼ *xn* � *<sup>p</sup>*½ � *<sup>x</sup>* . Show that *f x*ð Þ is irreducible over . Let *<sup>L</sup>* <sup>¼</sup> ð Þ *<sup>α</sup>* with *<sup>α</sup>* a complex root of *f x*ð Þ. Determine all absolute value of *L* extending ∣∣.

**Answer**. First *<sup>Γ</sup>* <sup>¼</sup> *pk*, *<sup>k</sup>*∈ � � is the value group of ∣∣. Since <sup>∣</sup>*p*<sup>∣</sup> <sup>¼</sup> *<sup>p</sup>*�1, *<sup>γ</sup>* <sup>¼</sup> j j *a*<sup>0</sup> <sup>1</sup>*=<sup>n</sup>* <sup>¼</sup> *<sup>p</sup>*�1*=n*, we conclude that the smallest integer satisfying *<sup>γ</sup><sup>e</sup>* <sup>∈</sup><sup>Γ</sup> is *<sup>n</sup>*. Thus, by Corollary 3.9, *f x*ð Þ is irreducible over *<sup>h</sup>* ,and so is over . Since *f x*ð Þ¼ *xn* in *p*½ � *<sup>x</sup>* , by Theorem 3.11, there is a unique absolute value of *<sup>L</sup>* extending ∣∣ and it is defined by j j *P*ð Þ *α <sup>L</sup>* ¼ max f∣*pi* ∣*γi* , *i* ¼ *o*, … , *l*g for every polynomial *<sup>P</sup>* <sup>¼</sup> <sup>P</sup>*<sup>l</sup> <sup>i</sup>*¼0*x<sup>i</sup>* with *<sup>l</sup>*<*n*.

2.Let <sup>∥</sup> be the *<sup>p</sup>*-adic absolute value and *f x*ð Þ¼ *xn*�*<sup>a</sup>* ∈½ � *x* such that *p* does not divide *νp*ð Þ *a* . Show that *f x*ð Þ is irreducible over . Let *L* ¼ ð Þ *α* with *α* a complex root of *f x*ð Þ. Determine all absolute value of *L* extending ∣∣.

**Answer**. First *<sup>Γ</sup>* <sup>¼</sup> *pk*, *<sup>k</sup>*∈ � � is the value group of ∣∣. Since <sup>∣</sup>*p*<sup>∣</sup> <sup>¼</sup> *<sup>p</sup>*�1, *<sup>γ</sup>* <sup>¼</sup> j j *a*<sup>0</sup> <sup>1</sup>*=<sup>n</sup>* <sup>¼</sup> *<sup>p</sup>*�1*=<sup>n</sup>*, we conclude that the smallest integer satisfying *<sup>γ</sup><sup>e</sup>* <sup>∈</sup><sup>Γ</sup> is *<sup>n</sup>*. Thus, by Corollary 3.9, *f x*ð Þ is irreducible over *<sup>h</sup>* ,and so is over . Since *f x*ð Þ¼ *xn* in *p*½ � *<sup>x</sup>* , by Theorem 3.11, there is a unique absolute value of *<sup>L</sup>* extending ∣∣ and it is defined by j j *P*ð Þ *α <sup>L</sup>* ¼ max f∣*pi* ∣*γi* , *i* ¼ *o*, … , *l*g for every polynomial *<sup>P</sup>* <sup>¼</sup> <sup>P</sup>*<sup>l</sup> <sup>i</sup>*¼<sup>0</sup>*x<sup>i</sup>* with *<sup>l</sup>*<*n*.

3.Let *f x*ð Þ¼ *<sup>ϕ</sup>*<sup>6</sup> <sup>þ</sup> <sup>24</sup>*xϕ*<sup>4</sup> <sup>þ</sup> <sup>24</sup>*ϕ*<sup>3</sup> <sup>þ</sup> 15 16 ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>32</sup> *<sup>ϕ</sup>* <sup>þ</sup> 48 with *<sup>ϕ</sup>*∈½ � *<sup>x</sup>* a monic polynomial whose reduction is irreducible in 2½ � *x* . In 2½ � *x* , how many monic irreducible factors *f x*ð Þ gets?, where <sup>2</sup> is the completion of ð Þ , jj and ∣∣ is the 2-adic absolute value.

**Answer.** It is easy to check that *f x*ð Þ satisfies the conditions of Theorem 3.8; *<sup>a</sup>*<sup>6</sup>�*<sup>i</sup>* j j<sup>∞</sup> <sup>≤</sup>*γ<sup>i</sup>* with *<sup>γ</sup>* <sup>¼</sup> <sup>2</sup>�4*=*<sup>6</sup> <sup>¼</sup> <sup>2</sup>�1*=*<sup>3</sup> � �<sup>2</sup> . Thus *e* ¼ 3 and *d* ¼ 2. By Theorem 3.8, *f x*ð Þ has at most 2 monic irreducible factors in 2½ � *x* .

### **Author details**

Lhoussain El Fadil\* and Mohamed Faris Faculty of Sciences Dhar El Mahraz, Sidi Mohamed ben Abdellah University, Morocco

\*Address all correspondence to: lhouelfadil2@gmail.com

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*On the Irreducible Factors of a Polynomial and Applications to Extensions of Absolute… DOI: http://dx.doi.org/10.5772/intechopen.100021*

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#### **Chapter 7**
