Reversed Dickson Polynomials of the Third Kind

Irreducible factorization of translates of reversed Dickson polynomials over finite fields

Let $F$ be a field of $q$ elements, where $q$ is a power of an odd prime. Fix $n = (q+1)/2$. For each $s \in F$, we describe all the irreducible factors over $F$ of the polynomial $g_s(y): = y^n + (1-y)^n -s$, and we give a necessary and sufficient condition on $s$ for $g_s(y)$ to be irreducible.

反向 Dickson 多项式

Reversed Dickson polynomials

本文考察有限域和整数剩余类环上第一、二类反向 Dickson 多项式的置换性质，分析相关置换与完全置换多项式的不动点、循环类型，并证明一个置换猜想的两个特例。

有限域上$(k+1)$-th kind的reversed Dickson polynomials，II

Reversed Dickson polynomials of the $(k+1)$-th kind over finite fields, II

本文研究了$(k+1)$-th kind的reversed Dickson polynomials $D_{n,k}(1,x)$在特定$n$值下的置换行为，找到了其非置换多项式的条件，推广了近期关于$D_{p^l-1,1}(1,x)$的结果，并给出了相关代数与算术性质。

Self-Reciprocal Polynomials and Coterm Polynomials

We classify all self-reciprocal polynomials arising from reversed Dickson polynomials over $\mathbb{Z}$ and $\mathbb{F}_p$, where $p$ is prime. As a consequence, we also obtain coterm polynomials arising from reversed Dickson polynomials.

Dickson多项式、Fibonacci多项式与Lucas多项式的一个共同推广及其应用

A common generalization of Dickson polynomials, Fibonacci polynomials, and Lucas polynomials and applications

本文定义了满足线性递推关系的更一般的多变量多项式族，给出了显式公式和行列式表达式，并将其应用于2阶递推多项式，揭示了Fibonacci、Lucas和Dickson多项式等的关系与恒等式，是对多项研究的统一推广。

Dembowski-Ostrom多项式与Dickson多项式

Dembowski-Ostrom polynomials and Dickson polynomials

本文完整分类了由Dickson多项式与单项式复合生成的Dembowski-Ostrom多项式，并利用Weil界变体讨论其平面性，这些多项式对编码与密码学至关重要。

Factorization of Dickson polynomials over Finite Fields

Let $D_n(x;a)$ and $E_n(x;a)\in\mathbb F_q[x]$ be Dickson polynomials of first and second kind respectively, where $\mathbb F_q$ is a finite field with $q$ elements. In this article we show explicitly the irreducible factors these polynomials in the case that every prime divisor of $n$ divides $q-1$. This result generalizes the results find in Chou, W.S., The Factorization of Dickson polynomials over finite fields. Finite Fields Appl. {\bf3} (1997) 84-96, Fitzgerald R. W., Yucas J. L., Explicit factorization of cyclotomic and Dickson polynomials over finite fields. Arithmetic of Finite Fields.…

Permutation properties of Dickson and Chebyshev polynomials and connections to number theory

The $k$th Dickson polynomial of the first kind, $D_k(x) \in {\mathbb Z}[x]$, is determined by the formula: $D_k(u+1/u) = u^k + 1/u^k$, where $k \ge 0$ and $u$ is an indeterminate. These polynomials are closely related to Chebyshev polynomials and have been widely studied. Leonard Eugene Dickson proved in 1896 that $D_k(x)$ is a permutation polynomial on ${\mathbb F}_{p^n}$, $p$ prime, if and only if GCD$(k,p^{2n}-1)=1$, and his result easily carries over to Chebyshev polynomials when $p$ is odd. This article continues on this theme, as we find special subsets of ${\mathbb F}_{p^n}$ that are st…

Special Function Representation Of Dickson Polynomials

Generating functions and functional equations of Dickson polynomials of the first and second kind are derived and continued analytically. These formulae are expressed in terms of the incomplete gamma function over complex variables of the parameters involved. Special cases are evaluated in terms of composite incomplete gamma functions and mathematical constants.

有限域中迪克森多项式的周期性和动力系统

Periodicity and Dynamical Systems of Dickson Polynomials in Finite Fields

研究有限域上第一类迪克森多项式的动力性质，重点是其迭代序列的周期性和代数结构。确定特定序列的精确周期，通过函数方程关联其动力学与幂映射，在置换和非置换情况均有成果，并得出多项式新恒等式和对称性质。

$x^{2p}+bx^p+c^p$ 在 $\mathbb{Q}$ 上的 Galois 群

The Galois Group of $x^{2p}+bx^p+c^p$ over $\mathbb{Q}$

本文研究有理系数多项式 $x^{2p}+bx^p+c^p$，借助第一类 Dickson 多项式建立不可约性判据，并显式给出相关多项式的 Galois 群结构。