S. V. Ludkovsky
The article is devoted to the investigation of transformation groups of polynomials over Cayley-Dickson algebras and their manifolds of zeros. The problems about expressibility of zeros with the help of roots and decomposibility of polynomials as products of linear terms are studied.
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A derivation of Dickson polynomials using the Cayley-Hamilton theorem
本文借助 Cayley-Hamilton 定理，从矩阵 n 次幂迹的表达式出发引入一阶 Dickson 多项式，并说明该框架可直接推广至 Carlitz 公式、二阶及多元情形。
Alternating Roots of Polynomials over Cayley-Dickson Algebras
本文引入了 Cayley-Dickson 代数上多项式的交错根与交错多项式概念，证明了交错根与对应交错多项式根的联系，并提供了计算 Hamilton 四元数上交错根的详细 Octave 代码。
In the recent years a lot of effort has been made to extend the theory of hyperholomorphic functions from the setting of associative Clifford algebras to non-associative Cayley-Dickson algebras, starting with the octonions. An important question is whether there appear really essentially different features in the treatment with Cayley-Dickson algebras that cannot be handled in the Clifford analysis setting. Here we give one concrete example. Cayley-Dickson algebras namely admit the construction of direct analogues of CM-lattices, in particular lattices that are closed under multiplication. Can…
Roots and right factors of polynomials and left eigenvalues of matrices over Cayley-Dickson algebras
本文研究一般 Cayley-Dickson 代数中多项式有根与存在右因子的等价性，分析其适用边界，并计算八元数矩阵的左特征值。
We give a complete classification of Dembowski-Ostram polynomials from reversed Dickson polynomials in odd characteristic.
We discuss the properties and the permutation behaviour of the reversed Dickson polynomials of the $(k+1)$-th kind $D_{n,k}(1,x)$ over finite fields. The results in this paper unify and generalize several recently discovered results on reversed Dickson polynomials over finite fields.
This article explains the similar appearance of two polynomial identities involving Dickson polynomials in char. 2, one found by Abhyankar, Cohen, and Zieve, and the other found by the author.
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.…
In this paper, we present several necessary conditions for the reversed Dickson polynomial $E_{n}(1, x)$ of the second kind to be a permutation of $\mathbb{F}_{q}$. In particular, we give explicit evaluation of the sum $\sum_{a\in \mathbb{F}_{q}}E_{n}(1,a)$.
We discuss the problem of classifying Dembowski-Ostrom polynomials from the composition of reversed Dickson polynomials of arbitrary kind and monomials over finite fields of odd characteristic. Moreover, by using a variant of the Weil bound for the number of points of affine algebraic curves over finite fields, we discuss the planarity of all such Dembowski-Ostrom polynomials. Planar Dembowski-Ostrom polynomials have applications in many areas including cryptography and coding theory.
Let $G_n$ be the group of permutations on $\mathbb{Z}_n$ that is induced by a Dickson polynomial, where $n$ is a positive integer. In this work, by solving special types of systems of linear congruences, we obtain $G_n$. In addition, we give an a algorithm that gets $|G_n|$.
In this paper, we obtain several results on the permutational behavior of the reversed Dickson polynomial $D_{n,3}(1,x)$ of the fourth kind over the finite field ${\mathbb F}_{q}$. Particularly, we present the explicit evaluation of the first moment $\sum_{a\in {\mathbb F}_{q}}D_{n,3}(1,a)$.
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 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。
We present here some results of applying the Cayley-Dickson process to certain alternative algebras (notably built upon Galois fields and congruence rings), in a manner which might yield new building blocks for cryptographic systems. We focus on enumeration properties rather than the classification and comparison questions which are extensively studied elsewhere, at least for algebras over fields. The results presented here are technical but not inherently difficult.
Let $p$ be a prime and $q=p^e$. We discuss the properties of the reversed Dickson polynomial $D_{n,2}(1,x)$ of the third kind. We also give several necessary conditions for the reversed Dickson polynomial of the third kind $D_{n,2}(1,x)$ to be a permutation of $\mathbb{F}_{q}$. In particular, we give explicit evaluation of the sum $\sum_{a\in \mathbb{F}_q}D_{n,2}(1,a)$.
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。
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.
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…
In this paper, we use the method developed previously by Hong, Qin and Zhao to obtain several results on the permutational behavior of the reversed Dickson polynomial $D_{n,k}(1,x)$ of the $(k+1)$-th kind over the finite field ${\mathbb F}_{q}$. Particularly, we present the explicit evaluation of the first moment $\sum_{a\in {\mathbb F}_{q}}D_{n,k}(1,a)$. Our results extend the known results from the case $0\le k\le 3$ to the general $k\ge 0$ case.
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.
On the Waring problem with Dickson polynomials modulo a prime
本文利用 Kloosterman 和与 Gauss 和的新界限，改进了素数有限域下 Dickson 多项式 Waring 问题的现有结果。
For an $[n,k]$ Reed-Solomon code $\mathcal{C}$, it can be shown that any received word $r$ lies a distance at most $n-k$ from $\mathcal{C}$, denoted $d(r,\mathcal{C})\leq n-k$. Any word $r$ meeting the equality is called a deep hole. Guruswami and Vardy (2005) showed that for a specific class of codes, determining whether or not a word is a deep hole is NP-hard. They suggested passingly that it may be easier when the evaluation set of $\mathcal{C}$ is large or structured. Following this idea, we study the case where the evaluation set is the image of a Dickson polynomial, whose values appear w…
Residue sums of Dickson polynomials over finite fields
本文研究奇特征有限域上任意次数Dickson多项式的剩余类和问题，给出了像中不同剩余类之和的闭式表达式并刻画像集大小，首次实现次数无界多项式此类和的非平凡分类。
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。
A new polynomial identity is found for Dickson polynomials in characteristic 2. The identity is used to prove that the two polynomials $x^{q+1}+x+1/a$ and $C(x)+a$ have the same splitting field over $F$, where $F$ is a field of characteristic 2, $a$ is a nonzero element of $F$, $q=2^n>2$, and $C(x) = x (\sum_{i=0}^{n-1} x^{2^i-1})^{q+1}$ is a Müller--Cohen--Matthews polynomial of degree $(q^2-q)/2$. In addition, a new proof is obtained for the known result that $C(x)$ induces a permutation on $F_{2^m}$ if $2m$ and $n$ are relatively prime.
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.
Wilson's Theorem states that the product of all nonzero elements of a finite field ${\mathbb F}_q$ is $-1$. In this article, we define some natural subsets $S \subset {\mathbb F}_q^\times$ and find formulas for the product of the elements of $S$, denoted $\prod S$. These new formulas are appealing for the simple, natural description of the sets $S$, and for the simplicity of the product. An example is $\prod\left\{ a \in {\mathbb F}_q^\times : \text{$1-a$ and $3+a$ are nonsquares} \right\} = 2$ if $q \equiv \pm 1 \pmod{12}$, or $-1$ otherwise.
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。
We improve upper bounds of F. R. K. Chung and of M. Lu, D. Wan, L.-P. Wang, X.-D. Zhang on the diameter of some Cayley graphs constructed from polynomials over finite fields.
By using the piecewise method, Lagrange interpolation formula and Lucas' theorem, we determine explicit expressions of the inverses of a class of reversed Dickson permutation polynomials and some classes of generalized cyclotomic mapping permutation polynomials over finite fields of characteristic three.
Sylvester-Cayley vector partitions algorithm and the Gaussian polynomials
本文扩展了 Sylvester 和 Cayley 提出的二重分割算法，通过受限分割函数的卷积推导出 Gaussian 多项式系数的显式表达式。
This paper provides an algorithmic generalization of Dickson's method of classifying permutation polynomials (PPs) of a given degree $d$ over finite fields. Dickson's idea is to formulate from Hermite's criterion several polynomial equations satisfied by the coefficients of an arbitrary PP of degree $d$. Previous classifications of PPs of degree at most $6$ were essentially deduced from manual analysis of these polynomial equations. However, these polynomials, needed for that purpose when $d>6$, are too complicated to solve. Our idea is to make them more solvable by calculating some radicals o…
Let $\mathbb{F}_q$ be the finite field of $q$ elements. Then a \emph{permutation polynomial} (PP) of $\mathbb{F}_q$ is a polynomial $f \in \mathbb{F}_q[x]$ such that the associated function $c \mapsto f(c)$ is a permutation of the elements of $\mathbb{F}_q$. In 1897 Dickson gave what he claimed to be a complete list of PPs of degree at most 6, however there have been suggestions recently that this classification might be incomplete. Unfortunately, Dickson's claim of a full characterisation is not easily verified because his published proof is difficult to follow. This is mainly due to antiquat…
In \cite{D1}, Dickson listed all permutation polynomials up to degree 5 over an arbitrary finite field, and all permutation polynomials of degree 6 over finite fields of odd characteristic. The classification of degree 6 permutation polynomials over finite fields of characteristic 2 was left incomplete. In this paper we complete the classification of permutation polynomials of degree 6 over finite fields of characteristic 2. In addition, all permutation polynomials of degree 7 over finite fields of characteristic 2 are classified.
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