Related papers: Chebyshev Action on Finite Fields

The number of linear independent algebraic relations among elementary symmetric polynomial functions over finite fields is computed. An algorithm able to find all such relations is described. It is proved that the basis of the ideal of…

The observed output of an interferometer is the result of interference among the parts of the input light beam traveling along each possible optical path. In complex systems, writing down all these possible optical paths and computing their…

Let $G$ be a finite group. The co-prime order graph of $G$ is the graph whose vertex set is $G$, and two distinct vertices $x,y$ are adjacent if gcd$(o(x),o(y))$ is either $1$ or a prime, where $o(x)$ and $o(y)$ are the orders of $x$ and…

This paper considers functional series whose terms are higher-order derivatives of Chebyshev polynomials of the second kind, where the degree of the polynomial is related to the order of the derivative. Analytic summation is used to…

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.

Let $G$ be a graph with $n$ vertices and $m$ edges. The energy $E$ of the graph $G$ is defined as the sum of the moduli of the adjacency eigenvalues $\lambda_{1} \geq \lambda_{2} \geq \ldots \geq \lambda_{n}$ of $G$: $$…

Let $G$ be a finite simple graph with edge ideal $I(G)$. Let $J(G)$ denote the Alexander dual of $I(G)$. We show that a description of all induced cycles of odd length in $G$ is encoded in the associated primes of $J(G)^2$. This result…

The graph polynomial for the number of independent sets of size $k$ in a general undirected graph is shown to be equal to an elementary symmetric polynomial of the vertex monomials, which are determined by the edges incident at the…

The notion of the weighted degree of a polynomial is a basic tool in Affine Algebraic Geometry. In this paper, we study the properties of the weighted multidegrees of polynomial automorphisms by a new approach which focuses on stable…

We give a direct and intuitive proof (via sliding some columns up and down) of the following interesting fact: if we write out the Chebyshev polynomials in a chart and take the sums of coefficients along certain diagonals, we obtain the…

We explore graph theoretical properties of minimal prime graphs of finite solvable groups. In finite group theory studying the prime graph of a group has been an important topic for the past almost half century. Recently prime graphs of…

Given a number field $K$ and a polynomial $f(z) \in K[z]$ of degree at least 2, one can construct a finite directed graph $G(f,K)$ whose vertices are the $K$-rational preperiodic points for $f$, with an edge $\alpha \to \beta$ if and only…

The estimates of the uniform norm of the Chebyshev polynomials associated with a compact set $K$ in the complex plane are established. These estimates are exact (up to a constant factor) in the case where $K$ consists of a finite number of…

We present a new structure theorem for finite fields of odd order that relates multiplicative and additive structure in an interesting way. This theorem has several applications, including an improved understanding of Dickson and Chebyshev…

Let $\mathbb{F}_q$ be the finite field with $q$ elements and $char(\mathbb{F}_q)$ odd. In this article we will describe completely the dynamics of the map $f(X)=c(X^{q+1}+aX^2)$, for $a=\{\pm1\}$ and $c\in\mathbb{F}_q^*$, over the finite…

Given a reciprocal/palindromic polynomial of even degree, we show that the gamma vector is essentially given by an inverted Chebyshev polynomial basis expansion. As an immediate consequence, we characterize real-rootedness of a linear…

Let $e$ be a positive integer, $p$ be an odd prime, $q=p^{e}$, and $\Bbb F_q$ be the finite field of $q$ elements. Let $f,g \in \Bbb F_q [X,Y]$. The graph $G=G_q(f,g)$ is a bipartite graph with vertex partitions $P=\Bbb F_q^3$ and $L=\Bbb…

This paper studies infinite graphs produced from a natural unfolding operation applied to finite graphs. Graphs produced via such operations are of finite degree and automatic over the unary alphabet (that is, they can be described by…

The Chudnovsky and Chudnovsky algorithm for the multiplication in extensions of finite fields provides a bilinear complexity which is uniformly linear whith respect to the degree of the extension. Recently, Randriambololona has generalized…

In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of…