Periodicity and Dynamical Systems of Dickson Polynomials in Finite Fields

TL;DR

This paper explores the periodicity and structural behavior of Dickson polynomials over finite fields, providing explicit formulas for their sequence periods and analyzing their dynamical properties using combinatorics and number theory.

Contribution

It introduces explicit formulas for the periods of Dickson polynomial sequences modulo finite field polynomials, especially when degrees are coprime to $q^2 - 1$, and examines their algebraic structure.

Findings

Derived explicit formulas for sequence periods

Identified symmetric properties of polynomial coefficients

Developed algorithms to compute exact periods

Abstract

This paper investigates the dynamical properties of Dickson polynomials over finite fields, focusing on the periodicity and structural behavior of their iterated sequences. We introduce and analyze the sequence $[D_n(x, \alpha) \mod (x^q - x)]_n$, where $D_n(x, \alpha)$ denotes a Dickson polynomial of the first kind, and explore its periodic nature when reduced modulo $x^q - x$. We derive explicit formulas for the period of these sequences, particularly in the case when $n$ is coprime to $q^2 - 1$. In addition, we identify a symmetric property of the polynomial coefficients that plays a crucial role in the analysis of these sequences. Using tools from combinatorics, elementary number theory, and finite fields, we present algorithms to compute the exact period and investigate the dynamical structure of these polynomials. We also highlight open problems in cases where the degree $n$ is not coprime to $q^2 - 1$. Our results offer deep insights into the algebraic structure of Dickson polynomials and their role in dynamical systems over finite fields.