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

TL;DR

This paper investigates the permutation properties of Dickson and Chebyshev polynomials over finite fields, identifies special subsets they permute, and derives new number theoretic results including polynomial factorizations and product formulas.

Contribution

It provides new insights into the permutation behavior of these polynomials, identifies stabilized subsets, and establishes novel factorization and number theory results.

Findings

Dickson polynomials permute specific subsets of finite fields

Derived a factorization formula for Dickson and Chebyshev polynomials

Established a new product formula involving nonsquares in finite fields

Abstract

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 stabilized or permuted by Dickson or Chebyshev polynomials. Our analysis also leads to a factorization formula for Dickson and Chebyshev polynomials and some new results in elementary number theory. For example, we show that if $q$ is an odd prime power, then $\prod\left\{ a \in{\mathbb F}_q^\times : \text{$a$ and $4-a$ are nonsquares} \right\} = 2$.