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

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$.

25 pages. First presented as "Permutations properties of Dickson polynomials and connections to number theory" at the Mathematical Congress of the Americas, MCA2017. Version of May 1, 2021 is shortened and has a new Theorem 8.1. Submitted to Finite Fields and Their Applications

References in corpus (3)

• New Wilson-like theorems arising from Dickson polynomials
• A structure theorem for finite fields
• Explicit Artin maps into ${\mathrm{PGL}}_2$

Cited by in corpus (2)

• Explicit Artin maps into ${\mathrm{PGL}}_2$
• Residue sums of Dickson polynomials over finite fields