Polynomials in $\mathbb{F}_p[x]$ which commute under composition

TL;DR

This paper studies polynomials over finite fields that commute under composition with a fixed linear polynomial, providing a simpler proof of a known result and exploring the structure of such commuting polynomials.

Contribution

It offers a new, simpler proof of a known characterization of polynomials commuting with a linear polynomial over finite fields.

Findings

Characterization of polynomials commuting with a fixed linear polynomial

Simplified proof of Park's result

Enumeration of such commuting polynomials

Abstract

Let $\mathbb{F}$ be a finite field and let $f$ be a linear polynomial in $\mathbb{F}[x]$. We investigate the number of polynomials of degree $d$ which commute with $f$ under composition. In so doing, we rediscover a result of Park, but with a conceptually simpler proof.