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

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.