The small value set polynomials over finite fields and monodromy groups

Abstract

In this article, we study polynomials over finite fields with small value sets through their monodromy groups. More specifically, we consider polynomials $f\left(T\right)\in F_q\left[T\right]$ such that $F_q\left(T\right)/F_q\left(f\right(T\left)\right)$ is Galois, which we call absolutely minimal value set polynomials (AMVSPs) over $F_q$. We examine their relationship with minimal value set polynomials (MVSPs) by directly utilizing tools and results from function field theory. In particular, we prove that if $f\left(T\right)\in F_q\left[T\right]$ satisfies $1<\mathrm{deg}\left(f\right)\le \sqrt{q}+1$, then $f\left(T\right)$ is an MVSP if and only if it is an AMVSP. As an application of our results, we provide a simple proof of the classification results of Mills (see [16, Theorem 2]) and Borges-Conceição (see [4, Theorem 2.3]) for MVSPs with degree $\le \sqrt{q}+1$ using a completely different approach.