Extending a result of Carlitz and McConnel to polynomials which are not permutations

Abstract

Let $D$ denote the set of directions determined by the graph of a polynomial $f$ of ${\mathbb{F}}_q[x]$, where $q$ is a power of the prime $p$. If $D$ is contained in a multiplicative subgroup $M$ of ${\mathbb{F}}_q^{\times }$, then by a result of Carlitz and McConnel it follows that $f(x)=a{x}^{p^k}+b$ for some $k\in \mathbb{N}$. Of course, if $D\subseteq M$, then $0\notin D$ and hence $f$ is a permutation. If we assume the weaker condition $D\subseteq M\cup \{0\}$, then $f$ is not necessarily a permutation, but Sziklai conjectured that $f(x)=a{x}^{p^k}+b$ follows also in this case. When $q$ is odd, and the index of $M$ is even, then a result of Ball, Blokhuis, Brouwer, Storme and Sz\H onyi combined with a result of McGuire and Göloğlu proves the conjecture. Assume $°f\ge 1$. We prove that if the size of $D^{-1}D=\{d^{-1}{d}^{\prime }:d\in D\setminus \{0\},d^{\prime }\in D\}$ is less than $q-°f+2$, then $f$ is a permutation of ${\mathbb{F}}_q$. We use this result to verify the conjecture of Sziklai.