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 $f(x)=ax^{p^k}+b$ 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 $f(x)=ax^{p^k}+b$ 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 G\"olo\u{g}lu proves the conjecture. Assume $\deg f\geq 1$. We prove that if the size of $D^{-1}D=\{d^{-1}d' : d\in D\setminus \{0\},\, d'\in D\}$ is less than $q-\deg f+2$, then $f$ is a permutation of $\mathbb{F}_q$. We use this result to verify the conjecture of Sziklai.