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 $F_q\left[x\right]$, where q is a power of the prime p. If D is contained in a multiplicative subgroup M of $F_q^{\times }$, then by a result of Carlitz and McConnel it follows that $f\left(x\right)=a{x}^{p^k}+b$ for some $k\in N$. Of course, if $D\subseteq M$, then $0\notin D$ and hence f is a permutation. If we assume the weaker condition $D\subseteq M\cup \left\{0\right\}$, then f is not necessarily a permutation, but Sziklai conjectured that $f\left(x\right)=a{x}^{p^k}+b$ 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 Szőnyi combined with a result of Göloğlu and McGuire proves the conjecture. Assume $\mathrm{deg}f\ge 1$. 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 $q-\mathrm{deg}f+2$, then f is a permutation of $F_q$. We use this result to prove the conjecture of Sziklai.