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

TL;DR

This paper extends a classical result on polynomials over finite fields, showing that under certain conditions on the directions determined by the polynomial's graph, the polynomial must be a permutation of the field.

Contribution

It proves a new criterion involving the set of directions and their ratios, confirming Sziklai's conjecture in broader cases beyond permutations.

Findings

If the set of ratios of directions is small, then the polynomial is a permutation.

The paper verifies Sziklai's conjecture for a wider class of polynomials.

Provides a new permutation criterion based on direction sets and their ratios.

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.