The Weil bound and non-exceptional permutation polynomials over finite fields

Abstract

A well-known result of von zur Gathen asserts that a non-exceptional permutation polynomial of degree $n$ over $\mathbb{F}_{q}$ exists only if $q<n^{4}$. With the help of the Weil bound for the number of $\mathbb{F}_{q}$-points on an absolutely irreducible (possibly singular) affine plane curve, Chahal and Ghorpade improved von zur Gathen's proof to replace $n^{4}$ by a bound less than $n^{2}(n-2)^{2}$. Also based on the Weil bound, we further refine the upper bound for $q$ with respect to $n$, by a more concise and direct proof following Wan's arguments.