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.

5 pages