Carlitz–Wan conjecture for permutation polynomials and Weil bound for curves over finite fields

Abstract

The Carlitz–Wan conjecture, which is now a theorem, asserts that for any positive integer n, there is a constant Cn such that if q is any prime power >Cn with GCD(n,q−1)>1, then there is no permutation polynomial of degree n over the finite field with q elements. From the work of von zur Gathen, it is known that one can take Cn=n4. On the other hand, a conjecture of Mullen, which asserts essentially that one can take Cn=n(n−2) has been shown to be false. In this paper, we use a precise version of Weil bound for the number of points of affine algebraic curves over finite fields to obtain a refinement of the result of von zur Gathen where n4 is replaced by a sharper bound. As a corollary, we show that Mullen's conjecture holds in the affirmative if n(n−2) is replaced by n2(n−2)2.