Carlitz-Wan conjecture for permutation polynomials and Weill bound for curves over finite fields

摘要

The Carlitz-Wan conjecture, which is now a theorem, asserts that for any positive integer n, there is a constant C n such that if q is any prime power C(n )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 C-n = n(4). On the other hand, a conjecture of Mullen, which asserts essentially that one can take C-n = 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 n(4) 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 n(2) (n - 2)(2). (C) 2018 Elsevier Inc. All rights reserved.