Carlitz-Wan Conjecture for Permutation Polynomials and Weil Bound for Curves over Finite Fields

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

Jasbir S. Chahal

Department of Mathematics
Brigham Young University
Provo, Utah 84602, USA



Sudhir R. Ghorpade

Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai 400076, India



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.

1 Introduction 1
2 Preliminaries 3
3 Main Theorem 5
References 7

This paper will be published in Finite Fields and Their Applications

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