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

E-Mail: jasbir@math.byu.edu

and

Sudhir R. Ghorpade

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

E-Mail: srg@math.iitb.ac.in



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.


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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