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 *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}.

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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