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 is published in Finite Fields and Their Applications, Vol. 54 (2018), pp. 366-375. The article is, in fact, a republication, with minor changes, of the article
published earlier in the same journal: Vol. 28 (July 2014), pp.282-291. The authors requested a withdrawal
of the original article shortly after it first appeared in press in 2014. This was
mishandled by
the publisher for which they have subsequently apologized publicly to the authors and
republished the article.. Those wishing to get more information on this matter, may write to
srg@math.iitb.ac.in
Download the full paper as:
Back to the List of Publications
Back to the Sudhir Ghorpade's Home Page