Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
11033145 | Finite Fields and Their Applications | 2018 | 10 Pages |
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,qâ1)>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(nâ2) 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(nâ2) is replaced by n2(nâ2)2.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Jasbir S. Chahal, Sudhir R. Ghorpade,