Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Nonexistence of permutation binomials of certain shapes. | Nonexistence of permutation binomials of certain shapes Ariane M. Masuda Department of Mathematics and Statistics University of Ottawa Ottawa ON K1N 6N5 Canada amasuda@ Michael E. Zieve Center for Communications Research 805 Bunn Drive Princeton NJ 08540 zieve@ Submitted Dec 23 2006 Accepted May 24 2007 Published Jun 21 2007 Mathematics Subject Classification 11T06 Abstract Suppose xm axn is a permutation polynomial over Fp where p 5 is prime and m n 0 and a 2 Fp. We prove that gcd m n p 1 2 2 4g. In the special case that either p 1 2 or p 1 4 is prime this was conjectured in a recent paper by Masuda Panario and Wang. 1 Introduction A polynomial over a finite field is called a permutation polynomial if it permutes the elements of the field. These polynomials have been studied intensively in the past two centuries. Permutation monomials are completely understood for m 0 xm permutes Fq if and only if gcd m q 1 1. However even though dozens of papers have been written about them permutation binomials remain mysterious. In this note we prove the following result Theorem . If p 5 is prime and f xm axn permutes Fp where m n 0 and a 2 Fp then gcd m n p 1 2 2 4g. This work proves the conjectures stated in the first author s talk at the November 2006 BIRS workshop on Polynomials over Finite Fields and Applications. The authors thank BIRS for providing wonderful facilities. The first author was at Carleton University when this research was performed. THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 N12 1 In case p 1 2 or p 1 4 is prime this was conjectured in the recent paper 2 by Panario Wang and the first author. It is well-known that the gcd is not 1 for in that case f has more than one root in Fp since xm n is a permutation polynomial. It is much more difficult to show that the gcd is not 2 or 4. In Section 2 we prove some general results about permutation binomials and in particular we show that it suffices to prove Theorem when m n .
