TY - JOUR

T1 - The difference between permutation polynomials over finite fields

AU - Cohen, Stephen D.

AU - Mullen, Gary L.

AU - Shiue, Peter Jau Shyong

N1 - Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 1995/7

Y1 - 1995/7

N2 - Recently S. D. Cohen resolved a conjecture of Chowla and Zassenhaus (1968) in the affirmative by showing that, if f(x) and g(x) are integral polynomials of degree n ≥ 2 and p is a prime exceeding (n2 - 3n + 4)2 for which f and g are both permutation polynomials of the finite field Fp, then their difference h = f - g cannot be such that h(x) = ex for some integer c not divisible by p. In this note we provide a significant generalization by proving that, if h is not a constant in Fp and t is the degree of h, then and, provided t and n are not relatively prime. In a sense this measures the isolation of permutation polynomials of the same degree over large finite prime fields.

AB - Recently S. D. Cohen resolved a conjecture of Chowla and Zassenhaus (1968) in the affirmative by showing that, if f(x) and g(x) are integral polynomials of degree n ≥ 2 and p is a prime exceeding (n2 - 3n + 4)2 for which f and g are both permutation polynomials of the finite field Fp, then their difference h = f - g cannot be such that h(x) = ex for some integer c not divisible by p. In this note we provide a significant generalization by proving that, if h is not a constant in Fp and t is the degree of h, then and, provided t and n are not relatively prime. In a sense this measures the isolation of permutation polynomials of the same degree over large finite prime fields.

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U2 - 10.1090/S0002-9939-1995-1196163-1

DO - 10.1090/S0002-9939-1995-1196163-1

M3 - Article

AN - SCOPUS:84966240925

VL - 123

SP - 2011

EP - 2015

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 7

ER -