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The difference between permutation polynomials over finite fields


Authors: Stephen D. Cohen, Gary L. Mullen and Peter Jau-Shyong Shiue
Journal: Proc. Amer. Math. Soc. 123 (1995), 2011-2015
MSC: Primary 11T06
DOI: https://doi.org/10.1090/S0002-9939-1995-1196163-1
MathSciNet review: 1196163
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Abstract: 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 \geq 2$ and p is a prime exceeding $ {({n^2} - 3n + 4)^2}$ for which f and g are both permutation polynomials of the finite field $ {F_p}$, then their difference $ h = f - g$ cannot be such that $ h(x) = cx$ 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 $ {F_p}$ and t is the degree of h, then $ t \geq 3n/5$ and, provided $ t \leq n - 3$, 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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DOI: https://doi.org/10.1090/S0002-9939-1995-1196163-1
Keywords: Finite field, permutation polynomial
Article copyright: © Copyright 1995 American Mathematical Society