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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The difference between permutation polynomials over finite fields
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by Stephen D. Cohen, Gary L. Mullen and Peter Jau-Shyong Shiue PDF
Proc. Amer. Math. Soc. 123 (1995), 2011-2015 Request permission

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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Additional Information
  • © Copyright 1995 American Mathematical Society
  • 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