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 and are integral polynomials of degree and *p* is a prime exceeding for which *f* and *g* are both permutation polynomials of the finite field , then their difference cannot be such that 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 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.

**[1]**S. Chowla and H. Zassenhaus,*Some conjectures concerning finite fields*, Norske Vid. Selsk. Forh. (Trondheim)**41**(1968), 34–35. MR**0233805****[2]**Stephen D. Cohen,*Proof of a conjecture of Chowla and Zassenhaus on permutation polynomials*, Canad. Math. Bull.**33**(1990), no. 2, 230–234. MR**1060378**, https://doi.org/10.4153/CMB-1990-036-3**[3]**Michael Fried,*On a conjecture of Schur*, Michigan Math. J.**17**(1970), 41–55. MR**0257033****[4]**L. Kuipers and H. Niederreiter,*Uniform distribution of sequences*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR**0419394****[5]**Rudolf Lidl and Harald Niederreiter,*Finite fields*, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 20, Cambridge University Press, Cambridge, 1997. With a foreword by P. M. Cohn. MR**1429394**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1995-1196163-1

Keywords:
Finite field,
permutation polynomial

Article copyright:
© Copyright 1995
American Mathematical Society