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 (38:2126)****[2]**S. D. Cohen,*Proof of a conjecture of Chowla and Zassenhaus on permutation polynomials*, Canad. Math. Bull.**33**(1990), 230-234. MR**1060378 (91g:11146)****[3]**M. Fried,*On a conjecture of Schur*, Michigan Math. J.**17**(1970), 41-55. MR**0257033 (41:1688)****[4]**L. Kuipers and H. Niederreiter,*Uniform distribution of sequences*, Wiley-Interscience, New York, 1974. MR**0419394 (54:7415)****[5]**R. Lidl and H. Niederreiter,*Finite fields*, Encyclopedia Math. Appl., vol. 20, Addison-Wesley, Reading, MA, 1983 (now distributed by Cambridge Univ. Press). MR**1429394 (97i:11115)**

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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