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The difference between permutation polynomials over finite fields. (English) Zbl 0827.11074
The authors generalize the recent solution, by S. D. Cohen, of the 1968 Chowla-Zassenhaus conjecture, which says that, if $$f(x)$$, $$g(x)$$ are integral polynomials of degree $$n\geq 2$$ and a prime $$p$$ exceeds $$(n^2- 3n+ 4)^2$$, for which $$f$$ and $$g$$ are both permutation polynomials over the finite field $$\mathbb{F}_p$$, then their difference $$h= f-g$$ cannot be such that $$h(x)= cx$$ for some integer $$c$$, not divisible by $$p$$. The authors of the paper under review prove that, if $$h$$ is not constant in $$\mathbb{F}_p$$ and $$t$$ is the degree of $$h$$, then $$t\geq 3n/5$$, and $$\text{gcd} (t,n)=1$$ whenever $$t\leq n-3$$. The authors observe that their result provides some kind of measure of the isolation of permutation polynomials of the same degree over large prime fields.
Reviewer: R.Mollin (Calgary)

##### MSC:
 11T06 Polynomials over finite fields 12E20 Finite fields (field-theoretic aspects)
##### Keywords:
finite field; permutation polynomials
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##### References:
 [1] S. Chowla and H. Zassenhaus, Some conjectures concerning finite fields, Norske Vid. Selsk. Forh. (Trondheim) 41 (1968), 34 – 35. · Zbl 0186.09203 [2] Stephen D. Cohen, Proof of a conjecture of Chowla and Zassenhaus on permutation polynomials, Canad. Math. Bull. 33 (1990), no. 2, 230 – 234. · Zbl 0722.11060 · doi:10.4153/CMB-1990-036-3 · doi.org [3] Michael Fried, On a conjecture of Schur, Michigan Math. J. 17 (1970), 41 – 55. · Zbl 0169.37702 [4] L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. · Zbl 0281.10001 [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. · Zbl 0866.11069
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