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Proof of a conjecture of Chowla and Zassenhaus on permutation polynomials. (English) Zbl 0722.11060
Let $$f(x)$$ be a polynomial with integral coefficients and degree $$n\geq 2$$. Then it is shown that for any prime $$p>(n^2-3n+4)^2$$ for which $$f$$, considered modulo $$p$$, is a permutation polynomial of degree $$n$$ of the finite field $$\mathbb F_p$$, there is no integer $$c$$ with $$1\leq c<p$$ for which $$f(x)+cx$$ is also a permutation polynomial of $$\mathbb F_p$$. This settles a conjecture of S. Chowla and H. Zassenhaus [Norske Vid. Selsk. Forhdl. 41, 34–35 (1968; Zbl 0186.09203)]. The proof makes clever use of the Lang-Weil theorem and of a result from the work of M. Fried [Mich. Math. J. 17, 41–55 (1970; Zbl 0169.37702)] on the resolution of Schur’s conjecture.

##### MSC:
 11T06 Polynomials over finite fields
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