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When does a polynomial over a finite field permute the elements of the field? II. (English) Zbl 0777.11054
A polynomial over a finite field \(F_ q\), \(q\) a power of a prime, is a permutation polynomial (PP) if it induces a 1-1 mapping on \(F_ q\). A brief survey of the main known classes of PP’s is given, following on from the earlier survey [R. Lidl and G. L. Mullen, Am. Math. Mon. 95, 243-246 (1988; Zbl 0653.12010)]. Progress on two of those problems has been particularly significant and is noted (using the notation of the earlier survey):
P8. Chowla and Zassenhaus conjecture: If \(p\) is a sufficiently large prime and \(f(x)\) of degree \(\geq 2\) permutes \(F_ p\), then \(f(x)+ax\) with \(0<a<p\) is not a PP of \(F_ p\).
P9. Carlitz conjecture: For each positive integer \(k\), there is a constant \(C_ k\) such that for each finite field of odd order \(q>C_ k\), there does not exist a PP of degree \(k\) over \(F_ q\).

11T06 Polynomials over finite fields
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