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Complete mappings and Carlitz rank. (English) Zbl 1408.11115
Summary: The well-known Chowla and Zassenhaus conjecture, proven by S. D. Cohen in [Can. Math. Bull. 33, No. 2, 230–243 (1990; Zbl 0722.11060)], states that for any $$d\geq 2$$ and any prime $$p>(d^2-3d+4)^2$$ there is no complete mapping polynomial in $$\mathbb {F}_p[x]$$ of degree $$d$$. For arbitrary finite fields $$\mathbb {F}_q$$, we give a similar result in terms of the Carlitz rank of a permutation polynomial rather than its degree. We prove that if $$n<\lfloor q/2\rfloor$$, then there is no complete mapping in $$\mathbb {F}_q[x]$$ of Carlitz rank $$n^*$$ of small linearity. We also determine how far permutation polynomials $$f$$ of Carlitz rank $$n<\lfloor q/2\rfloor$$ are from being complete, by studying value sets of $$f+x$$. We provide examples of complete mappings if $$n=\lfloor q/2\rfloor$$, which shows that the above bound cannot be improved in general.

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