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On a problem of Niederreiter and Robinson about finite fields. (English) Zbl 0607.12009
A polynomial f(x) over a finite field $${\mathbb{F}}_ q$$ is called a permutation polynomial of $${\mathbb{F}}_ q$$ if the mapping induced by f(x) is a permutation of $${\mathbb{F}}_ q$$. If both f(x) and $$f(x)+x$$ are permutation polynomials of $${\mathbb{F}}_ q$$, then f(x) is called a complete mapping polynomial of $${\mathbb{F}}_ q$$. The degree of the reduction of f(x) modulo $$x^ q-x$$ is called the reduced degree of f(x).
The reviewer and K. H. Robinson [ibid. 33, 197-212 (1982; Zbl 0495.12018)] have shown that for a finite field $${\mathbb{F}}_ q$$ with odd order $$q>3$$, any complete mapping polynomial has reduced degree at most q-3. In the present paper this result is proved for finite fields $${\mathbb{F}}_ q$$ of even order $$q>3$$. The proof is based on a clever extension of the method for odd q.
Reviewer: H.Niederreiter

MSC:
 11T06 Polynomials over finite fields