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A simple proof of a conjecture about complete mapping over finite fields. (Chinese. English summary) Zbl 1007.11075
Summary: In [J. Aust. Math. Soc., Ser. A 33, 197-212 (1982; Zbl 0495.12018)] H. Niederreiter and K. H. Robinson conjectured that if \(F=F_q\) is a finite field of \(q\) elements with \(\operatorname {ch}F=2\), \(q>3\), and \(f(x)\) is a complete mapping polynomial of \(F\) (i.e. both \(f(x)\) and \(f(x)+x\) are permutation polynomials of \(F\)), then the reduction of \(f(x) \operatorname {mod} (x^q-x)\) has degree \(\leq q-3\). In [J. Aust. Math. Soc., Ser. A 41, 336-338 (1986; Zbl 0607.12009)], D. Wan proved this conjecture to be true. In this note, by an application of the 2-adic number field, the authors give a simple proof for this conjecture.

11T06 Polynomials over finite fields
12E20 Finite fields (field-theoretic aspects)