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Permutation binomials over finite fields. (English) Zbl 0817.11056
For \(q\) a prime power let \(\mathbb{F}_ q\) denote the finite field of order \(q\). A polynomial \(f\in \mathbb{F}_ q [x]\) is a permutation polynomial (PP) if \(f\) induces a 1-1 mapping of \(\mathbb{F}_ q\) onto itself. In a previous paper [Acta Math. Sin., New Ser. 3, 1-5 (1987; Zbl 0636.12011)], the author proved that for odd \(q\), the polynomial \(x^{1+ (q-1) /3}+ ax\) with \(a\neq 0\in \mathbb{F}_ q\) is not a PP over \(\mathbb{F}_{q^ r}\) with \(r\geq 2\). In the present paper the author proves that the same conclusion holds in the case when \(q\) is even. For related results see the author and R. Lidl [Monatsh. Math. 112, 149-163 (1991; Zbl 0737.11040)].

11T06 Polynomials over finite fields
Full Text: DOI
[1] Ambos-Spies, K. and Lerman, M.,Lattice embeddings into the recursively enumerable degrees, J. Symbolic Logic,51 (1986), 257–272. · Zbl 0633.03037 · doi:10.1017/S0022481200031133
[2] Ding, D.,The distribution of the generic recursively enumerable degrees, Arch. for Math. Logic;32 (1992), 113–135. · Zbl 0790.03043 · doi:10.1007/BF01269953
[3] Fejer, P. A.,Branching degrees above low degrees, Trans. Amer. Math. Soc.,273 (1982), 157–180. · Zbl 0498.03028 · doi:10.1090/S0002-9947-1982-0664035-X
[4] Fejer, P. A.,The density of the non-branching degrees, Ann. Pure Appl. Logic,24 (1983), 113–130. · Zbl 0521.03026 · doi:10.1016/0168-0072(83)90028-3
[5] Friedberg, R.M.,Two recursively enumerable sets of incomparable degrees of unsolvability, Proc. Natl. Acad. Sci. USA,43 (1957), 236–238. · Zbl 0080.24302 · doi:10.1073/pnas.43.2.236
[6] Ingrassia, M.,P-genericity for recursively enumerable sets, Ph. D. Dissertation, University of Illinois at Urbana-Champaign, 1981.
[7] Jockusch, C. G.,Degrees of generic sets, In: Drake and Wainer [1980], 110–139. · Zbl 0457.03042
[8] Jockusch, C. G.,Genericity for recursively enumerable sets, In: Ebbinghaus, Müller, and Sacks [1985], 203–232. · Zbl 0594.03024
[9] Robinson, R. W.,Interpolation and embedding in the recursively enumerable degrees, Ann. of Math.,93 (1971), 285–314. · Zbl 0259.02033 · doi:10.2307/1970776
[10] Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967. · Zbl 0183.01401
[11] Sacks, G. E.,The recursively enumerable degrees are dense, Ann. of Math.,80 (1964), 300–312. · Zbl 0135.00702 · doi:10.2307/1970393
[12] Soare, R. I., Recursively Enumerable sets and Degrees, Springer-Verlag, Berlin, Heidelberg, New York, 1987. · Zbl 0667.03030
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