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On some permutation polynomials over finite fields. (English) Zbl 1092.11046
Let \(q\) be a prime power of the form \(q=7s+1\). The authors classify the permutation binomials \(P(X)=X^r(1+X^{es})\), \(1\leq e\leq 6\), in terms of the sequence \((a_n)\) defined by \(a_n=a_{n-1}+2a_{n-2}-a_{n-3}\), \(n\geq 3\), with initial values \(a_0=3\), \(a_1=1\), and \(a_2=5\). For prime powers of the form \(q=3s+1\) and \(q=5s+1\) the permutation behaviour of binomials was studied by L. Wang [Finite Fields Appl. 8, No. 3, 311–322 (2002; Zbl 1044.11103)]. For results on arbitrary permutation binomials see the authors’ paper [Proc. Am. Math. Soc. 134, No. 1, 15–22 (2006; Zbl 1137.11355)].

11T06 Polynomials over finite fields
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