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Further results on permutation polynomials of the form \((x^{p^m} - x + \delta)^s + L(x)\) over \(\mathbb{F}_{p^{2 m}}\). (English) Zbl 1352.05009
Summary: In this paper, some classes of permutation polynomials of the form \((x^{p^m} - x + \delta)^s + L(x)\) over the finite field \(\mathbb{F}_{p^{2 m}}\) are investigated by determining the number of solutions of certain equations, where \(L(x) = x\) or \(x^{p^m} + x\). More precisely, for an integer \(s\) satisfying \(s(p^m + 1) \equiv p^m + 1 \pmod {p^{2m}-1}\), we give four classes of permutation polynomials of the form \((x^{2^m} + x + \delta)^s + x\) over \(\mathbb{F}_{2^{2 m}}\), and five classes of permutation polynomials of the form \((x^{3^m} - x + \delta)^s + x^{3^m} + x\) over \(\mathbb{F}_{3^{2 m}}\), respectively.

MSC:
05A05 Permutations, words, matrices
11T06 Polynomials over finite fields
11T55 Arithmetic theory of polynomial rings over finite fields
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