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A new approach to permutation polynomials over finite fields. (English) Zbl 1273.11169
Summary: Let \(p\) be a prime and \(q=p^{\kappa }\). We study the permutation properties of the polynomial \(g_{n,q}\in \mathbb F_{p}[x]\) defined by the functional equation \(\sum _{a\in \mathbb F_{q}}(x+a)^{n} = g_{n,q}(x^{q} - x)\). The polynomial \(g_{n,q}\) is a q-ary version of the reversed Dickson polynomial in characteristic 2. We are interested in the parameters \((n,e;q)\) for which \(g_{n,q}\) is a permutation polynomial (PP) of \(\mathbb F_{q^{e}}\). We find several families of such parameters and obtain various necessary conditions on such parameters. Initial results, both theoretical and numerical, indicate that the class \(g_{n,q}\) contains an abundance of PPs over finite fields, many of which are yet to be explained and understood.

MSC:
11T06 Polynomials over finite fields
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