Permutation and local permutation polynomials of maximum degree

Abstract

Let \(\mathbb {F}_q\) be the finite field with q elements and \(\mathbb {F}_q[x_1,\ldots , x_n]\) the ring of polynomials in n variables over \(\mathbb {F}_q\). In this paper we consider permutation polynomials and local permutation polynomials over \(\mathbb {F}_q[x_1,\ldots , x_n]\), which define interesting generalizations of permutations over finite fields. We are able to construct permutation polynomials in \(\mathbb {F}_q[x_1,\ldots , x_n]\) of maximum degree \(n(q-1)-1\) and local permutation polynomials in \(\mathbb {F}_q[x_1,\ldots , x_n]\) of maximum degree \(n(q-2)\) when \(q>3\), extending previous results.