Abstract: Let Fq be the finite field with q elements and Fq [x1, . . . , xn] the ring of polynomials in n variables over Fq . In this paper we consider permutation polynomials and local permutation polynomials over Fq [x1, . . . , xn], which define interesting generalizations of permutations over finite fields. We are able to construct permutation polynomials in Fq [x1, . . . , xn] of maximum degree n(q - 1) -1 and local permutation polynomials in Fq [x1, . . . , xn] of maximum degree n(q - 2) when q > 3, extending previous results
