Permutation and local permutation polynomials of maximum degree

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