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