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.

翻译：设$F_q$为含$q$个元素的有限域，$F_q[x_1,\ldots, x_n]$为$F_q$上$n$元多项式环。本文研究$F_q[x_1,\ldots, x_n]$上的置换多项式与局部置换多项式，它们刻画了有限域上置换的有趣推广形式。当$q>3$时，我们能够构造出$F_q[x_1,\ldots, x_n]$中最大度数为$n(q-1)-1$的置换多项式，以及最大度数为$n(q-2)$的局部置换多项式，从而推广了先前的结果。