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 $F_q$ 为含有 $q$ 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，这两类多项式定义了有限域上置换的有趣推广。我们能够在 $F_q[x_1,\ldots, x_n]$ 中构造次数为 $n(q-1)-1$ 的置换多项式，并在 $q>3$ 时构造次数为 $n(q-2)$ 的局部置换多项式，从而推广了先前的结果。