设 $\mathbb{Z}$ 表整数集, $p$ 为一给定奇素数, $k$ 为一正整数. 张起帆(1995)得到了一类 模 $p$ 奇异的二元多项式成为剩余类环 $\mathbb{Z}/p^k\mathbb{Z}$ 上的置换多项式的一个充要条件, 胡永忠(2001)将张起帆的这一结果的充分条件推广到了一般 $n$ 元的情形. 本文得到了一类模 $p$ 奇异的 $n$ 元多项式成为剩余类环 $\mathbb{Z}/p^k\mathbb{Z}$ 上的置换多项式的一个充要 条件, 所得结论是对张文$(n=2)$和胡文$(n>2)$的自然推广和改进.

Let $\mathbb{Z}$ be a ring of integers, $p$ be an odd prime and $k$ be a positive integer. Zhang Qifan${}^{[1]}$ obtained a sufficient and necessary condition that some singular polynomials mod $p$ in two variables over $\mathbb{Z}$ should be permutation polynomials over the residue class ring $\mathbb{Z}/p^k\mathbb{Z}$. And Hu Yongzhong${}^{[2]}$ generalized the sufficient condition of this result to the case where the polynomials are in $n$ variables. In this note the authors obtain a sufficient and necessary condition that some singular polynomials mod $p$ in $n$ variables over $\mathbb{Z}$ should be permutation polynomials over $\mathbb{Z}/p^k\mathbb{Z}$, which is a natural generalization and an improvement of the cases [1]($n=2$) and [2]($n>2$).