有限交换环上的多项式置换群

Groups of Polynomial Permutations over Finite Commutative Rings

摘要:

Sophie Frisch描述了$\mathbb{Z}/p^2\mathbb{Z}$上多项式置换群的结构。 张起帆找到$\mathbb{Z}/p^2\mathbb{Z}$上多项式函数与$\mathbb{Z}/p\mathbb{Z}$上多项式函数的3维向量之间的对应关系。 本文先证明在任意有限交换环$R$上，多项式置换群同构于多项式函数形成的$R$-代数的自同构群。 然后我们用张起帆的对应对Sophie Frisch描述给出一个新的证明。

Abstract:

Sophie Frisch characterized the structure of the group of polynomial permutations over $\mathbb{Z}/p^2\mathbb{Z}$. Qifan Zhang found a correspondence between polynomial functions over $\mathbb{Z}/p^2\mathbb{Z}$ and 3-tuples of polynomial functions over $\mathbb{Z}/p\mathbb{Z}$, this paper is giving another proof of [1]. In this paper, we first prove that over any finite commutative ring $R$, the group of polynomial permutations is isomorphic to the automorphism group of the $R$-algebra of the polynomial functions. Then we give an easy proof to the characterization of Sophie Frisch using the correspondence set found by Zhang.