标题： 关于与简单导子交换的多项式自同构 显示英文标题

标题： On polynomial automorphisms commuting with a simple derivation

摘要： 设$D$为多项式环$\mathbb{k}[x_1,\dots,x_n]$的一个简单导子，其中$\mathbb{k}$是特征为零的代数闭域，并记$\operatorname{Aut}(D)\subset\operatorname{Aut}(\mathbb{k}[x_1,\dots,x_n])$为与$D$交换的$\mathbb{k}$-自同构的子群。 我们证明通过单位的$\operatorname{Aut}(D)$的连通分支是一个幂零代数群，其维数至多为$n-2$，此界是紧的。 此外，$\operatorname{Aut}(D)$是一个代数群当且仅当它是一个连通的ind-群。 给定一个简单的导子$D$，我们刻画当$\operatorname{Aut}(D)$包含一个平移的正规子群时的情形。 作为我们技术的应用，我们证明如果$n=3$，则$\operatorname{Aut}(D)$要么是一个离散群，要么与通过平移作用的加法群同构，并对$n=4$的情况提供一些见解。 显示英文摘要

摘要： Let $D$ be a simple derivation of the polynomial ring $\mathbb{k}[x_1,\dots,x_n]$, where $\mathbb{k}$ is an algebraically closed field of characteristic zero, and denote by $\operatorname{Aut}(D)\subset\operatorname{Aut}(\mathbb{k}[x_1,\dots,x_n])$ the subgroup of $\mathbb{k}$-automorphisms commuting with $D$. We show that the connected component of $\operatorname{Aut}(D)$ passing through the identity is a unipotent algebraic group of dimension at most $n-2$, this bound being sharp. Moreover, $\operatorname{Aut}(D)$ is an algebraic group if and only if it is a connected ind-group. Given a simple derivation $D$, we characterize when $\operatorname{Aut}(D)$ contains a normal subgroup of translations. As an application of our techniques we show that if $n=3$, then either $\operatorname{Aut}(D)$ is a discrete group or it is isomorphic to the additive group acting by translations, and give some insight on the case $n=4$.