On polynomial automorphisms commuting with a simple derivation

TL;DR

This paper investigates the structure of automorphisms commuting with a simple derivation on polynomial rings, revealing bounds on their algebraic group structure and conditions for containing translation subgroups, with specific results in dimensions three and four.

Contribution

It characterizes the automorphism group of a simple derivation, establishing bounds on its dimension, conditions for algebraic group structure, and the presence of translation subgroups, especially in low dimensions.

Findings

The connected component of automorphisms is a unipotent algebraic group of dimension at most n-2.

Automorphism group is algebraic iff it is a connected ind-group.

In dimension three, automorphism groups are either discrete or additive groups of translations.

Abstract

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 $n=4$.