A Class of simple derivations of polynomial ring $k[x_1,x_2, \ldots ,x_n]$

TL;DR

This paper introduces a new class of simple derivations on polynomial rings over fields of characteristic zero, generalizing previous results and analyzing their automorphism groups.

Contribution

It constructs a family of simple derivations on polynomial rings and characterizes their isotropy groups, extending prior work by D. A. Jordan.

Findings

Proves $d_n$ is a simple derivation for specified parameters.

Shows $d_n(R_n)$ contains no units.

Identifies the isotropy group as conjugate to a subgroup of translations.

Abstract

Let $k$ be a field of characteristic zero. Let $m$ and $\alpha$ be positive integers. For $n\geq 2$, let $R_n=k[x_1,x_2,\dots,x_n]$ with the $k$-derivation $d_n$ given by $d_n=(1-x_1x_2^{\alpha})\partial_{x_1}+x_1^m\partial_{x_2}+x_2\partial_{x_3}+\dots+x_{n-1}\partial_{x_n}$. We prove that for integers $m\geq 2$ and $\alpha \geq 1$, $d_n$ is a simple derivation on $R_n$ and $d_n(R_n)$ contains no units. This generalizes a result of D. A. Jordan. We also show that the isotropy group of $d_n$ is conjugate to a subgroup of translations.