On simple derivations and the group of polynomial automorphisms commuting with certain derivations

TL;DR

This paper investigates the automorphisms of polynomial rings that commute with simple derivations, showing they are mostly trivial or translations, and confirms a conjecture for the case of two variables.

Contribution

It characterizes automorphisms commuting with simple derivations and proves the conjecture for two-variable polynomial rings.

Findings

Automorphisms commuting with simple derivations are translations under certain conditions.

The subgroup of automorphisms commuting with simple derivations in two variables is trivial.

The conjecture from prior work is affirmed for the case n=2.

Abstract

In the paper, we first study the subgroup of $ K$-automorphisms of $K[x_1,\allowbreak \ldots,x_n]$ which commutes with a simple derivation of $K[x_1,\ldots,x_n]$. We show that the subgroup of $ K$-automorphisms of $K[x_1,\ldots,x_n]$ which commutes with a simple derivation of $K[x_1,\ldots,x_n]$ consists of translations under certain hypothesis. We also prove that the subgroup of $K$-automorphisms of $K[x_1,x_2]$ which commutes with a simple derivation is trivial under the same hypothesis, and the subgroup of $ K$-automorphisms of $K[x_1,\ldots,x_n]$ which commutes with simple Shamsuddin derivations is trivial under the same hypothesis. Then we give an affirmative answer to the conjecture proposed in \cite{13} for $n=2$.