On simple Shamsuddin derivations in two variables

TL;DR

This paper investigates the subgroup of automorphisms commuting with simple derivations in two variables, proving triviality for Shamsuddin simple derivations and exploring properties in the general case.

Contribution

It establishes that the subgroup commuting with a Shamsuddin simple derivation is trivial and analyzes properties of such subgroups for general simple derivations.

Findings

The subgroup is trivial for Shamsuddin simple derivations.

Properties of commuting automorphisms are characterized for general simple derivations.

Provides insights into the structure of automorphisms related to simple derivations.

Abstract

We study the subgroup of $k$-automorphisms of $k[x,y]$ which commute with a simple derivation $D$ of $k[x,y].$ We prove, for example, that this subgroup is trivial when $D$ is a Shamsuddin simple derivation. In the general case of simple derivations, we obtain properties for the elements of this subgroup.