On plane polynomial automorphisms commuting with simple derivations

TL;DR

This paper investigates automorphisms of polynomial rings that commute with simple derivations, proving that such automorphisms are trivial in the simple case and describing their structure for specific derivations.

Contribution

It establishes that automorphisms commuting with simple derivations are trivial, extending previous results and characterizing automorphisms for Shamsuddin type derivations.

Findings

Aut(D)=1 for simple derivations D

Describes Aut(D) for Shamsuddin type derivations

Aut(D)=1 implies D is simple for general derivations

Abstract

We consider the subgroup Aut(D) consisting of automorphisms of K[x,y] commuting with a derivation D, where K is an algebraically closed field of characteristic 0. We prove that if D is simple (i.e. D does not stabilize non-trivial ideals), then Aut(D)=1, in the case where D is of Shamsuddin type this result was proven by R.Baltazar in 2014 (arXiv:1412.8373). Moreover, we describe Aut(D) for Shamsuddin type derivations and deduce that Aut(D)=1 for a general such derivation implies D is simple.