On the Automorphism Group of a Polynomial Differential Ring in Two Variables

TL;DR

This paper investigates the automorphism groups of polynomial differential rings in two variables over algebraically closed fields, providing criteria for their algebraic group structure and classifying these rings up to conjugation.

Contribution

It offers new criteria to determine when the automorphism group is algebraic and classifies polynomial differential rings in two variables up to conjugation.

Findings

Criteria for automorphism groups to be algebraic

Classification of differential rings up to conjugation

Most cases lead to a primary classification

Abstract

We consider differential rings of the form (K[x; y];D), where K is an algebraically closed field of characteristic zero and D : K[x; y] \to K[x; y] is a K-derivation. We study the Automorphism Group of such a ring and give criteria for deciding whether that group is an algebraic group. In most cases, from that study we deduce a primary classification of this type of differential ring up to conjugation with a polynomial automorphism.