On the Automorphism Group of a polynomial differential ring in two variables

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]→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.