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

We consider differential rings of the form (k[x, y], D), where kis 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. (C) 2021 Published by Elsevier Inc.