Tame and wild automorphisms of differential polynomial algebras of rank 2

TL;DR

This paper investigates the structure of automorphism groups of differential polynomial algebras in two variables, proving the tame automorphisms form a free product with amalgamation and providing an example of a wild automorphism for multiple derivations.

Contribution

It establishes the algebraic structure of tame automorphisms and constructs the first known example of a wild automorphism in this context.

Findings

Tame automorphism group is a free product with amalgamation.

Existence of wild automorphisms for multiple derivations.

Structural insights into automorphism groups of differential polynomial algebras.

Abstract

It is proved that the tame automorphism group of a differential polynomial algebra $k\{x,y\}$ over a field $k$ of characteristic $0$ in two variables $x,y$ with $m$ commuting derivations $\delta_1, \ldots, \delta_m$ is a free product with amalgamation. An example of a wild automorphism of the algebra $k\{x,y\}$ in the case of $m\geq 2$ derivations is constructed.