Some Stably Tame Polynomial Automorphisms

TL;DR

This paper investigates the structure of length three polynomial automorphisms over a UFD and demonstrates their stable tameness under certain algebraic conditions involving special linear and elementary groups.

Contribution

It establishes that under specific conditions, all length three polynomial automorphisms over a UFD are stably tame, linking group-theoretic properties to automorphism structure.

Findings

Length three automorphisms are stably tame under certain conditions.

Conditions involve equality of special linear and elementary groups over polynomial rings.

Results connect algebraic group properties to polynomial automorphism structure.

Abstract

We study the structure of length three polynomial automorphisms of $R[X,Y]$ when $R$ is a UFD. These results are used to prove that if $\text{SL}_m(R[X_1,X_2,..., X_n]) = \text{E}_m(R[X_1,X_2,..., X_n])$ for all $n,\ge 0$ and for all $m \ge 3$ then all length three polynomial automorphisms of $R[X,Y]$ are stably tame.