Length Four Polynomial Automorphisms

TL;DR

This paper investigates the structure of length four polynomial automorphisms over a UFD and demonstrates their stable tameness under certain algebraic conditions.

Contribution

It provides a structural analysis of length four automorphisms and proves their stable tameness assuming a specific equality of special linear and elementary groups.

Findings

Length four automorphisms are structurally characterized.

Under certain conditions, all conjugate length four automorphisms are stably tame.

The results connect automorphism properties with algebraic group equalities.

Abstract

We study the structure of length four polynomial automorphisms of $R[X,Y]$ when $R$ is a UFD. The results from this study 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 $n, m \ge 0$ then all length four polynomial automorphisms of $R[X,Y]$ that are conjugates are stably tame.