Length Four Polynomial Automorphisms

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.