| I study the stable tameness of polynomial automorphisms which was first studied by Smith and then later by Drensky, Yu and Edo. Let R be a domain and F be a polynomial automorphism of a polynomial ring in two variables over R, R[X,Y]. Such an F is said to be tame if it is a product of elementary and affine automorphisms. A longstanding open question asks whether F is stably tame. i.e. Does there exists m ≥ 0 and new variables Xn+1,...,Xn +m such that the extended map ( F, X n+1,...,Xn +m) is tame?;Given such an F that fixes the origin, we can decompose it into elementary automorphisms of special kind of K[ X,Y] where K is the fraction field of R . The minimal number of elementary automorphisms appearing in this decomposition is known as the length of F. If F is of length one or two, it is tame and so the first non trivial case is when F has length three. This length three case was solved by Edo. I have studied length four automorphisms and proved that all length four automorphisms that are commutators are stably tame. I have also discovered an intriguing example of length four which is stably tame. |
