Polydegree properties of polynomial automorphisms

The group of automorphisms of the affine plane has the structure of an amalgamated free product of the triangular and affine subgroups. This leads us to the polydegree: the unique sequence of degrees of the triangular automorphisms in the amalgamated free product decomposition of the automorphism. This group is also endowed with the structure of an infinite dimensional algebraic variety. The interaction between these two structures is not well understood. We use the Valuation Criterion, due to Furter, to study the interaction between these structures; in particular, it allows us to see if an automorphism in G is also in the closure of $Gd$, where d is the polydegree sequence. In this paper, we will discuss a method that gives us new results concerning a class of automorphisms with a polydegree of length one being contained in the closure (in the Zariski topology) of a set of automorphisms with a polydegree of length 2.