  We study the interaction between two structures on the group of polynomial
automorphisms of the affine plane: its structure as an amalgamated free product
and as an infinite-dimensional algebraic variety. We introduce a new
conjecture, and show how it implies the Polydegree Conjecture. As the new
conjecture is an ideal membership question, this shows that the Polydegree
Conjecture is algorithmically decidable. We further describe how this approach
provides a unified and shorter method of recovering existing results of Edo and
Furter.
