An algorithmic approach to the Polydegree Conjecture for plane polynomial automorphisms

TL;DR

This paper introduces a new conjecture linking algebraic structures and algorithmic methods to address the Polydegree Conjecture for polynomial automorphisms, providing a decidable approach and unifying previous results.

Contribution

It proposes a novel conjecture that reduces the Polydegree Conjecture to an ideal membership problem, enabling an algorithmic decision process and simplifying existing proofs.

Findings

The new conjecture implies the Polydegree Conjecture.

The approach shows the conjecture is algorithmically decidable.

Provides a unified method to recover known results.

Abstract

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.