The two-dimensional Jacobian conjecture and the lower side of the Newton polygon

TL;DR

This paper investigates the two-dimensional Jacobian conjecture, establishing new restrictions on the Newton polygon of polynomial pairs if the conjecture fails, thereby refining the understanding of potential counterexamples.

Contribution

It introduces novel restrictions on the Newton polygon for minimal pairs, narrowing the search for counterexamples to the Jacobian conjecture in two variables.

Findings

Restrictions on the shape of the Newton polygon HH(P)

Elimination of certain corners previously considered possible

Refinement of the structure of potential counterexamples

Abstract

We prove that if the Jacobian Conjecture in two variables is false and (P,Q) is a standard minimal pair, then the Newton polygon HH(P) of P must satisfy several restrictions that had not been found previously. This allows us to discard some of the corners found in [GGV, Remark 7.14] for HH(P), together with some of the infinite families found in [H, Theorem~2.25]