On the Newton polytope of a Jacobian pair

TL;DR

This paper introduces a Newton polytope associated with potential counterexamples to the Jacobian conjecture, providing sharper degree estimates and a new proof for a specific case of the conjecture.

Contribution

It defines a Newton polytope for Jacobian pairs and uses it to improve degree bounds and offer a novel proof for Abhyankar's two characteristic pair case.

Findings

Sharper estimate for the geometric degree of polynomial mappings

New proof of Abhyankar's two characteristic pair case

Introduction of a Newton polytope related to Jacobian pairs

Abstract

The Newton polytope related to a ``minimal" counterexample to the Jacobian conjecture is introduced and described. This description allows to obtain a sharper estimate for the geometric degree of the polynomial mapping given by a Jacobian pair and to give a new proof of the Abhyankar's two characteristic pair case.