On the two-dimensional Jacobian conjecture: Magnus' formula revisited, IV

TL;DR

This paper revisits Magnus' formula in the context of the two-dimensional Jacobian conjecture, focusing on the structure of inner polynomials and their Newton polygons to develop new approaches and conjectures.

Contribution

It introduces a novel approach to the Jacobian conjecture using inner polynomials and Newton polygons, and proves key properties for specific cases.

Findings

The northeastern vertex of the Newton polygon of inner polynomials is within a specific region.

Several conjectures are proposed based on the geometric properties of inner polynomials.

Some conjectures are proved for special cases, advancing understanding of the Jacobian conjecture.

Abstract

Let $(F,G)$ be a Jacobian pair with $d=w\text{-deg}(F)$ and $e=w\text{-deg}(G)$ for some direction $w$. A generalized Magnus' formula approximates $G$ as $\sum_{\gamma\ge 0} c_\gamma F^{\frac{e-\gamma}{d}}$ for some complex numbers $c_\gamma$. We develop an approach to the two-dimensional Jacobian conjecture, aiming to minimize the use of terms corresponding to $\gamma>0$. As an initial step in this approach, we define and study the inner polynomials of $F$ and $G$. The main result of this paper shows that the northeastern vertex of the Newton polygon of each inner polynomial is located within a specific region. As applications of this result, we introduce several conjectures and prove some of them for special cases.