Plane Jacobian conjecture for simple polynomials

TL;DR

This paper proves that polynomial maps with constant Jacobian determinant in two variables are invertible if the polynomial component is simple, characterized by its extension to a compactification with certain degree conditions.

Contribution

It establishes a new criterion for invertibility of polynomial maps based on the simplicity of one component and its behavior at infinity.

Findings

Polynomial maps with constant Jacobian are invertible when one component is simple.

The simplicity condition involves the extension to a compactification and degree restrictions.

This provides a new approach to the Jacobian conjecture in two variables.

Abstract

A non-zero constant Jacobian polynomial map $F=(P,Q):\mathbb{C}^2 \longrightarrow \mathbb{C}^2$ has a polynomial inverse if the component $P$ is a simple polynomial, i.e. if, when $P$ extended to a morphism $p:X\longrightarrow \mathbb{P}^1$ of a compactification $X$ of $\mathbb{C}^2$, the restriction of $p$ to each irreducible component $C$ of the compactification divisor $D = X-\mathbb{C}^2$ is either degree 0 or 1.