A new proof of a known special case of the Jacobian Conjecture

TL;DR

This paper presents a new proof for a special case of the Jacobian Conjecture, extending known results and applying novel ideas to the general conjecture without additional assumptions.

Contribution

It generalizes a known invertibility criterion for polynomial maps with invertible Jacobian and applies these ideas to the Jacobian Conjecture.

Findings

Established a new invertibility criterion under a generalized condition.

Extended known results to a broader class of polynomial maps.

Applied methods to the Jacobian Conjecture without extra assumptions.

Abstract

The famous Jacobian Conjecture asks if a morphism $f:K[x,y]\to K[x,y]$ with invertible Jacobian, is invertible ($K$ is a characteristic zero field). A known result says that if $K[f(x),f(y)] \subseteq K[x,y]$ is an integral extension, then $f$ is invertible. We slightly generalize this known result to the following: If for some "good" $\lambda \in K$ (in a sense that will be explained) $m K[x,y] \neq K[x,y]$ for every maximal ideal $m$ of $K[f(x),f(y)][x+ \lambda y]$, then $f$ is invertible. We also apply our ideas to the Jacobian Conjecture, without any further assumptions.