A special case of the two-dimensional Jacobian Conjecture

TL;DR

This paper identifies a specific algebraic condition under which a polynomial endomorphism with invertible Jacobian in two variables is guaranteed to be an automorphism, contributing to the understanding of the Jacobian Conjecture.

Contribution

It establishes a new criterion involving invertible elements in localized polynomial rings that ensures an endomorphism is an automorphism, advancing the study of the Jacobian Conjecture in two dimensions.

Findings

The proposed condition guarantees automorphism status for certain polynomial maps.

It provides a new perspective on Keller's theorem and the Jacobian Conjecture.

The result simplifies the verification of invertibility in specific algebraic settings.

Abstract

Let $f: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$ be a $\mathbb{C}$-algebra endomorphism having an invertible Jacobian. We show that for such $f$, if, in addition, the group of invertible elements of $\mathbb{C}[f(x),f(y),x][1/v] \subset \mathbb{C}(x,y)$ is contained in $\mathbb{C}(f(x),f(y))-0$, then $f$ is an automorphism. Here $v \in \mathbb{C}[f(x),f(y)]-0$ is such that $y = u/v$, with $u \in \mathbb{C}[f(x),f(y),x]-0$. Keller's theorem (in dimension two) follows immediately, since Keller's condition $\mathbb{C}(f(x),f(y))=\mathbb{C}(x,y)$ implies that the group of invertible elements of $\mathbb{C}[f(x),f(y),x][1/v]$ is contained in $\mathbb{C}(x,y)-0 = \mathbb{C}(f(x),f(y))-0$.