The two-dimensional Jacobian Conjecture and unique factorization

TL;DR

This paper investigates the two-dimensional Jacobian Conjecture, showing that under certain conditions involving prime factorizations in an associated ring, an endomorphism with invertible Jacobian is an automorphism.

Contribution

It introduces a new criterion involving prime element factorizations in an extended ring to establish when a Jacobian invertible endomorphism is an automorphism.

Findings

Established a condition linking prime factorizations to automorphism status.

Extended the understanding of the Jacobian Conjecture in two dimensions.

Provided a new approach to verify invertibility of polynomial maps.

Abstract

The two-dimensional Jacobian Conjecture says that a $\mathbb{C}$-algebra endomorphism $F:\mathbb{C}[x,y] \to \mathbb{C}[x,y]$ that has an invertible Jacobian is an automorphism. We show that if a $\mathbb{C}$-algebra endomorphism $F:\mathbb{C}[x,y] \to \mathbb{C}[x,y]$ has an invertible Jacobian and if $v \in \mathbb{C}[F(x),F(y),x]$ is a product of prime elements of $\mathbb{C}[F(x),F(y),x]$, then $F$ is an automorphism, where $v$ is such that $y = u/v$, where $u \in \mathbb{C}[F(x),F(y),x]$.