The two-dimensional Jacobian Conjecture and unique factorization

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 $u\in \mathbb{C}[F(x),F(y),x]$.