A special case of the two-dimensional Jacobian Conjecture

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 $\mathbb{C}(x,y)-0 = \mathbb{C}(f(x),f(y))-0$.