A slight generalization of Keller's theorem

Abstract

The famous Jacobian problem asks: Is a morphism $f:\mathbb{C}[x,y]\to \mathbb{C}[x,y]$ having an invertible Jacobian, invertible? If we add the assumption that $\mathbb{C}(f(x),f(y))=\mathbb{C}(x,y)$, then $f$ is invertible; this result is due to O. H. Keller (1939). We suggest the following slight generalization of Keller's theorem: If $f:\mathbb{C}[x,y]\to \mathbb{C}[x,y]$ is a morphism having an invertible Jacobian, and if there exist $n \geq 1$, $a \in \mathbb{C}(f(x),f(y))^*$ and $b \in \mathbb{C}(f(x),f(y))$ such that $(ax +b)^n \in \mathbb{C}(f(x),f(y))$, then $f$ is invertible. A similar result holds for $\mathbb{C}[x_1,\ldots,x_m]$.