A new proof of a known special case of the Jacobian Conjecture

Abstract

The famous Jacobian Conjecture asks if a morphism $f:K[x,y]\to K[x,y]$ with invertible Jacobian, is invertible ($K$ is a characteristic zero field). A known result says that if $K[f(x),f(y)]\subseteq K[x,y]$ is an integral extension, then $f$ is invertible. We slightly generalize this known result to the following: If for some "good" $\lambda \in K$ (in a sense that will be explained) $mK[x,y]\ne K[x,y]$ for every maximal ideal $m$ of $K[f(x),f(y)][x+\lambda y]$, then $f$ is invertible. We also apply our ideas to the Jacobian Conjecture, without any further assumptions.

This paper has been withdrawn by the author due to an error in the proof of Theorem 2.1 (since Corollary 9 of Wang may not be applicable here)