Jacobian Conjecture in two dimension

Abstract

Let $(P,Q)$ be a pair of Jacobian polynomials. We can show that $<P,Q>+l+2g(P)-2=0=<P,[P,Q]>$, where $<f,g>$ is the intersection number of $f, g\in \CC[x, y]$ in the affine plane, $l$ is the number of branch at point at infinity and $g(P)$ is the geometric genus of affine curve defined by $P$. Hence we can show that every Jacobian polynomial defines a smooth rational curve with one point at infinity. It is sufficient to fix the Jacobian conjecture in two dimension by the Abhyankar theorem or the Abhyankar-Moh-Suzuki theorem.

This paper has been withdrawn by the author due to a crucial error in the proposition 1