  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.

Algebraic Geometry;Commutative Algebra