Jacobian Conjecture in two dimension

TL;DR

This paper investigates properties of Jacobian polynomials in two dimensions, showing they define smooth rational curves with one point at infinity, and discusses implications for the Jacobian conjecture.

Contribution

It establishes a geometric characterization of Jacobian polynomials and links their properties to the Jacobian conjecture using classical theorems.

Findings

Jacobian polynomials define smooth rational curves with one point at infinity

Intersection numbers relate to the geometric genus and branch points

Results support approaches to the Jacobian conjecture in two dimensions

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.