The plane Jacobian conjecture for rational curves

TL;DR

This paper proves that under certain conditions involving rational curves and a nonzero constant Jacobian, the polynomial f generates the entire polynomial ring with some polynomial g.

Contribution

It establishes the plane Jacobian conjecture for cases where the generic fiber is a rational curve, linking rationality to the invertibility of polynomial maps.

Findings

If the generic fiber of f is rational and J(f,g) is a nonzero constant, then K[f,g] = K[x,y]

The result confirms the Jacobian conjecture for a new class of polynomial maps involving rational curves

Provides a criterion connecting rational fibers and invertibility of polynomial maps.

Abstract

Let K be an algebraically closed field of characteristic zero and let f(x,y) be a nonzero polynomial of K[x,y]. We prove that if the generic element of the family $(f-\lambda)\_{\lambda}$ is a rational polynomial, and if the Jacobian J(f,g) is a nonzero constant for some polynomial g in K[x,y], then K[f,g] =K[x,y].