The plane Jacobian conjecture for rational curves

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].