Meromorphic plane curves
Let \(f(x,y)\) be a reduced polynomial of \(C[x,y]\) and consider the family \((f_{\lambda})_{\lambda \in C}\) in \(C[x,y]\), where for any \(\lambda \in C, f_{\lambda} = f- \lambda\) is a reduced polynomial of \(C[x,y]\). Let \(f_x, f_y\) be the derivatives of \(f\) with respect to \(x\) and \(y\) and assume that \(f\) has only one place at infinity. S. S. Abhyankar and T. T. Moh proved that if the intersection multiplicity int(\(f_x, f_y\)) of \(f_x\) and \(f_y\) in \(C^2\) was 0, then there was an automorphism \(\sigma\) of \(C^2\) such that \(f\circ \sigma\) is a coordinate of \(C^2\). The author of this paper generalizes Abhyankar-Moh's result to the families \((f_{\lambda})_{\lambda \in C}\) having only one irregular value and revisites the result under a constructive point of view, by including it into an algebraic classification of plane curves having only one place at infinity. By using a result of Abhyankar the author also proves that if the family \((f_{\lambda})_{\lambda \in C}\) has only one irregular value, then \(J(f, g) = f_x g_y -f_y g_x \not\in C^*\) for any other polynomial \(g \in C[x, y]\). This answers in this particular case the plane Jacobian conjecture.
scientific article; zbMATH DE number 1336818