Meromorphic plane curves
type: Item
instance of: scholarly article
title: Meromorphic plane curves
zbMATH Open document ID: 0934.32017
DOI: 10.1007/PL00004682
author: Abdallah Assi
published in: Mathematische Zeitschrift
publication date: 2000-04-11T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 32S15
Mathematics Subject Classification ID: 14E15
zbMATH DE Number: 1336818
zbMATH Keywords: meromorphic plane curve
zbMATH Keywords: intersection multiplicity
zbMATH Keywords: equisingularity
zbMATH Keywords: Jacobian conjecture
reviewed by: Jiancheng Zou
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1336818