Chains of the Eggers tree and polar curves
type: Item
instance of: scholarly article
title: Chains of the Eggers tree and polar curves
zbMATH Open document ID: 1057.14032
DOI: 10.4171/RMI/367
published in: Revista Matemática Iberoamericana
publication date: 2004-10-25T00:00:00Z
full work available at URL: https://eudml.org/doc/39583
review text: Let us consider a curve \(C \subset {\mathbb C}^{2}\) defined by an holomorphic function \(f(x,y)=0\). We may assume that \(f(0,0)=0\). Its polar curve has equation \(P \frac{\partial f}{\partial x} +Q \frac{\partial f}{\partial y} + Rf=0\), where \(P\) and \(Q\) are functions, at least one of them non-vanishing at \((0,0)\). Up to a correct choice of the function \(f\), we have that its polar curve is defined by \(\frac{\partial f}{\partial y}=0\). Let us assume we have this correct choice for \(f\). We say that the polar curve is transverse to \(C\) if the line \(x=0\) is transverse to \(C\). In the transversal situation, the point \((0,0)\) is a singular point of the polar curve with multiplicity one less than for \(C\). One would like to say something more about the type of singularity of the polar curve if one knows the type of singularity of \(C\). A general statement is given by \textit{M. Merle} [Invent. Math. 41, 103--111 (1977; Zbl 0371.14003)] in the case that \(C\) consists of a single branch. In the case of several branches a general theorem is given by \textit{E. García Barroso} [Invariants des singularités de courbes planes et courbure des fibres de Milnor. Doctoral thesis, La Laguna (1996)], but the known proofs involve messy combinatorial arguments. In this paper the author reduces these to easy calculations by introducing additional structure on the Eggers tree.
Mathematics Subject Classification ID: 14H20
zbMATH DE Number: 2109615
zbMATH Keywords: exponent of contact
zbMATH Keywords: decomposition
author: C. T. C. Wall
reviewed by: Rubén A. Hidalgo
MaRDI profile type: Publication
cites work: Q3715247
cites work: On the singularities of polar curves
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cites work: Q4285672
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cites work: On the polar curves of a reduced plane curve
cites work: On analytic function germs of two complex variables
cites work: Courbes polaires et topologie des courbes planes
cites work: Invariants polaires des courbes planes
OpenAlex ID: W2024596415
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description: scientific article; zbMATH DE number 2109615