Complex algebraic plane curves via the Poincaré-Hopf formula. I. Parametric lines
type: Item
instance of: scholarly article
title: Complex algebraic plane curves via the Poincaré-Hopf formula. I. Parametric lines
zbMATH Open document ID: 1153.14026
author: Henryk Żołądek
author: Maciej Borodzik
published in: Pacific Journal of Mathematics
publication date: 2008-11-14T00:00:00Z
review text: There do not exist simple rules to determine the topology of a plane algebraic curve, whether given in implicit or parametric form. This paper presents a new approach to this problem. The authors consider parametric polynomial curves; they use a Poincaré-Hopf formula applied to a suitable Hamiltonian vector field to calculate the number of double points and the Milnor numbers of singular points of the curve and study finite-dimensional spaces of curves with given asymptotics at infinity. This method is then applied to classify parametric lines (i.e. plane curves \(\xi: {\mathbb C} \rightarrow {\mathbb C}^2, \xi(t)=((\phi(t), \phi(t)))\) where \((\phi, \phi)\) are polynomials) with one self-intersection.
reviewed by: Caterina Cumino
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 32S05
Mathematics Subject Classification ID: 14H15
Mathematics Subject Classification ID: 14H20
zbMATH DE Number: 5366196
zbMATH Keywords: affine algebraic curve
zbMATH Keywords: Puiseux expansion
MaRDI profile type: Publication
full work available at URL: pjm.2007.229.307
OpenAlex ID: W2121276628
DOI: 10.2140/PJM.2007.229.307
description: scientific article; zbMATH DE number 5366196