Un analogue local du théorème de Harnack. (A local analogue of Harnack's theorem)
type: Item
instance of: scholarly article
title: Un analogue local du théorème de Harnack. (A local analogue of Harnack's theorem)
zbMATH Open document ID: 0672.14020
DOI: 10.1007/BF01404674
author: Jean-Jacques Risler
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143474
review text: The result of Harnack referred to in the title of this paper, says that any smooth, real algebraic plane curve of degree d, has at most \((d-1)(d- 2)+1=g+1\) connected components, and for every d there exists a curve with that many connected components. The author shows in this paper, that any irreducible real plane curve singularity, at the origin, with Milnor number \(\mu\), may be deformed into a smooth curve with \(1/2\;\mu\) compact and 1 non-compact, connected components, in the neighborhood of the origin [see also the result of \textit{N. A'Campo}, in Math. Ann. 213, 1-32 (1975; Zbl 0316.14011)]. Since an easy argument shows that for a real plane curve singularity f, with r irreducible components, and Milnor number \(\mu\), the Milnor fibre \(f_{\epsilon}^{-1}(0)\cap B\) has at most \((\mu-r+1)\) compact components (ovals), and exactly r noncompact ones, the result above is, in the irreducible case, the best possible. - In the general situation, the author obtains a best possible result only when f has r distinct tangents. The proofs consist of a succession of intelligent applications of Harnack's construction.
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14N10
zbMATH DE Number: 4099474
zbMATH Keywords: number of components of curve
zbMATH Keywords: Harnack theorem
zbMATH Keywords: Milnor number
zbMATH Keywords: real plane curve singularity
reviewed by: Olav Arnfinn Laudal
MaRDI profile type: Publication
cites work: Le groupe de monodromie du déploiement des singularités isolées de courbes planes. I
cites work: Q3714198
cites work: Singular Points of Complex Hypersurfaces. (AM-61)
cites work: Studies in Equisingularity I Equivalent Singularities of Plane Algebroid Curves
Recommended article: Q4843079
Recommended article: Knotted Milnor fibres
Recommended article: Milnor numbers in deformations of homogeneous singularities
Recommended article: Maximal smoothings of real plane curve singular points
Recommended article: Q4263311
Recommended article: Generalization of Arnold-Viro inequalities for real singular algebraic curves
Recommended article: A bound for the Milnor sum of projective plane curves in terms of GIT
Recommended article: Q3731748
Recommended article: Q4865931
Recommended article: Q4011141
description: scientific article; zbMATH DE number 4099474