On the number of branches of a plane curve germ
type: Item
instance of: scholarly article
title: On the number of branches of a plane curve germ
zbMATH Open document ID: 0611.58010
author: Takuo Fukuda
author: K. Aoki
author: Weizhi Sun
published in: Kodai Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: Let f: (\({\mathbb{R}}^ 2,0)\to ({\mathbb{R}},0)\) be a \(C^{\infty}\) function with an isolated critical point 0. The connected components of the intersection \(f^{-1}(0)\) with a small circuit \(S^ 1_{\epsilon}\) at 0 are called branches of \(f^{-1}(0)\). Let \(J_ f\) be \(J_ f=y\frac{\partial f}{\partial x}-x\frac{\partial f}{\partial y}.\) Main theorem. Let \(J_ f\) be a function with isolated critical point at 0. Then the number of branches of \(f^{-1}(0)=2| \deg (f,J_ f)|\), where \(| \deg (f,J_ f)|\) denotes the absolute value of the topological degree of the mapping \((f,J_ f)/\| f,J_ f)\|: S^ 1_{\epsilon}\to S^ 1\). The proof of the main theorem is very simple and based on the following two lemmas. Lemma 2.1. Let f,g: (\({\mathbb{R}}^ 2,0)\to ({\mathbb{R}},0)\) be \(C^{\infty}\) function germs with isolated critical points such that \(f^{-1}(0)\cap g^{-1}(0)=\{0\}\). Suppose that for any two adjacent branches of \(f^{- 1}(0)\), \(g^{-1}(0)\) has an odd number of branches between them. Then the number of branches of \(f^{-1}(0)=2| \deg (f,g)|.\) Lemma 2.2. In conditions of the main theorem \(J_ f^{-1}(0)\) has an odd number of branches between any pair of adjacent branches of \(f^{- 1}(0).\) The last paragraph of the article is devoted to a generalization of the notion of the degree of map germ on the germ: \(({\mathbb{R}}^ n,0)\to ({\mathbb{R}}^ p,0)\), with \(n>p\). The authors define degree as the number of connected components of \(f^{-1}(0)\cap S_{\epsilon}^{n-1}\). The main result of this paragraph is a statement that the homological behavior of f is completely determined by the homology groups of \(f^{- 1}(0)\cap S_{\epsilon}^{n-1}\) and the orientations. This result is a motivation for studying the number of branches.
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 58K99
Mathematics Subject Classification ID: 57R45
zbMATH DE Number: 3988296
zbMATH Keywords: functional germ
zbMATH Keywords: critical point
zbMATH Keywords: branch
zbMATH Keywords: degree of mapping
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138037200
OpenAlex ID: W2049854640
cites work: An algebraic formula for the degree of a \(C^\infty\) map germ. Sur une inegalite a la Minkowski pour les multiplicites
cites work: Q5507495
cites work: Topological invariance of the Milnor number mod 2
DOI: 10.2996/KMJ/1138037200
description: scientific article; zbMATH DE number 3988296