On the number of branches of an 1-dimensional semianalytic set
type: Item
instance of: scholarly article
title: On the number of branches of an 1-dimensional semianalytic set
zbMATH Open document ID: 0652.58011
author: Zbigniew Szafraniec
published in: Kodai Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: Let \(F=(F_ 1,...,F_{n-1}): ({\mathbb{R}}^ n,0)\to ({\mathbb{R}}^{n-1},0)\) be a germ of an analytic map and assume that 0 is an isolated singular point of \(F^{-1}(0)=X\). Let b(F) be the number of branches of X-\(\{\) \(0\}\). Let G: (\({\mathbb{R}}^ n,0)\to ({\mathbb{R}},0)\) be another analytic germ and suppose that 0 is isolated in \(G^{-1}(0)\cap X\). Let \(b_+(G,F)\) (resp. \(b_-(G,F))\) be the number of branches of X-\(\{\) \(0\}\) where G is positive (resp. negative). Letting \(\Delta\) be the Jacobean of (G,F), \(H=(\Delta,F)\) and taking suitable representatives of the above map-germs the author proves that \(b_+(G,F)-b_-(G,F)=2 \deg (H).\) Choosing \(G=x^ 2_ 1+...+x^ 2_ n\), a formula for b(F) is obtained. If \(X\cap \{x_ n=0\}=\{0\},\) a formula for the difference between the number of branches of X-\(\{\) \(0\}\) contained in \(\{x_ n>0\}\) and the number of branches contained in \(\{x_ n<0\}\) is also obtained. These two formulas have also been proved in recent papers of Aoki, Fukuda, Sun and Nishimura. Finally, the author considers a more general class of 1- dimensional semianalytic sets, and it is shown how the number of branches of such sets can be computed in terms of the topological degrees of some finite families of map-germs.
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 58K99
zbMATH DE Number: 4064090
zbMATH Keywords: real analytic map
zbMATH Keywords: number of branches
zbMATH Keywords: topological degrees
reviewed by: Hans Brodersen
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138038822
OpenAlex ID: W2094492650
cites work: On the number of branches of a plane curve germ
cites work: An algebraic formula for the topological types of one parameter bifurcation diagrams
cites work: Q3474749
DOI: 10.2996/KMJ/1138038822
description: scientific article; zbMATH DE number 4064090