On \(p\)-parameter bifurcation of an \(n\)-dimensional function-germ
type: Item
instance of: scholarly article
title: On \(p\)-parameter bifurcation of an \(n\)-dimensional function-germ
zbMATH Open document ID: 0902.32015
DOI: 10.1007/BF02677836
author: Nicolas Dutertre
published in: Manuscripta Mathematica
publication date: 1998-12-03T00:00:00Z
full work available at URL: https://eudml.org/doc/156321
review text: Let \(f,g : (\mathbb{R}^{n+p},0) \to (\mathbb{R},0)\) be analytic germs with \(0\) an isolated singularity of \(f\) and grad \(g(0) \neq 0\). For \(\varepsilon > 0\) and \(| \delta| \) small set \(F_\delta := f^{-1}(\delta) \cap B_\varepsilon(0)\). The author computes the difference of Euler characteristics, \(\chi(F_\delta \cap \{g\geq 0\})-\chi(F_\delta \cap \{g \leq 0\})\), in terms of the degree of an associated map \(H=H(g,f) : B_\varepsilon (0) \to \mathbb{R}^{n+p}\), provided \(H^{-1}(0)=0\). The components of \(H\) are given by \(f\) and certain products of the first partial derivatives of \(f\) and \(g\). The proof uses arguments from Morse theory.
Mathematics Subject Classification ID: 32S05
Mathematics Subject Classification ID: 32S50
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 1138671
zbMATH Keywords: isolated singularity
zbMATH Keywords: Euler characteristic
zbMATH Keywords: local bifurcation
reviewed by: Thomas J. Bartsch
MaRDI profile type: Publication
cites work: Mapping degree formula for 2-parameter bifurcation of function-germs
cites work: Morse Theory. (AM-51)
cites work: A formula for the Euler characteristic of a real algebraic manifold
description: scientific article; zbMATH DE number 1138671