Topological invariance of weights for weighted homogeneous singularities
type: Item
instance of: scholarly article
title: Topological invariance of weights for weighted homogeneous singularities
zbMATH Open document ID: 0612.32001
author: Takahashi Nishimura
published in: Kodai Mathematical Journal
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://ynu.repo.nii.ac.jp/
review text: A polynomial \(f(z_ 1,...,z_ n)\) is called weighted homogeneous with weights \((r_ 1,...,r_ n)\in {\mathbb{Q}}^ n\) if \(i_ 1r_ 1+...+i_ nr_ n=1\) for any monomial \(\alpha z_ 1^{i_ 1}...z_ n^{i_ n}\) of f, and non-degenerate if \(\{(\partial f/\partial z_ 1)(z)=...=(\partial f/\partial z_ n)(z)=0\}=\{0\}\) as germs at the origin of \({\mathbb{C}}^ n\). The author gives here a simple proof of the following theorem: Let \(f_ i(z_ 1,z_ 2)\) \((i=1,2)\) be non- degenerate weighted homogeneous polynomials with weights \((r_{i1},r_{i2})\) such that \(0<r_{i1}<r_{i2}<\). If \(({\mathbb{C}}^ 2,f_ 1^{-1}(0))\) is relatively homeomorphic to \(({\mathbb{C}}^ 2,f_ 2^{-1}(0))\), then \((r_{11},r_{12})=(r_{21},r_{22}).\) The method used is more geometric and makes clear the topological structure of non-degenerate weighted homogeneous singularities.
Mathematics Subject Classification ID: 32A15
Mathematics Subject Classification ID: 32A05
zbMATH DE Number: 3989759
zbMATH Keywords: non-degenerate weighted homogeneous polynomials
zbMATH Keywords: topological structure of non-degenerate weighted homogeneous singularities
reviewed by: Arun Kumar Agarwal
MaRDI profile type: Publication
OpenAlex ID: W2064542933
cites work: Topological type of isolated critical points
cites work: Isolated singularities defined by weighted homogeneous polynomials
cites work: Weighted homogeneous polynomials and fundamental groups
cites work: Q4107635
cites work: Quasihomogeneous isolated singularities of hyperplanes.
cites work: Topological types of quasihomogeneous singularities in \(\mathbb{C}^2\)
DOI: 10.2996/KMJ/1138037201
description: scientific article; zbMATH DE number 3989759