Topological invariance of weights for weighted homogeneous singularities
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.
non-degenerate weighted homogeneous polynomials
topological structure of non-degenerate weighted homogeneous singularities
scientific article; zbMATH DE number 3989759