Treffer: 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