Lower estimate of Milnor number and characterization of isolated homogeneous hypersurface singularities
From authors' abstract: A classical result of K. Saito characterizes weighted homogeneous germs of holomorphic germs with an isolated singularity (up to change of coordinates). The present paper studies the analogous problem for a homogeneous isolated singularity germ \(f : (\mathbb C^{n},0) \rightarrow (\mathbb C,0)\). In 2005, S. Yau formulates the following conjecture: Let \(\mu\) and \(\nu\) be respectively the Milnor number and the multiplicity of \(f\). Then: 1) \(\mu \geq (\nu - 1)^n\) and the equality holds if and only if f is a semihomogeneous polynomial after a change of coordinates; 2) If f is a quasihomogeneous function then \(\mu = (\nu -1)^n\) if and only if f is homogeneous after a suitable change of coordinates. The authors prove here the point 1) for any \(n\) and the point 2) for the values \(n=5\) and \(n=6\).
scientific article; zbMATH DE number 6121954