Complete characterization of isolated homogeneous hypersurface singularities

Abstract

Dedicated to Professor Michael Artin on the occasion of his 80th birthday Let X be a nonsingular projective variety in ‫ސރ‬ n−1. Then the cone over X in ‫ރ‬ n is an affine variety V with an isolated singularity at the origin. It is a very natural and important question to ask when an affine variety with an isolated singularity at the origin is a cone over nonsingular projective variety. This problem is very hard in general. In this paper we shall treat the hypersurface case. Given a function f with an isolated singularity at the origin, we can ask whether f is a weighted homogeneous polynomial or a homogeneous polynomial after a biholomorphic change of coordinates. The former question was answered in a celebrated 1971 paper by Saito. However, the latter question had remained open for 40 years until Xu and Yau solved it for f with three variables. Recently, Yau and Zuo solved it for f with up to six variables. However, the methods they used are hard to generalize. In this paper, we solve the latter question for general n completely; i.e., we show that f is a homogeneous polynomial after a biholomorphic change of coordinates if and only if µ = τ = (ν − 1) n , where µ, τ and ν are the Milnor number, Tjurina number and multiplicity of the singularity respectively. We also prove that there are at most µ 1/n + 1 multiplicities within the same topological type of the isolated hypersurface singularity, while the famous Zariski multiplicity problem asserts that there is only one multiplicity.

Proposing a New Theorem to Determine If an Algebraic Polynomial Is Nonnegative in an Interval

“…[4]); characterization of weighted homogeneity (cf. [5][6][7]); Bernoulli Polynomials (cf. [8]); PDE (cf.…”