Lower estimate of Milnor number and characterization of isolated homogeneous hypersurface singularities

Abstract

Let f : ‫ރ(‬ n , 0) → ‫,ރ(‬ 0) be a germ of a complex analytic function with an isolated critical point at the origin. Let V = { z ∈ ‫ރ‬ n : f (z) = 0 }. A beautiful theorem of Saito [1971] gives a necessary and sufficient condition for V to be defined by a weighted homogeneous polynomial. It is a natural and important question to characterize (up to a biholomorphic change of coordinates) a homogeneous polynomial with an isolated critical point at the origin. For a two-dimensional isolated hypersurface singularity V , Xu and Yau [1992; 1993] found a coordinate-free characterization for V to be defined by a homogeneous polynomial. and Chen, Lin, Yau, and Zuo [2001] gave necessary and sufficient conditions for 3-and 4-dimensional isolated hypersurface singularities with p g ≥ 0 and p g > 0, respectively. However, it is quite difficult to generalize their methods to give characterization of homogeneous polynomials. In 2005, Yau formulated the Yau Conjecture 1.1: (1) Let µ and ν be the Milnor number and multiplicity of (V, 0), respectively. Then µ ≥ (ν−1) n , and the equality holds if and only if f is a semihomogeneous function. (2) If f is a quasihomogeneous function, then µ = (ν − 1) n if and only if f is a homogeneous polynomial after change of coordinates. In this paper we solve part (1) of Yau Conjecture 1.1 for general n. We introduce a new method, which allows us to solve the part (2) of Yau Conjecture 1.1 for n = 5 and 6. As a result we have shown that for n = 5 or 6, f is a homogeneous polynomial after a biholomorphic change of coordinates if and only if µ = τ = (ν − 1) n . As a by-product we have also proved Yau Conjecture 1.2 in some special cases.

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.…”