Let f : (ℂn,0) → (ℂ,0) be a
germ of a complex analytic function with an isolated critical point at the origin. Let
V = {z ∈ ℂn: 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. Lin and Yau [2004] and Chen, Lin, Yau, and Zuo [2001] gave
necessary and sufficient conditions for 3- and 4-dimensional isolated hypersurface
singularities with pg≥ 0 and pg> 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.
