Coordinate-free characterization of homogeneous polynomials with isolated singularities

Abstract

The Durfee conjecture, proposed in 1978, relates two important invariants of isolated hypersurface singularities by a famous inequality; however, the inequality in this conjecture is not sharp. In 1995, Yau announced his conjecture which proposed a sharp inequality. The Yau conjecture characterizes the conditions under which an affine hypersurface with an isolated singularity at the origin is a cone over a nonsingular projective hypersurface; in other words, the conjecture gives a coordinate-free characterization of when a convergent power series is a homogeneous polynomial after a biholomorphic change of variables. In this paper, we have proved that if p g > 0, then 54) and p g , μ and v are, respectively, the geometric genus, the Milnor number, and the multiplicity of the isolated singularity at the origin of a weighted homogeneous polynomial. As a consequence, we prove that the Yau conjecture holds for n = 5 if p g > 0. In the process, we have also defined yet another sharp upper bound for the number of positive integral points within a five-dimensional simplex.

A sharp estimate of positive integral points in 6-dimensional polyhedra and a sharp estimate of smooth numbers

“…The Yau geometric conjecture was answered affirmatively for n = 3, 4, 5 by [3,22,37], respectively. In order to overcome the difficulty that the GLY sharp estimate conjecture is only true if a n is larger than y(n), a positive integer depending on n, Yau proposes to prove a new sharp polynomial estimate conjecture which is motivated from the Yau geometric conjecture.…”

“…It follows that Φ 2 > 0 for A 1 5, A 2 4, A 3 3, A 4 2, A 5 1 and β 2 ∈ (0, 1 3 ]. Case (3) a 6 ∈ (3,4]. The four levels of Case (3) are k = 1, k = 2, k = 3 and k = 4.…”

Complete characterization of isolated homogeneous hypersurface singularities

“…For a two-dimensional isolated hypersurface singularity V , Xu and Yau [1992;1993] found a coordinate-free characterization when V is defined by a homogeneous polynomial. Recently, necessary and sufficient conditions were given for threedimensional isolated hypersurface singularities with p g ≥ 0 [Lin and Yau 2004;Lin et al 2006a;Xu and Yau 1996] and four-dimensional isolated hypersurface singularities with p g > 0 [Chen et al 2011], where p g is the geometric genus of the singularity. Based on the classification of weighted homogeneous singularities, Yau and Zuo [2012] solved the problem for f with up to six variables.…”

“…Using p g , µ and ν, Yau made another conjecture in 1995 (see [Lin and Yau 2004;Chen et al 2011]) describing when a weighted homogeneous singularity is a homogeneous singularity. Let f : ‫ރ(‬ n , 0) → ‫,ރ(‬ 0) be a weighted homogeneous polynomial with an isolated singularity at the origin.…”

“…Theorem 1.5 [Chen et al 2011]. Let (V, 0) be a four-dimensional isolated hypersurface singularity defined by a weighted homogeneous polynomial f (x, y, z, w, t) = 0.…”