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.

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Abstract.Recently there has been tremendous interest in counting the number of integral points in n-dimensional tetrahedra with non-integral vertices due to its applications in primality testing and factoring in number theory and in singularities theory. The purpose of this note is to formulate a conjecture on sharp upper estimate of the number of integral points in n-dimensional tetrahedra with non-integral vertices. We show that this conjecture is true for low dimensional cases as well as in the case of homogeneous n-dimensional tetrahedra. We also show that the Bernoulli polynomials play a role in this counting.

Let f : (C n+1 , 0) → (C, 0) be a holomorphic germ defining an isolated hypersurface singularity V at the origin. Let µ and ν and p g be the Milnor number, multiplicity and geometric genus of (V, 0), respectively. We conjecture that µ ≥ (ν − 1) n+1 and the equality holds if and only if f is a semi-homogeneous function. We prove that this inequality holds for n = 1, and also for n = 2 or 3 with additional assumption that f is a quasihomogeneous function. For n = 1, if V has at most two irreducible branches at the origin, or if f is a quasi-homogeneous function, then µ = (ν − 1) 2 if and only if f is a homogeneous polynomial. For n = 2, if f is a quasihomogeneous function, then µ = (ν − 1) 3 iff 6p g = ν(ν − 1)(ν − 2) iff f is a homogeneous polynomial after biholomorphic change of variables. For n = 3, if f is a quasi-homogeneous function, then µ = (ν − 1) 4 iff 24p g = ν(ν − 1)(ν − 2)(ν − 3) iff f is a homogeneous polynomial after biholomorphic change of variables.