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 characteriza-tion of when a convergent power series is a homogeneous polyno-mial after a biholomorphic change of variables. In this paper, we have proved that if p_g > 0, then 5!p_9 <μ p(v), p(v) = (v – 1)~5 – v(v – 1) ... (v – 4) and p_g, μ and v are, respectively, the geometric genus, the Milnor number, and the multiplicity of the isolated sin-gularity 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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