Sharp upper estimate of geometric genus and coordinate-free characterization of isolated homogeneous hypersurface singularities

Abstract

The subject of counting positive lattice points in n-dimensional simplexes has interested mathematicians for decades due to its applications in singularity theory and number theory. Enumerating the lattice points in a right-angled simplex is equivalent to determining the geometric genus of an isolated singularity of a weighted homogeneous complex polynomial. It is also a method to shed insight into large gaps in the sequence of prime numbers. Seeking to contribute to these applications, in this paper, we prove the Yau Geometric Conjecture in six dimensions, a sharp upper bound for the number of positive lattice points in a six-dimensional tetrahedron. The main method of proof is summing existing sharp upper bounds for the number of points in 5-dimensional simplexes over the cross sections of the six-dimensional simplex. Our new results pave the way for the proof of a fully general sharp upper bound for the number of lattice points in a simplex. It also sheds new light on proving the Yau Geometric and Yau Number-Theoretic Conjectures in full generality.

Proposing a New Theorem to Determine If an Algebraic Polynomial Is Nonnegative in an Interval

“…Stephen S.-T. Yau proposes Yau Number Theoretic Conjecture and Yau Geometric Conjecture (cf. [14]) to solve the above-mentioned problems. The novelty of Yau Number Theoretic Conjecture is that we can count the integral points in a polytope whose vertices are not necessarily integer points.…”

“…The Yau Geoemtric Conjecture has been proven for 3 ≤ n ≤ 6 (cf. [14,18]). We believe that our Main Theorem (Theorem 1) will play an important role in the proof of these two Yau conjectures.…”