Asian Journal of Mathematics

Volume 22 (2018)

Number 4

Special issue in honor of Ngaiming Mok (3 of 3)

Guest Editors: Jun-Muk Hwang, Korea Institute for Advanced Study; Yum-Tong Siu, Harvard University; Wing-Keung To, National University of Singapore; Stephen S.-T. Yau, Tsinghua University; Sai-Kee Yeung, Purdue University

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

Pages: 599 – 646

DOI: https://dx.doi.org/10.4310/AJM.2018.v22.n4.a1

Authors

Ke-Pao Lin (Chang Gung University of Science & Technology, and Chang Gung Memorial Hospital, Tao-Yuan, Taiwan)

Sarvasva Raghuvanshi (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Stephen S.-T. Yau (Department of Mathematical Sciences, Tsinghua University, Beijing, China)

Huaiqing Zuo (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)

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.

Keywords

sharp estimate, integral points, simplex

2010 Mathematics Subject Classification

Primary 11P21. Secondary 11Y99.

Full Text (PDF format)

This work was partially supported by NSFC (grant no. 11531007, 11771231), start-up fund from Tsinghua University, Tsinghua University Initiative Scientific Research Program and Ministry of Science and Technology R.O.C. and Chang Gung Memorial Hospital (grant no. MOST 105-2115-M-255-001, NMRPF3F0241).

Received 29 February 2016

Accepted 7 June 2017

Published 20 September 2018