On the GLY Conjecture of upper estimate of positive integral points in real right-angled simplices

Abstract

The GLY (Granville–Lin–Yau) Conjecture is a generalization of Lin, Xu and Yau's results. An important application of GLY is its use in characterizing an affine hypersurface in C n as a cone over a nonsingular projective variety. In addition, the Rough Upper Estimate Conjecture in GLY, recently proved by Yau and Zhang, implies the Durfee Conjecture in singularity theory. This paper develops a unified approach to prove the Sharp Upper Estimate Conjecture for general n. Using this unified approach, we prove that the Sharp Upper Estimate Conjecture is true for n = 4 , 5 , 6 . After giving a counter-example to show that the Sharp Upper Estimate Conjecture is not true for n = 7 , we propose a Modified GLY Conjecture. For each fixed n, our unified approach can be used to prove this Modified GLY Conjecture.