Sharp polynomial estimate of integral points in right-angled simplices

Characterization of homogeneous polynomials with isolated critical point at the origin follows from a study of complex geometry. Yau previously proposed a Numerical Characterization Conjecture. A step forward in solving this conjecture, the Granville–Lin–Yau Conjecture was formulated, with a sharp estimate that counts the number of positive integral points in n-dimensional (n⩾3) real right-angled simplices with vertices whose distances to the origin are at least n−1. The estimate was proven for n⩽6 but has a counterexample for n=7. In this project we come up with an idea of forming a New Sharp Estimate Conjecture where we need the distances of the vertices to be n. We have proved this New Sharp Estimate Conjecture for n⩽9.