Abstract
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 .
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