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Sharp polynomial estimate of integral points in right-angled simplices. (English) Zbl 1284.11130

Summary: 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 \((n\geq 3)\) real right-angled simplices with vertices whose distances to the origin are at least \(n-1\). The estimate was proven for \(n\leq 6\) 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 \(n\leq 9\).

MSC:

11P21 Lattice points in specified regions
11H06 Lattices and convex bodies (number-theoretic aspects)
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
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