Sharp Polynomial Estimate of Integral Points in Real-Angled Simplices

Linda Zhao University of Chicago Laboratory Schools

S.-T. Yau High School Science Awarded Papers mathscidoc:1608.35011

2008
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 ndimensional (n≥3) real right-angled simplices with vertices whose distance 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≤7 and are in the process of proving the general n case.
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@inproceedings{linda2008sharp,
  title={Sharp Polynomial Estimate of Integral Points in Real-Angled Simplices},
  author={Linda Zhao},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160813215154577696058},
  year={2008},
}
Linda Zhao. Sharp Polynomial Estimate of Integral Points in Real-Angled Simplices. 2008. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160813215154577696058.
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