On the sharp polynomial upper estimate conjecture in eight-dimensional simplex

Ke-Pao Lin Chang Gung University of Science and Technology Andrew Yang, Queen's College Stephen S.-T. YAU Tsinghua University Huaiqing Zuo Tsinghua University

Number Theory mathscidoc:1712.24001

Pure and Applied Mathematics Quarterly, 2017
The estimate of integral points in an n-dimensional polyhedron has many applications in singularity theory, number theory and toric geometry. The third named author formulated a conjecture (i.e., Yau Number Theoretic Conjecture) which gives a sharp polynomial upper estimate on the number of positive integral points in n-dimensional (n  3) real right-angled simplices. The previous results on the conjecture in low dimension cases (n  6) have been proved by using the sharp GLY conjecture. However, it is only valid in low dimension. The Yau Number Theoretic Conjecture for n = 7 has been shown with a completely new method in [22]. In this paper, on the one hand, the similar method has been applied to prove the conjecture for n = 8, but with more meticulous analyses. The main method of proof is summing existing sharp upper bounds for the number of points in seven-dimensional simplex over the cross sections of eight-dimensional simplex. This is a signi cant progress since it sheds light on proving the Yau Number Theoretic Conjecture in full generality. On the other hand, we give a new sharper estimate of the Dickman-De Bruijn function (x; y) for 5  y < 23, compared with the result obtained by Ennola.
Sharp upper estimate, integral points, prime decomposition, simplex
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  author={Ke-Pao Lin, Andrew Yang,, Stephen  S.-T. YAU, and Huaiqing Zuo},
  booktitle={ Pure and Applied Mathematics Quarterly},
Ke-Pao Lin, Andrew Yang,, Stephen S.-T. YAU, and Huaiqing Zuo. ON THE SHARP POLYNOMIAL UPPER ESTIMATE CONJECTURE IN EIGHT-DIMENSIONAL SIMPLEX. 2017. In Pure and Applied Mathematics Quarterly. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20171229094056457416863.
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