On the polynomial sharp upper estimate conjecture in 7-dimensional simplex

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• 作者：Stephen S.-T. Yau^a ; ^{yau@uic.edu" class="auth_mail" title="E-mail the corresponding author} ; Beihui Yuan^b ; ^{by238@cornell.edu" class="auth_mail" title="E-mail the corresponding author} ; Huaiqing Zuo^c ; ^{hqzuo@math.tsinghua.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 11P21 ; secondary ; 11Y99
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：160
• 期：Complete
• 页码：254-286
• 全文大小：497 K

文摘

Because of its importance in number theory and singularity theory, the problem of finding a polynomial sharp upper estimate of the number of positive integral points in an n  -dimensional (n≥3$n\ge 3$) polyhedron has received attention by a lot of mathematicians. The first named author proposed the Number Theoretic Conjecture for the upper estimate. The previous results on the Number Theoretic Conjecture in low dimension cases (n<7$n<7$) are proved by using the sharp GLY conjecture which is true only for low dimensional case. Thus the proof cannot be generalized to high dimension. In this paper, we offer a uniform approach to prove the Number Theoretic Conjecture for all dimensions by simply using the induction method and the Yau–Zhang [19] estimates (see ,  and ). As a result, the Number Theoretic Conjecture is proven for n=7$n=7$. An important estimate for all dimensions is also obtained (  and ) which will be useful to prove the general case of the Number Theoretic Conjecture. As an application, we give a sharper estimate of the Dickman–De Bruijn function ψ(x,y)$\psi (x,y)$ for 5≤y<19$5\le y<19$, compared with the result obtained by Ennola.