Premium
A sharp polynomial estimate of positive integral points in a 4‐dimensional tetrahedron and a sharp estimate of the Dickman‐de Bruijn function
Author(s) -
Luo Xue,
Yau Stephen S.T.,
Zuo Huaiqing
Publication year - 2015
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201010081
Subject(s) - mathematics , conjecture , de bruijn sequence , tetrahedron , combinatorics , polynomial , function (biology) , geometry , mathematical analysis , evolutionary biology , biology
The estimate of integral points in right‐angled simplices has many applications in number theory, complex geometry, toric variety and tropical geometry. In [24], [25], [27], the second author and other coworkers gave a sharp upper estimate that counts the number of positive integral points in n dimensional ( n ≥ 3 ) real right‐angled simplices with vertices whose distance to the origin are at least n − 1 . A natural problem is how to form a new sharp estimate without the minimal distance assumption. In this paper, we formulate the Number Theoretic Conjecture which is a direct correspondence of the Yau Geometry conjecture. We have proved this conjecture for n = 4 . This paper gives hope to prove the new conjecture in general. As an application, we give a sharp estimate of the Dickman‐de Bruijn function ψ ( x , y ) for y < 11 .