Integral Points in n-dimensional Tetrahedra Based on GLY Conjecture

GLY猜想於n維多面體之格點數估算

林克保(Ke-Pao Lin) ； 丘成棟(Stephen S.-T. Yau)

《長庚科技學刊》 17期 (2012/12) Pp. 27-37

https://doi.org/10.6192/CGUST.2012.12.17.3

摘要

近來學術界對n維非整數座標頂點多面體之格點數之估算有極大之興趣。此問題在數論如質數檢驗與質因數分解及奇異理論都有很重要之應用。我們在[Li-Ya 1] 對n維非格頂點多面體之格點數提出一精確上估計猜想。我們指出在n=3, 4, 5維及n維齊次情形上估計猜想為真。我們於本文先回顧GLY猜想接著我們針對GLY猜想於n維多面體之格點數在a1 = a2=......=an情形之估算提出一新證明。用組合學方法重證此情形是有趣的。

關鍵字

GLY猜想 ； Ehrhart多項式 ； 格點 ； 多面體 ； 數論 ； 奇異理論

並列摘要

Counting the number of integral points in n-dimensional tetrahedra with non-integral vertice has been tremendous interest recently. It has very important applications in primality testing and factoring in number theory and in singularities theory. In [Li-Ya 1] we proposed a conjecture on sharp upper estimate of the number of integral points in n-dimensional tetrahedra with non-integral vertice. We demonstrated that this conjecture is true for dimension n = 3, 4, 5 cases as well as in the case of homogeneous n-dimensional tetrahedra. In this paper we review the GLY conjecture first and then we propose a new proof for GLY Conjecture on counting the number of integral points in n-dimensional tetrahedra for the case a1 = a2 = .... = an. It's interesting to use a combinatorial way to prove it again.

並列關鍵字

GLY Conjecture ； Ehrhart polynomial ； integral points ； tetrahedra ； number theory ； singularities theory