曹锡华代数论坛
报告人：丘成栋教授 （伊利诺伊大学）
时间：2010年5月25日(周二)上午10:00-11:00)(比原定的时间5月18日推迟了一周)
地点：闵行校区数学楼102报告厅
Number theoretic conjecture of positive integral points in real simplices and a sharp estimate of DickmanDe Bruijn function
实单纯形内部正整点的数论猜想
及DickmanDe Bruijn函数之最佳估计
Abstract
It is well known that getting the estimate of integral points in right-angled simplices is equivalent to getting the estimate of DickmanDe Bruijn function D(x, y) which is the number of positive integersy. The estimate of integral points in right-angled simplices has many applications in complex geometry toric variety and tropical geometry. Previously we 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. Motivating the Yau Geometric Conjecture, we formulated the Number Theoretic Conjecture which gives a sharp polynomial upper estimate that counts the number of positive integral points in n-dimensional (n >= 3) real right-angled simplices. We prove this Number Theoretic Conjecture for n = 3, 4 and 5. As an application, we give a sharp estimate of DickmanDe Bruijn function D(x, y) for 5.
