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Counting the Number of Integral Points in General $n$-Dimensional Tetrahedra and Bernoulli Polynomials

Published online by Cambridge University Press:  20 November 2018

Ke-Pao Lin  and
Stephen S.-T. Yau
Ke-Pao Lin
Affiliation:
Department of Information Management, Chang Gung Institute of Technology, 261 Wen-Hwa 1 Road, Kwei-Shan, Tao-Yuan, Taiwan 333-03, Republic of China, e-mail: kplin@cc.cgin.edu.tw
Stephen S.-T. Yau
Affiliation:
Department of Mathematics, Statistics and Computer Science (M/C 249), University of Illinois at Chicago, 851 South Morgan Street, Chicago, Illinois 60607-7045, U.S.A., e-mail: yau@uic.edu
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Abstract

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Recently there has been tremendous interest in counting the number of integral points in $n$-dimensional tetrahedra with non-integral vertices due to its applications in primality testing and factoring in number theory and in singularities theory. The purpose of this note is to formulate a conjecture on sharp upper estimate of the number of integral points in $n$-dimensional tetrahedra with non-integral vertices. We show that this conjecture is true for low dimensional cases as well as in the case of homogeneous $n$-dimensional tetrahedra. We also show that the Bernoulli polynomials play a role in this counting.

Keywords

11B7511H0611P2111Y99
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 46 , Issue 2 , 01 June 2003 , pp. 229 - 241
Copyright
Copyright © Canadian Mathematical Society 2003

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