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

http://dx.doi.org/10.4153/CMB-2003-023-0
Canad. Math. Bull. 46(2003), 229-241

  Published:2003-06-01
 Printed: Jun 2003

Recently there has been tremendous interest in counting the number of integral points in $n$-dimen\-sional 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.