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

Abstract

AbstractRecently 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.