Lattice points in rational ellipsoids

Counting Lattice Points of Rational Polyhedra

The generating function F (P ) = ∑ α∈P∩ZN xα for a rational polytope P carries all essential information of P . In this paper we show that for any positive integer n, the generating function F (P, n) of nP = {nx : x ∈ P} can be written as F (P, n) = ∑ α∈A Pα(n)x, where A is the set of all vertices of P and each Pα(n) is a certain periodic function of n. The Ehrhart reciprocity law follows autom...

Discrete Truncated Powers And Lattice Points In Rational Polytope ∗

Discrete truncate power is very useful for studying the number of nonnegative integer solutions of linear Diophantine equations. In this paper, some detail information about discrete truncated power is presented. To study the number of integer solutions of linear Diophantine inequations, the generalized truncated power and generalized discrete truncated power are defined and discussed respectiv...

A Closer Look at Lattice Points in Rational Simplices

We generalize Ehrhart’s idea ([Eh]) of counting lattice points in dilated rational polytopes: Given a rational simplex, that is, an n-dimensional polytope with n+ 1 rational vertices, we use its description as the intersection of n+ 1 halfspaces, which determine the facets of the simplex. Instead of just a single dilation factor, we allow different dilation factors for each of these facets. We ...

A Combinatorial Property of Points anf Ellipsoids

For each d-1 there is a constant Ca > 0 such that any finite set X c R a contains a subset YcX, IYI<-[~d(d+3)J +1 having the following property: if E = Y is an ellipsoid, then IE nXt-> c~JXl.

Distribution of Rational Points: A Survey