Lattice Points in Polytopes, Box Splines, and Todd Operators

Abstract

Let X be a list of vectors that is totally unimodular. In a previous article the author proved that every real-valued function on the set of interior lattice points of the zonotope defined by X can be extended to a function on the whole zonotope of the form p(D)B X in a unique way, where p(D) is a differential operator that is contained in the so-called internal P-space. In this paper we construct an explicit solution to this interpolation problem in terms of Todd operators. As a corollary we obtain a slight generalisation of the Khovanskii-Pukhlikov formula that relates the volume and the number of integer points in a smooth lattice polytope.

Classification of External Zonotopal Algebras

“…The first definition was used by F. Ardila and A. Postnikov [2]; it originates from the algebras generated by the curvature forms of tautological Hermitian linear bundles [4,29], see also papers [5,6,15,16,17,23,24,27,28,30], where the quotient algebras by these ideals were discussed by details. At the same time a similar definition and the term were established by O. Holtz and A. Ron [13]; Their definitions originates from Box-Splines and from Dahmen-Micchelli's space [1,9,11], see also the papers [10,12,14,19,20,21,22,31].…”

Multidimensional Analog of the Bernoulli Polynomials and its Properties

“…is called the Todd operator (see, for example, [18]). The Todd operator [19] helps to connect volumes and a number of lattice points of convex polytopes.…”

Classification of external Zonotopal algebras

“…We will work with the definition from F. Ardila and A. Postnikov [2]; it comes from algebras generated by the curvature forms of tautological Hermitian linear bundles [3,28], see also papers [4,5,14,15,16,22,23,26,27,29], where people work with quotients algebras by these ideals. At the same time the definition and the name was established by O. Holtz and A. Ron [12]; it comes from Box-Splines and from Dahmen-Micchelli space [1,8,10], see also the papers [9,11,13,18,19,20,21,30].…”