16 research outputs found
Hierarchical Zonotopal Power Ideals
Zonotopal algebra deals with ideals and vector spaces of polynomials that are related to several combinatorial and geometric structures defined by a finite sequence of vectors. Given such a sequence , an integer and an upper set in the lattice of flats of the matroid defined by , we define and study the associated . This ideal is generated by powers of linear forms. Its Hilbert series depends only on the matroid structure of . It is related to various other matroid invariants, the shelling polynomial and the characteristic polynomial. This work unifies and generalizes results by Ardila-Postnikov on power ideals and by Holtz-Ron and Holtz-Ron-Xu on (hierarchical) zonotopal algebra. We also generalize a result on zonotopal Cox modules due to Sturmfels-Xu
Interpolation, box splines, and lattice points in zonotopes
Given a finite list of vectors , one can define the box spline . Box splines are piecewise polynomial functions that are used in approximation theory. They are also interesting from a combinatorial point of view and many of their properties solely depend on the structure of the matroid defined by the list . The support of the box spline is the zonotope . We show that if the list is totally unimodular, any real-valued function defined on the set of lattice points in the interior of can be extended to a function on of the form in a unique way, where is a differential operator that is contained in the so-called internal -space. This was conjectured by Olga Holtz and Amos Ron. We also point out connections between this interpolation problem and matroid theory, including a deletion-contraction decomposition
Ehrhart polynomial and multiplicity Tutte polynomial
We prove that the Ehrhart polynomial of a zonotope is a specialization of the
multiplicity Tutte polynomial. We derive some formulae for the volume and the
number of integer points of the zonotope.Comment: 6 pages, 1 pictur
A Tutte polynomial for toric arrangements
We introduce a multiplicity Tutte polynomial M(x,y), with applications to
zonotopes and toric arrangements. We prove that M(x,y) satisfies a
deletion-restriction recurrence and has positive coefficients. The
characteristic polynomial and the Poincare' polynomial of a toric arrangement
are shown to be specializations of the associated polynomial M(x,y), likewise
the corresponding polynomials for a hyperplane arrangement are specializations
of the ordinary Tutte polynomial. Furthermore, M(1,y) is the Hilbert series of
the related discrete Dahmen-Micchelli space, while M(x,1) computes the volume
and the number of integral points of the associated zonotope.Comment: Final version, to appear on Transactions AMS. 28 pages, 4 picture
Lattice points in polytopes, box splines, and Todd operators
Let 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 can be extended to a
function on the whole zonotope of the form in a unique way, where
is a differential operator that is contained in the so-called internal
\Pcal-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.Comment: 15 pages, 4 figure
Geometric realizations and duality for Dahmen-Micchelli modules and De Concini-Procesi-Vergne modules
We give an algebraic description of several modules and algebras related to
the vector partition function, and we prove that they can be realized as the
equivariant K-theory of some manifolds that have a nice combinatorial
description. We also propose a more natural and general notion of duality
between these modules, which corresponds to a Poincar\'e duality-type
correspondence for equivariant K-theory.Comment: Final version, to appear on Discrete and Computational Geometr
Splines, lattice points, and (arithmetic) matroids
Let be a -matrix. We consider the variable polytope . It is known that the function that assigns to a parameter the volume of the polytope is piecewise polynomial. Formulas of Khovanskii-Pukhlikov and Brion-Vergne imply that the number of lattice points in can be obtained by applying a certain differential operator to the function . In this extended abstract we slightly improve the formulas of Khovanskii-Pukhlikov and Brion-Vergne and we study the space of differential operators that are relevant for (ıe operators that do not annihilate ) and the space of nice differential operators (ıe operators that leave continuous). These two spaces are finite-dimensional homogeneous vector spaces and their Hilbert series are evaluations of the Tutte polynomial of the (arithmetic) matroid defined by