External zonotopal algebra

Nan, Li; Ron, Amos

doi:10.48550/arxiv.1104.2072

Cited by 4 publications

(3 citation statements)

References 8 publications

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“…The space P(X) first appeared in approximation theory [1,9,14]. These two P-spaces and generalisations were later studied by various authors, including [2,5,16,19,21,22].…”

Section: Introductionmentioning

confidence: 99%

Lattice Points in Polytopes, Box Splines, and Todd Operators

Lenz

2014

Int Math Res Notices

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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.

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“…The space P(X) first appeared in approximation theory [1,9,14]. These two P-spaces and generalisations were later studied by various authors, including [2,5,16,19,21,22].…”

Section: Introductionmentioning

confidence: 99%

Lattice Points in Polytopes, Box Splines, and Todd Operators

Lenz

2014

Int Math Res Notices

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“…The space P(X) first appeared in approximation theory [1,10,16]. Later, spaces of this type and generalisations were also studied by authors in other fields, e. g. [2,4,19,22,24,26,29].…”

Section: Introductionmentioning

confidence: 99%

Interpolation, box splines, and lattice points in zonotopes

Lenz

2013

Discrete Mathematics &Amp; Theoretical Computer Science

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Given a finite list of vectors $X \subseteq \mathbb{R}^d$, one can define the box spline $B_X$. 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 $X$. The support of the box spline is the zonotope $Z(X)$. We show that if the list $X$ is totally unimodular, any real-valued function defined on the set of lattice points in the interior of $Z(X)$ can be extended to a function on $Z(X)$ 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 $\mathcal{P}$-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. Etant donné une liste finie de vecteurs $X \subseteq \mathbb{R}^d$, on peut définir la box spline $B_X$. Les box splines sont des fonctions continues par morceaux qui sont utilisées en théorie de l’approximation. Elles sont aussi intéressantes d’un point de vue combinatoire et beaucoup de leurs propriétés dépendent uniquement de la structure du matroïde défini par la liste $X$. Le support de la box spline est le zonotope $Z(X)$. Si la liste $X$ est totalement unimodulaire, nous démontrons que toute fonction à valeurs réelles définie sur l’ensemble des points du réseau à l’intérieur de $Z(X)$ peut être étendue à une fonction sur $Z(X)$ de la forme $p(D)B_X$ de manière unique, où $p(D)$ est un opérateur différentiel qui est contenu dans l’espace appelé $\mathcal{P}$-interne. Cela a été conjecturé par Olga Holtz et Amos Ron. Nous indiquons aussi des relations entre ce problème d’interpolation et la théorie des matroïdes, en plus d’une décomposition suppressions-contractions.

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“…The central case was treated in an algebraic setting in [12,13]. Other related results include [4,29,30,38].…”

Section: Introductionmentioning

confidence: 99%

Hierarchical zonotopal power ideals

Lenz¹

2012

European Journal of Combinatorics

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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 X, an integer k>=-1 and an upper set in the lattice of flats of the matroid defined by X, we define and study the associated hierarchical zonotopal power ideal. This ideal is generated by powers of linear forms. Its Hilbert series depends only on the matroid structure of X. Via the Tutte polynomial, it is related to various other matroid invariants, e.g. 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 that were introduced by Sturmfels-Xu.Comment: 22 pages, 1 figure; small changes reflecting referees' comments; to appear in European Journal of Combinatoric

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External zonotopal algebra

Cited by 4 publications

References 8 publications

Lattice Points in Polytopes, Box Splines, and Todd Operators

Lattice Points in Polytopes, Box Splines, and Todd Operators

Interpolation, box splines, and lattice points in zonotopes

Hierarchical zonotopal power ideals

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