Abstract
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 $k \geq -1$ and an upper set in the lattice of flats of the matroid defined by $X$, we define and study the associated $\textit{hierarchical zonotopal power ideal}$. This ideal is generated by powers of linear forms. Its Hilbert series depends only on the matroid structure of $X$. It is related to various other matroid invariants, $\textit{e. g.}$ 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 due to Sturmfels-Xu. La théorie de l'algèbre "zonotopique'' s'occupe d'idéaux et d'espaces vectoriels de polynômes qui ont un rapport avec plusieurs structures combinatoires et géométriques définies par des suites finies de vecteurs. Étant donné une telle suite $X$, un nombre entier $k \geq -1$ et un ensemble supérieur dans le treillis des plans du matroïde défini par $X$, nous définissons et étudions l'$\textit{idéal hiérarchique zonotopique}$, engendré par des puissances de formes linéaires. Sa série de Hilbert dépend seulement de la structure matroïdale de $X$. Il existe des relations avec d'autres invariants de matroïdes, tels que le polynôme d'épluchage et le polynôme caractéristique. Ce travail unifie et généralise des résultats d'Ardila-Postnikov sur les idéaux de puissances et de Holtz-Ron et Holtz-Ron-Xu sur l'algèbre zonotopique (hiérarchique). Nous généralisons aussi un résultat sur les modules de Cox zonotopiques, dû à Sturmfels-Xu.
Highlights
Let X = (x1, . . . , xN ) ⊆ Rr be a sequence of vectors that span Rr
We show that a statement as in Theorem 1.1 holds in a far more general setting: we study the kernel of the hierarchical zonotopal power ideal
The choice of a sequence of vectors X defines a large number of objects in various mathematical fields which are all related to zonotopal algebra [11]
Summary
We show that a statement as in Theorem 1.1 holds in a far more general setting: we study the kernel of the hierarchical zonotopal power ideal. The choice of a sequence of vectors X defines a large number of objects in various mathematical fields which are all related to zonotopal algebra [11]. Olga Holtz and Amos Ron coined the term zonotopal algebra [11] They introduced internal (k = −1) and external (k = +1) P-spaces and D-spaces. ([8] and A Tutte polynomial for toric arrangements (Luca Moci) [16], presented at FPSAC 2010) In this theory, the connection with lattice points in zonotope remains valid even if X is not unimodular. As an example for the connections between zonotopal algebra and combinatorics, we explain various relationships between zonotopal spaces and matroid/graph polynomials. The full paper is available on the arXiv [14]
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