Hierarchical zonotopal power ideals

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>=-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.

22 pages, 1 figure; small changes reflecting referees' comments; to appear in European Journal of Combinatorics

References in corpus (8)

• Products of Linear Forms and Tutte Polynomials
• Hierarchical zonotopal spaces
• Hierarchical zonotopal power ideals
• Vector partition function and generalized Dahmen-Micchelli spaces
• Zonotopal algebra and forward exchange matroids
• Matroids and log-concavity
• External zonotopal algebra
• Monomization of Power Ideals and Parking Functions

Cited by in corpus (11)

• Hierarchical zonotopal power ideals
• Zonotopal algebra and forward exchange matroids
• Interpolation, box splines, and lattice points in zonotopes
• Splines, lattice points, and arithmetic matroids
• Arithmetic matroids, Tutte polynomial, and toric arrangements
• Matroids and log-concavity
• External zonotopal algebra
• Geometric realizations and duality for Dahmen-Micchelli modules and De Concini-Procesi-Vergne modules
• Lattice points in polytopes, box splines, and Todd operators
• A Tutte polynomial for toric arrangements
• Ehrhart polynomial and multiplicity Tutte polynomial