Hierarchical zonotopal power ideals

TL;DR

This paper introduces hierarchical zonotopal power ideals linked to matroid structures, unifying and extending previous results in zonotopal algebra and related combinatorial invariants.

Contribution

It defines and studies hierarchical zonotopal power ideals, generalizing prior work and connecting them to matroid invariants via the Tutte polynomial.

Findings

Hilbert series depends only on matroid structure

Relates power ideals to Tutte polynomial and other invariants

Unifies previous results in zonotopal algebra

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.