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 XX, an integer k≥−1k≥−1 and an upper set in the lattice of flats of the matroid defined by XX, 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 XX. 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 and Postnikov on power ideals and by Holtz and Ron, and Holtz et al. on (hierarchical) zonotopal algebra. We also generalize a result on zonotopal Cox modules that were introduced by Sturmfels and Xu.