Matthias Lenz

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 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 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.

Zonotopal algebra and forward exchange matroids

Abstract Zonotopal algebra is the study of a family of pairs of dual vector spaces of multivariate polynomials that can be associated with a list of vectors X. It connects objects from combinatorics, geometry, and approximation theory. The origin of zonotopal algebra is the pair ( D ( X ) , P ( X ) ) , where D ( X ) denotes the Dahmen–Micchelli space that is spanned by the local pieces of the box spline and P ( X ) is a space spanned by products of linear forms. The first main result of this paper is the construction of a canonical basis for D ( X ) . We show that it is dual to the canonical basis for P ( X ) that is already known. The second main result of this paper is the construction of a new family of zonotopal spaces that is far more general than the ones that were recently studied by Ardila–Postnikov, Holtz–Ron, Holtz–Ron–Xu, Li–Ron, and others. We call the underlying combinatorial structure of those spaces forward exchange matroid. A forward exchange matroid is an ordered matroid together with a subset of its set of bases that satisfies a weak version of the basis exchange axiom.