Zonotopal algebra and forward exchange matroids

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))(D(X),P(X)), where D(X)D(X) denotes the Dahmen–Micchelli space that is spanned by the local pieces of the box spline and P(X)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)D(X). We show that it is dual to the canonical basis for P(X)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.