An emerging theory of **Zonotopal Algebra** is a framework for studying various algebraic, combinatorial, and analytic objects associated to a linear map from a higher-dimensional space to a lower-dimensional one. It provides connections between polynomial structures such as partition functions and positive structures such as f- and h-vectors of matroids.
This perspective gives formulas for volumes and lattice point enumerators of zonotopes, hence the name.

This framework was inspired by the theory of **Box Splines**, piecewise polynomial functions whose chambers are determined by such a linear map. There are wall-crossing formulas that explain how box splines change when moving from one chamber to another. Bos splines can be thought of as fiber volume functions, as they measure the volume of the preimage of their argument under the linear map in question, when the preimage is restricted to the unit cube.

Zonotopal algebra thus connects analytic, algebraic and matroidal structures generated by linear maps. It examines a family of polynomial ideals, spaces of polynomials annihilated by these ideals, geometry of the associated hyperplane arrangements, tilings of the associated zonotope, and many related phenomena from a unified point of view.

- List of papers on polynomial ideals, commutative algebra and approximation theory maintained by Carl de Boor.
- Topics in Hyperplane Arrangements, Polytopes and Box-Splines, a book by Corrado de Concini and Claudio Procesi.
- Front for the arXiv list of all things zonotopal.

- Federico Ardila
- Carl de Boor
- Corrado De Concini
- Olga Holtz
- Matthias Lenz
- Nan Li
- Luca Moci
- Alexander Postnikov
- Claudio Procesi
- Amos Ron
- Bernd Sturmfels
- Michele Vergne
- Zhiqiang Xu

Information on the NSF DMS grant Zonotopal Algebra and Combinatorics at UC Berkeley.

Last modified: Jun 30, 2014