We provide a general, unified, framework for external zonotopal algebra. The approach is critically based on employing simultaneously the two dual algebraic constructs and invokes the underlying matroidal and geometric structures in an essential way. This general theory makes zonotopal algebra an applicable tool for a larger class of polytopes.
Key words. Multivariate polynomials, polynomial ideals, duality, grading, kernels of differential operators, polynomial interpolation, box splines, zonotopes, hyperplane arrangements, matroids, graphs, Hilbert series
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