Zonotopes, toric arrangements, and generalized Tutte polynomials

We introduce a multiplicity Tutte polynomial M(x, y), which generalizes the ordinary one and has applications to zonotopes and toric arrangements. We prove that M(x, y) satisfies a deletion-restriction recurrence and has positive coefficients. The characteristic polynomial and the Poincaré polynomial of a toric arrangement are shown to be specializations of the associated polynomial M(x, y), likewise the corresponding polynomials for a hyperplane arrangement are specializations of the ordinary Tutte polynomial. Furthermore, M(1, y) is the Hilbert series of the related discrete Dahmen-Micchelli space, while M(x, 1) computes the volume and the number of integral points of the associated zonotope. © 2010 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.