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Zonotopes, toric arrangements, and generalized Tutte polynomials

Abstract : 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.
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Luca Moci. Zonotopes, toric arrangements, and generalized Tutte polynomials. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.413-424, ⟨10.46298/dmtcs.2878⟩. ⟨hal-01186307⟩

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