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. On introduit un polynôme de Tutte avec multiplicité $M(x, y)$, qui généralise le polynôme de Tutte ordinaire et a des applications aux zonotopes et aux arrangements toriques. Nous prouvons que $M(x, y)$ satisfait une récurrence de "deletion-restriction'' et a des coefficients positifs. Le polynôme caractéristique et le polynôme de Poincaré d'un arrangement torique sont des spécialisations du polynôme associé $M(x, y)$, de même que les polynômes correspondants pour un arrangement d'hyperplans sont des spécialisations du polynôme de Tutte ordinaire. En outre, $M(1, y)$ est la série de Hilbert de l'espace discret de Dahmen-Micchelli associé, et $M(x, 1)$ calcule le volume et le nombre de points entiers du zonotope associé.
Highlights
The Tutte polynomial is an invariant naturally associated to a matroid and encoding many of its features, such as the number of bases and their internal and external activity ([21], [3], [6])
If the matroid is defined by a finite list of vectors, it is natural to consider the arrangement obtained by taking the hyperplane orthogonal to each vector
We consider two finite dimensional vector spaces: a space of polynomials D(X), defined by differential equations, and a space of quasipolynomials DM (X), defined by difference equations. These spaces were introduced by Dahmen and Micchelli to study respectively box splines and partition functions, and are deeply related respectively with the hyperplane arrangement and the toric arrangement defined by X, as explained in the forthcoming book [6]
Summary
The Tutte polynomial is an invariant naturally associated to a matroid and encoding many of its features, such as the number of bases and their internal and external activity ([21], [3], [6]). To the poset of the intersections of the hyperplanes one associates its characteristic polynomial, which provides a rich combinatorial and topological description of the arrangement ([19], [22]) This polynomial can be obtained as a specialization of the Tutte polynomial. We consider two finite dimensional vector spaces: a space of polynomials D(X), defined by differential equations, and a space of quasipolynomials DM (X), defined by difference equations These spaces were introduced by Dahmen and Micchelli to study respectively box splines and partition functions, and are deeply related respectively with the hyperplane arrangement and the toric arrangement defined by X, as explained in the forthcoming book [6]. Remark 1.1 This paper is an extended abstract of [17], which contains more details and all the proofs, which are omitted here
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