Arithmetic matroids, Tutte polynomial, and toric arrangements

D'Adderio, Michele; Moci, Luca

doi:10.48550/arxiv.1105.3220

Cited by 4 publications

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“…Our definition of arithmetic matroid is equivalent to the one given in [7]. In [7], the function m satisfies five axioms.…”

Section: Arithmetic Matroids and Multivariate Tutte Polynomialsmentioning

confidence: 99%

“…Our definition of arithmetic matroid is equivalent to the one given in [7]. In [7], the function m satisfies five axioms. The first three correspond precisely to (A1) and (A2), while the others are:…”

Section: Arithmetic Matroids and Multivariate Tutte Polynomialsmentioning

confidence: 99%

“…We also give a new proof of the nonnegativity of the coefficients of the (bivariate) arithmetic Tutte polynomial in the more general framework of pseudo-arithmetic matroids (Theorem 4.5). Moreover, if A is represented by a list L of elements in a finitely generated abelian group G, then in Theorem 6.3 we give a geometrical interpretation of such coefficients, generalizing various formulae proved in [7,10].…”

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confidence: 96%

“…This vector space was introduced in order to study vector partition functions (see [10,Section 6]); • the Ehrhart polynomial of the zonotope Z(L) (see [10,Section 4] and [8]). Furthermore, the arithmetic Tutte polynomial has applications to graph theory [8], and it has nonnegative coefficients, as proved in [7] by providing a combinatorial interpretation that extends Crapo's theorem.…”

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confidence: 98%

“…Recently an arithmetic version of this polynomial has been studied [10,7]. Namely, to a finite list L of elements in a finitely generated abelian group G, one associates an arithmetic Tutte polynomial.…”

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The multivariate arithmetic Tutte polynomial

Brändén

Moci

2014

Trans. Amer. Math. Soc.

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Abstract. We introduce an arithmetic version of the multivariate Tutte polynomial recently studied by Sokal, and a quasi-polynomial that interpolates between the two. We provide a generalized Fortuin-Kasteleyn representation for representable arithmetic matroids, with applications to arithmetic colorings and flows. We give a new proof of the positivity of the coefficients of the arithmetic Tutte polynomial in the more general framework of pseudo-arithmetic matroids. In the case of a representable arithmetic matroid, we provide a geometric interpretation of the coefficients of the arithmetic Tutte polynomial.Résumé. Nous introduisons une version arithmétique du polynôme de Tutte multivariée récemmentétudié par Sokal, et un quasi-polynôme qui interpole entre les deux. Nous proposons une représentation de Fortuin-Kasteleyn generalisée pour les matroïdes arithmétiques représentables, avec des applications aux colorations et flux arithmétiques. Nous donnons une nouvelle preuve de la positivité des coefficients du polynôme de Tutte arithmétique dans le cadre plus général des matroïdes pseudo-arithmétiques. Dans le cas d'un matroïde arithmétique représentable, nous proposons une interpretation geometrique des coefficients du polynôme de Tutte arithmetique.

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“…Our definition of arithmetic matroid is equivalent to the one given in [7]. In [7], the function m satisfies five axioms.…”

Section: Arithmetic Matroids and Multivariate Tutte Polynomialsmentioning

confidence: 99%

Section: Arithmetic Matroids and Multivariate Tutte Polynomialsmentioning

confidence: 99%

mentioning

confidence: 96%

mentioning

confidence: 98%

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confidence: 99%

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The multivariate arithmetic Tutte polynomial

Brändén

Moci

2014

Trans. Amer. Math. Soc.

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Hierarchical zonotopal power ideals

Lenz¹

2012

European Journal of Combinatorics

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Zonotopal algebra deals with ideals and vector spaces of polynomials that are related to several combinatorial and geometric structures defined by a finite sequence of vectors. Given such a sequence X, an integer k>=-1 and an upper set in the lattice of flats of the matroid defined by X, we define and study the associated hierarchical zonotopal power ideal. This ideal is generated by powers of linear forms. Its Hilbert series depends only on the matroid structure of X. Via the Tutte polynomial, it is related to various other matroid invariants, e.g. the shelling polynomial and the characteristic polynomial. This work unifies and generalizes results by Ardila-Postnikov on power ideals and by Holtz-Ron and Holtz-Ron-Xu on (hierarchical) zonotopal algebra. We also generalize a result on zonotopal Cox modules that were introduced by Sturmfels-Xu.Comment: 22 pages, 1 figure; small changes reflecting referees' comments; to appear in European Journal of Combinatoric

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