Arithmetic matroids, the Tutte polynomial and toric arrangements

We introduce the notion of an arithmetic matroid whose main example is a list of elements of a finitely generated abelian group. In particular, we study the representability of its dual, providing an extension of the Gale duality to this setting. Guided by the geometry of generalized toric arrangeme...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2013-01, Vol.232 (1), p.335-367
Main Authors: D’Adderio, Michele, Moci, Luca
Format: Article
Language:English
Subjects:
Arithmetic matroids
Combinatorial interpretation
Toric arrangements
Tutte polynomial
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Summary:We introduce the notion of an arithmetic matroid whose main example is a list of elements of a finitely generated abelian group. In particular, we study the representability of its dual, providing an extension of the Gale duality to this setting. Guided by the geometry of generalized toric arrangements, we provide a combinatorial interpretation of the associated arithmetic Tutte polynomial, which can be seen as a generalization of Crapo’s formula for the classical Tutte polynomial.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2012.09.001