Arithmetic matroids, the Tutte polynomial and toric arrangements

By Michele D’Adderio, Luca Moci. 2011

95 citations recorded by the source corpus.

Abstract 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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/42eb33ff69fa9c0fdd978ee03afd7b41980614a2

How researchers see this paper

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In Algebra and Number Theory

1. Alex Fink
2. Petter Brand'en
3. Luca Moci
4. Federico Ardila
5. Eric Katz

In Discrete Mathematics and Combinatorics

1. Alex Fink
2. Petter Brand'en
3. Luca Moci
4. Julius Borcea
5. Federico Ardila
6. Eric Katz

In Mathematical Physics

1. Alex Fink
2. Petter Brand'en
3. Luca Moci
4. Julius Borcea

Citations in the corpus

• The multivariate arithmetic Tutte polynomial (2012, background): “In [11] the multivariate Tutte polynomial of a graph was defined as ZG(q,v) := ∑ A⊆E qk(A) ∏ e∈A ve, where k(A) denotes the number of connected components in the subgraph (V,A).”
• Matroids over a ring (2012, methodology): “ demands are, in the realizable case, numbers of components of certain strata in the corresponding toric arrangement. Its coefficients also have two natural but nontrivial combinatorial interpretations [6, 3]. The additional property (A2a) appeared in an earlier choice of the axioms [6, first arXiv ver”