Abstract
We introduce the notion of arithmetic matroid, whose main example is provided by a list of elements in a finitely generated abelian group. We study the representability of its dual, and, guided by the geometry of toric arrangements, we give a combinatorial interpretation of the associated arithmetic Tutte polynomial, which can be seen as a generalization of Crapo's formula. Nous introduisons la notion de matroï de arithmètique, dont le principal exemple est donnè par une liste d'élèments dans un groupe abèlien fini. Nous ètudions la reprèsentabilitè de son dual, et, guidè par la gèomètrie des arrangements toriques, nous donnons une interprètation combinatoire du polynôme de Tutte arithmètique associèe, ce qui peut être vu comme une gènèralisation de la formule de Crapo.
Highlights
Several mathematical constructions arise from a finite list of vectors X: hyperplane arrangements and zonotopes in geometry, box splines in numerical analysis, root systems and parking functions in combinatorics are only some of the most well-known examples
If the list X lies in Zn, an even wider spectrum of mathematical objects appears. In their recent book [8], De Concini and Procesi explored the connection between the toric arrangement associated to such a list and the vector partition function
While the spline and the hyperplane arrangement only depend on the “linear algebra” of X, the partition function and the toric arrangement are influenced by its “arithmetics”
Summary
Several mathematical constructions arise from a finite list of vectors X: hyperplane arrangements and zonotopes in geometry, box splines in numerical analysis, root systems and parking functions in combinatorics are only some of the most well-known examples. If the list X lies in Zn, an even wider spectrum of mathematical objects appears In their recent book [8], De Concini and Procesi explored (among other things) the connection between the toric arrangement associated to such a list and the vector partition function. This object axiomatizes both the linear algebra (via the matroid) and the arithmetics (via the multiplicity function) of a list of elements in a finitely generated abelian group. Our combinatorial ideas have their roots in the notion of a generalized toric arrangement, which provides the geometric inspiration and motivation of our work (see [4, Section 4])
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