Luca Moci

Matroids over a ring

We introduce the notion of a matroid M over a commutative ring R, assigning to every subset of the ground set an R-module according to some axioms. When R is a field, we recover matroids. When \(R = \mathbb{Z}\), and when R is a DVR, we get (structures which contain all the data of) quasi-arithmetic matroids, and valuated matroids i.e. tropical linear spaces, respectively.More generally, whenever R is a Dedekind domain, we extend all the usual properties and operations holding for matroids (e.g., duality), and we explicitly describe the structure of the matroids over R. Furthermore, we compute the Tutte-Grothendieck ring of matroids over R. We also show that the Tutte quasi-polynomial of a matroid over \(\mathbb{Z}\) can be obtained as an evaluation of the class of the matroid in the Tutte-Grothendieck ring.

Ehrhart polynomial and arithmetic Tutte polynomial

We prove that the Ehrhart polynomial of a zonotope is a specialization of the arithmetic Tutte polynomial introduced by Moci (2012) 16]. We derive some formulae for the volume and the number of integer points of the zonotope.

Graph colorings, flows and arithmetic Tutte polynomial

We introduce the notions of arithmetic colorings and arithmetic flows over a graph with labelled edges, which generalize the notions of colorings and flows over a graph. We show that the corresponding arithmetic chromatic polynomial and arithmetic flow polynomial are given by suitable specializations of the associated arithmetic Tutte polynomial, generalizing classical results of Tutte (1954) [9].

Products of arithmetic matroids and quasipolynomial invariants of CW-complexes

Abstract In this note we prove that the product of two arithmetic multiplicity functions on a matroid is again an arithmetic multiplicity function. This allows us to answer a question by Bajo–Burdick–Chmutov [2] , concerning the modified Tutte–Krushkal–Renhardy polynomials defined by these authors. Furthermore, we show that the Tutte quasi-polynomial introduced by Branden and Moci encompasses invariants defined by Beck–Breuer–Godkin–Martin [3] and Duval–Klivans–Martin [11] and can thus be considered as a dichromate for CW complexes.