A survey on arithmetic Tutte polynomials: motivations, applications, open problems.

Several objects can be associated to a list of vectors with integer coordinates: a toric arrangement, a zonotope, a vector partition function. The linear algebra of the list is encoded by the notion of a matroid, but the topology of the toric arrangement, as well as several properties of the other objects mentioned above, depend also on the arithmetics of the list: this is retained by the notions of a "arithmetic matroid" and of a "matroid over Z". After introducing briefly these structures, we will focus on two of their invariants: the arithmetic Tutte polynomial and the Tutte quasi-polynomial. Among their applications, we will show one to colorings and flows on CW complexes, which can be seen as a higher-dimensional generalization of Tutte's theorem for graphs. Finally we will show that the set of arithmetic matroids on a given matroid is endowed by a natural product, which corresponds to a convolution product of the corresponding arithmetic Tutte polynomials.