A Tutte polynomial for toric arrangements

We introduce a multiplicity Tutte polynomial M(x,y), with applications to zonotopes and toric arrangements. We prove that M(x,y) satisfies a deletion-restriction recurrence and has positive coefficients. The characteristic polynomial and the Poincare' polynomial of a toric arrangement are shown to be specializations of the associated polynomial M(x,y), likewise the corresponding polynomials for a hyperplane arrangement are specializations of the ordinary Tutte polynomial. Furthermore, M(1,y) is the Hilbert series of the related discrete Dahmen-Micchelli space, while M(x,1) computes the volume and the number of integral points of the associated zonotope.


Introduction
The Tutte polynomial is an invariant naturally associated to a matroid and encoding many of its features, such as the number of bases and their internal and external activity ( [25], [5], [9]). If the matroid is realized by a finite list of vectors, it is natural to consider the arrangement obtained by taking the hyperplane orthogonal to each vector. One associates to the poset of the intersections of the hyperplanes its characteristic polynomial, which provides a rich combinatorial and topological description of the arrangement ( [22], [26]). This polynomial can be obtained as a specialization of the Tutte polynomial.
Given a complex torus T = (C * ) n and a finite list X of characters, i.e. elements of Hom(T, C * ), we consider the arrangement of hypersurfaces in T obtained by taking the kernel of each element of X. To understand the geometry of this toric arrangement one needs to describe the poset C(X) of the layers, i.e. connected components of the intersections of the hypersurfaces ( [8], [12], [19], [21]). Clearly this poset depends also on the arithmetics of X, and not only on its linear algebra: for example, the kernel of the identity character λ of C * is the point t = 1, but the kernel of 2λ has equation t 2 = 1, hence is made of two points. Therefore we have no chance to get the characteristic polynomial of C(X) as a specialization of the ordinary Tutte polynomial T X (x, y) of X. In this paper we define a polynomial M X (x, y) that specializes to the characteristic polynomial of C(X) (Theorem 5.6) and to the Poincaré polynomial of the complement R X of the toric arrangement (Theorem 5.11). In particular M X (1, 0) is equal to the Euler characteristic of R X , and also to the number of connected components of the complement of the arrangement in the compact torus T = (S 1 ) n .
We call M X (x, y) the multiplicity Tutte polynomial of X, since it coincides with T X (x, y) when X is unimodular and, in general, it satisfies the same deletion-restriction recursion that holds for T X (x, y). By this formula (Theorem 3.4) we prove that M X (x, y) has positive coefficients (Theorem 3.5). We leave open the problem of explaining the meaning of these coefficients (Problem 3.7).
A similar polynomial can be defined more generally for matroids, if we enrich their structure in order to encode some "arithmetic data"; we call such objects multiplicity matroids. We hope to develop in a future paper an axiomatic theory of these matroids, as well as applications to graph theory, which are only outlined here. The focus of the present paper is in the case we have a list X of vectors with integer coordinates.
Given such a list, we consider two finite dimensional vector spaces: a space of polynomials D(X) defined by differential equations, and a space of quasipolynomials DM (X) defined by difference equations. These spaces were introduced by Dahmen and Micchelli to study box splines and partition functions, and are deeply related respectively with the hyperplane arrangement and the toric arrangement defined by X, as explained in the recent book [9]. In particular, T X (1, y) is known to be the Hilbert series of D(X), and we will show that M X (1, y) is the Hilbert series of DM (X) (Theorem 6.3).
On the other hand, by Theorem 4.1 the coefficients of M X (x, 1) count integer points in some faces of a convex polytope, the zonotope defined by X. The relations between zonotopes and Dahmen-Micchelli spaces is being studied intensively over the last years, giving arise to many algebraic and combinatorial constructions (see in particular [9], [13], [10], [2], [14], [18]). In particular M X (1, 1) is equal to the volume of the zonotope (Proposition 2.2), while M X (2, 1) is the number of its integer points (Proposition 4.5).
Finally, we focus on the case in which X is a root system. We will then show some connections with the theory of Weyl groups (see for instance Corollary 7.4). D'Adderio, Emanuele Delucchi, Matthias Lenz, Mario Marietti, Claudio Procesi, Mimi Tsuruga and Michele Vergne for stimulating discussions and remarks.
2 Multiplicity matroids and multiplicity Tutte polynomials

Definitions
We start by recalling the notions we will be generalizing.
A matroid M is a pair (X, I), where X is a finite set and I is a family of subsets of X (called the independent sets) with the following properties: 1. The empty set is independent; 2. every subset of an independent set is independent; 3. let A and B be two independent sets and assume that A has more elements than B. Then there exists an element a ∈ A \ B such that B ∪ {a} is still independent.
A maximal independent set is called a basis. The last axiom implies that all bases have the same cardinality, which is called the rank of the matroid. Every A ⊆ X has a natural structure of matroid, defined by considering a subset of A independent if and only if it is in I. Then each A ⊆ X has a rank which we denote by r(A).
The Tutte polynomial of the matroid is then defined as From the definition it is clear that T (1, 1) is equal to the number of bases of the matroid.
The interested reader may refer for instance to [23], [9] In the next sections we will recall the two most important examples of matroid and some properties of their Tutte polynomials.
We now introduce the following definitions. A multiplicity matroid M is a triple (X, I, m), where (X, I) is a matroid and m is a function (called multiplicity) from the family of all subsets of X to the positive integers.
We say that m is the trivial multiplicity if it is identically equal to 1. We define the multiplicity Tutte polynomial of a multiplicity matroid as Let us remark that we can endow every matroid with the trivial multiplicity, and then M (x, y) = T (x, y).
Remark 2.1. Given any two matroids M 1 = (X 1 , I 1 ) and M 2 = (X 2 , I 2 ), we can naturally define a matroid M 1 ⊕M 2 = (X, I), where X is the disjoint union of X 1 and X 2 , and A ∈ I if and only if A 1 .
We notice that the rank of a subset A is just the sum of the ranks of A 1 and A 2 , and so it is easily seen that the (multiplicity) Tutte polynomial of M 1 ⊕ M 2 is the product of the (multiplicity) Tutte polynomials of M 1 and M 2 .

Lists of vectors and zonotopes
Let X be a finite list of vectors spanning a real vector space U , and I be the family of its linearly independent subsets. Then (X, I) is a matroid, and the rank of a subset A is just the dimension of the spanned subspace. We denote by T X (x, y) the associated Tutte polynomial.
We associate to the list X a zonotope, that is a convex polytope in U defined as follows: Zonotopes play an important role in the theory of hyperplane arrangements, and also in that of splines, a class of functions studied in Approximation Theory. (see [9]).
We recall that a lattice Λ of rank n is a discrete subgroup of R n which spans R n as a real vector space. Every such Λ can be generated from some basis of the vector space by forming all linear combinations with integer coefficients; hence the group Λ is isomorphic to Z n . We will use always the term lattice with this meaning, and not in the combinatorial sense (a poset with join and meet).
Now let X be a finite list of elements in a lattice Λ, and let I and r be as above. For every A ⊂ X, we denote by A Z and A R respectively the sublattice of Λ and the subspace of U . = Λ ⊗ R spanned by A. Let us define the largest sublattice of Λ in which A Z has finite index. We define m as this index: This defines a multiplicity matroid and then a multiplicity Tutte polynomial M X (x, y), which is the main subject of this paper.
Concretely, if we identify Λ with Z n , we have that for every A ⊂ X of maximal rank, m(A) is equal to the GCD of the determinants of the bases extracted from A.
We say that the list X is unimodular if every B ⊂ X that is a basis for U over R spans Λ over Z (i.e, in the identification above, B has determinant ±1). In this case the multiplicity is trivial and M X (x, y) = T X (x, y).
We now start by showing some relations with the zonotope Z(X) generated by X in U .
We already observed that T X (1, 1) is equal to the number of bases that can be extracted from X. On the other hand we have Further relations between the polynomial M X (x, y) and the zonotope Z(X) will be shown in Section 4.

Graphs
Let G be a finite graph, V (G) be the set of its vertices, and X be the set of its edges. For every A ⊆ X, we consider the subgraph of G having set of vertices V (G) and set of edges A. By abuse of notation, we denote this subgraph by A. We define I as the set of the forests in G ( subgraphs whose connected components are simply connected). Then (X, I) is a matroid with rank function where c(A) is the number of connected components of A.
Remark 2.3. If G has neither loops nor multiple edges, let us take a vector space U with basis e 1 , . . . , e n bijective to V (G), and associate to the edge connecting two vertices i and j the vector e i − e j . In this way we get a list X G of vectors bijective to X and spanning a hyperplane U in U . Since in this bijection the rank is preserved and forests correspond to linearly independent sets, G and X G define the same matroid and have the same Tutte polynomial. Now let us assume every edge e ∈ X to have an integer label m e > 0. By defining we get a multiplicity matroid and then a multiplicity Tutte polynomial M G (x, y). We may view the labels m e as multiplicities of the edges in the following way. Let us define a new graph G m with the same vertices as G, but with m e edges connecting the two vertices incident to e ∈ X. Now let S(G m ) be the set of simple subgraphs of G m , i.e subgraphs with at most one edge connecting any two vertices, and at most one loop on every vertex. It is then clear that In particular, M G (2, 1) is equal to the number of forests of G m , and M G (1, 1) the number of spanning trees (i.e., trees connecting all the vertices) of G m .

Deletion-restriction formula and positivity
The central idea that inspired Tutte in defining the polynomial T (x, y) was to find the most general invariant satisfying a recursion known as deletionrestriction. Such a recursion allows to reduce the computation of the Tutte polynomial to some trivial cases.
In the two examples above (when the matroid is defined by a list of vectors or by a graph), the polynomial M (x, y) satisfies a similar recursion. We will explain and prove the algorithm in the first case. The case of graphs (where the recursion is known as deletion-contraction) is described in [20, Section 2.3.1].

Lists of vectors
Let X be a finite list of elements spanning a vector space U , and let λ ∈ X be a nonzero element. We define two new lists: the list X 1 . = X \ {λ} of elements of U and the list X 2 of elements of U/ λ obtained by reducing X 1 modulo λ. Assume that λ is dependent in X, i.e. λ ∈ X 1 R . We have the following well-known formula: It is now clear why we defined X as a list, and not as a set: even if we start with X composed by distinct (nonzero) elements, some vectors in X 2 may appear many times (and some vectors may be zero).
Notice that by applying the above formula recursively, our problem reduces to computing T Y (x, y) when Y is the union of a list Y 1 of k linearly independent vectors and of a list Y 0 of h zero vectors (k, h ≥ 0). In this case the Tutte polynomial is easily computed.

Hence by induction we get that
Thus we get: is a polynomial with positive coefficients.

Lists of elements in abelian groups.
We now want to show a similar recursion for the polynomial M X (x, y). Inspired by [11], we noticed that we need to work in a larger category. Indeed, whereas the quotient of a vector space by a subspace is still a vector space, the quotient of a lattice by a sublattice is not a lattice, but a finitely generated abelian group. For example, in the 1-dimensional case, the quotient of Z by mZ is the cyclic group of order m. Then let Γ be a finitely generated abelian group. For every subset S of Γ we denote by S the generated subgroup. We recall that Γ is isomorphic to the direct product of a lattice Λ and of a finite group Γ t , which is called the torsion subgroup of Γ. We denote by π the projection π : Γ → Λ.
Let X be a finite subset of Γ. For every A ⊆ X we set In other words, Γ A is the largest subgroup of Γ in which A has finite index.
We also define r(A) as the rank of π(A). In this way we defined a multiplicity matroid, to which we associate its multiplicity Tutte polynomial: It is clear that if Γ is a lattice, these definitions coincide with those given in the previous sections.
If on the opposite case, Γ is a finite group, then M (x, y) is a polynomial in which only the variable y appears. Furthermore, this polynomial evaluated at y = 1, gives the order of Γ. Indeed the only summand that does not vanish is the contribution of the empty set, which generates the trivial subgroup. Now let λ ∈ X be a nonzero element such that We set For every A ⊆ X, we denote by A its image under the natural projection Since Γ/ λ is a finitely generated abelian group and A is a subset of it, We denote by X 2 the subset X 1 of Γ/ λ . We have the following deletionrestriction formula.
Proof. The sum expressing M X (x, y) splits into two parts. The first is over the sets A ⊆ X 1 since clearly r(X) = r(X 1 ). The second part is over the sets A such that λ ∈ A. For such sets we have that

Now we can prove
Theorem 3.5. M X (x, y) is a polynomial with positive coefficients.
Proof. By applying recursively the formula above, we need only consider lists that do not contain any λ satisfying condition (2). Any such list Y is made of elements of some quotient Γ(Y ) of Γ, and is the disjoint is the product of the lattice and of the torsion subgroup). We first notice that Furthermore, it is easily seen that Finally, the positivity of M Y 1 (x, y) will be proved in Lemma 4.4.

Statistics
Usually, polynomials with positive coefficients encode some statistics. In other words, their coefficients count something. For example, the Tutte polynomial embodies two statistics on the set of the bases called internal and external activity. Although they can be stated for an abstract matroid (see for example [9, Section 2.2.2]), we give such definitions for a list X of vectors. Let B be a basis extracted from X.
1. We say that v ∈ X \ B is externally active for B if v is a linear combination of the elements of B following it (in the total ordering fixed on X); 3. the number e(B) of externally active elements is called the external activity of B; 4. the number i(B) of internally active elements is called the internal activity of B.
The following result is proved in [5]: .
Hence the coefficients of T X (x, y) count the number of bases having a given internal and external activity.
Since M X (x, y) has positive coefficients too, it is natural to wonder what the statistics involved are. We say that a basis B of X is a no-broken circuit if e(B) = 0. We denote by nbc(X) the number of no-broken circuit bases of X. It is clear from Theorem 3.6 that nbc(X) = T X (1, 0).
We will use this formula in the following sections.

Integer points in zonotopes
Let X be a finite list of vectors contained in a lattice Λ and generating the vector space U = Λ⊗R. We say that a point of U is integer if it is contained in Λ. In this section we prove that M X (2, 1) is equal to the number of integer points of the zonotope Z(X). Moreover, we compare this number with the volume. To do this, we have to move the zonotope to a "generic position". Following [9, Section 1.3], we define the cut-locus of the couple (Λ, X) as the union of all hyperplanes in U that are translations, under elements of Λ, of the linear hyperplanes spanned by subsets of X. Let ε be a vector of U that does not lie in the cut-locus and has length ε << 1. Let Z(X) − ε be the polytope obtained by translating Z(X) by −ε, and let I(X) be the set of its integer points: It is intuitive (and proved in [9,Prop 2.50]) that this number is equal to the volume: |I(X)| = vol (Z(X)) = M X (1, 1) by Proposition 2.2. We now prove a stronger result. Let us choose ε so that Z(X) − ε contains the origin 0. We partition I(X) as follows: set I n (X) = {0}, and for every k = n − 1, . . . , 0, let I k (X) be the set of elements of I(X) that are contained in some k−codimensional face of Z(X) and that are not contained in I h (X) for h > k. Then we have: Then M X (x, y) = (x − 1) 2 + (3 + 1 + 2)(x − 1) + (6 + 6 + 2) + 2(y − 1).

Hence
M X (x, 1) = x 2 + 4x + 9 and M X (2, 1) = 21. Indeed the zonotope Z(X) has area 14 and contains 21 integer points, 14 of which lie in Z(X) − ε, represented by the shaded portion in the image below. The sets I 2 (X), I 1 (X), and I 0 (X) contain 1, 4 and 9 points, respectively, each marked by the subscript of its set.
In order to prove the theorem above, we first assume X to be linearly independent (and hence a basis for U ). In this case the zonotope is called a parallelepiped. In particular, for every A ⊆ X the zonotope Z(A) is a face of Z(X). Moreover, we can choose ε so that the faces of Z(X) whose interior is contained in Z(X) − ε are precisely those of type Z(A) for some A ⊆ X: for instance take ε = λ∈X ε n λ.
We say that an integer point is internal to a face of Z(X) if that face is the smallest face containing this point. We denote by h(A) the number of integer points that are internal to Z(A).  Proof. By definition The coefficient of x k in this expression is By the previous Lemma, our claim amounts to proving that the coefficient of In this way the theorem is proved for linearly independent set of vectors, since |I k (X)| = A⊆X, |A|=n−k h(A).
As in Section 3.2, given a nonzero element λ ∈ X we set X 1 . = X \ {λ} and we denote by X 2 the image of X 1 under the natural projection Note that X 1 defines a zonotope Z(X 1 ) ⊂ Z(X), and X 2 defines a zonotope Z(X 2 ) in the quotient space p(U ). We briefly recall some results from [9, Section 2.3]. The reader is suggested to look at the picture below (in which Z(X 1 ) is the white rectangle, and λ is the vector of coordinates (2, 0)). The 1-parameter group of translations u → u + tλ, acting on U , has orbits that are the fibers of p. For every u ∈ Z(X 2 ), let s(u) be the point u + tλ where the translation group "exits" the zonotope Z(X 1 ). Then s(Z(X 2 )) (the bold line in the picture) is a piece of the boundary of Z(X 1 ) naturally identified to Z(X 2 ). Furthermore, we have the following decomposition: where Z(X 1 ) ∩ Z(X 2 ) = s(Z(X 2 )), and the polytope Z(X 2 ) (shaded in the picture) is a product of the polytope s(Z(X 2 )) and the segment [0, m λ ]. So p maps Z(X 2 ) on Z(X 2 ) with fiber [0, m λ ] ([9, Section 2.3]).
We can now conclude the proof of Theorem 4.1.
Proof. We can assume X contains only nonzero elements. If X is independent, we have Lemma 4.4. If X has rank 0, the statement is trivial. Thus we assume X to be dependent and of positive rank, we choose a vector λ that lies in the cone spanned by the other vectors, and we proceed inductively by deletion-restriction. We notice that the restriction of p to I k (X) \ I k (X 1 ) maps this set onto I k (X 2 ), and the fiber of every point has cardinality m λ . Thus by inductive hypothesis, since m λ is the torsion of Λ/ λ Z . Hence the claim follows by Theorem 3.4.
In the same way we can prove Proposition 4.5. The number |Z(X) ∩ Λ| of integer points in the zonotope is equal to M X (2, 1).
Proof. By applying deletion-restriction as in the previous proof, we can reduce to the case in which X is a basis of U . Then in this basis, Z(X) is a parallelepiped. For every face F we define A F as the subset of X corresponding to the coordinates which are not constant on F . Since all the other coordinates are identically equal either to 0 or to 1, for every A ⊆ X there are exactly 2 k faces F such that A F = A, k = |X \ A|. Among these faces, the only one contributing to M X (1, 1) is the one whose constant coordinates are all equal to 0, i. e., Z(A). On the other hand, to compute the total number of integer points we have to take all these 2 k faces. Since any two of them are disjoint and contain the same number of points in their interior, by Theorem 4.1 we get the claim.

Application to arrangements
In this Section we describe some geometrical objects related to the lists considered in Section 2.2, and show that many of their features are encoded in the polynomials T X (x, y) and M X (x, y).

Recall on hyperplane arrangements
Let X be a finite list of elements of a vector space U . A hyperplane arrangement H(X) is defined in the dual space V = U * , by taking the orthogonal hyperplane of each element of X. Conversely, given an arrangement of hyperplanes in a vector space V , let us choose for each hyperplane a nonzero vector in V * orthogonal to it. Let X be the list of such vectors. Since every element of X is determined up to scalar multiples, the matroid associated to X is well defined. In this way a Tutte polynomial is naturally associated to the hyperplane arrangement.
The importance of the Tutte polynomial in the theory of hyperplane arrangements is well known. Here we recall some results which will be generalized in the next sections.
To every sublist A ⊆ X we associate the subspace A ⊥ of V that is the intersection of the corresponding hyperplanes of H(X). In other words, A ⊥ is the subspace of vectors that are orthogonal to every element of A. Let L(X) be the set of such subspaces, partially ordered by reverse inclusion, and having as minimal element (which we denote by 0) the whole space V = ∅ ⊥ . L(X) is called the intersection poset of the arrangement, and is the main combinatorial object associated to a hyperplane arrangement.
We also recall that to every finite poset P there is an associated Moebius function µ : P × P → Z recursively defined as follows: Notice that the poset L(X) is ranked by the dimension of the subspaces. We define characteristic polynomial of the poset as This is an important invariant of H(X). Indeed, let M X be the complement in V of the union of the hyperplanes of H(X). Let P (q) be Poincaré polynomial of M X , i.e. the polynomial having the k−th Betti number of M X as the coefficient of q k . If V is a complex vector space, by [22] we have the following theorem. If, on the other hand, V is a real vector space, by [26] the number Ch(X) of chambers (i.e., connected components of M X ) is: Ch(X) = (−1) n χ(−1).
The Tutte polynomial T X (x, y) turns out to be a stronger invariant, in the following sense. Assume that 0 / ∈ X; then The proof of these theorems can be found, for example, in [9, Theorems 10.5, 2.34 and 2.33].
Notice that in the present section we have only considered central arrangements, i.e. arrangements in which all the hyperplanes contain the origin. More generally, one may consider arrangements of affine hyperplanes in affine spaces. In this setting, Ardila defined a notion of semimatroid and a Tutte polynomial (see [1]).

Toric arrangements and their generalizations
Let Γ = Λ × Γ t be a finitely generated abelian group, and define T Γ has a natural structure of abelian linear algebraic group: indeed it is the direct product of a complex torus T Λ of the same rank as Λ and of the finite group Γ t * dual to Γ t (and isomorphic to it). Moreover, Γ is identified with the group of characters of T Γ : indeed given λ ∈ Λ and t ∈ T Γ we can take any representative ϕ t ∈ Hom(Γ, C) of t and set λ(t) . = e 2πiϕt(λ) .
When this is not ambiguous we will denote T Γ by T . Let X ⊂ Λ be a finite subset spanning a sublattice of Λ of finite index. The kernel of every character χ ∈ X is a (non-connected) hypersurface in T : The collection T (X) = {H χ , χ ∈ X} is called the generalized toric arrangement defined by X on T . We denote by R X the complement of the arrangement: and by C X the set of all the connected components of all the intersections of the hypersurfaces H χ , ordered by reverse inclusion and having as minimal elements the connected components of T .
Since rank(Λ) = dim(T ), the maximal elements of C(X) are 0-dimensional, hence (since they are connected) they are points. We denote by C 0 (X) the set of such layers, which we call the points of the arrangement.
Given A ⊆ X let us define induces a bijection between the connected components of H A and the points of T (A).
In particular, when Γ is a lattice, T is a torus and T (X) is called the toric arrangement defined by X. Such arrangements have been studied for example in [17], [8], [19], [21]; see [9] for a complete reference. In particular, the complement R X has been described topologically and geometrically. In this description the poset C(X) plays a major role, for many aspects analogous to that of the intersection poset for hyperplane arrangements (see [8], [21]).
We will now explain the importance in this framework of the polynomial M X (x, y) defined in Section 3.3.

Characteristic polynomial
Let µ be the Moebius function of C(X). Notice that we have a natural rank function given by the dimension of the layers. For every C ∈ C(X), let T C be the connected component of T that contains C. We can define the characteristic polynomial of C(X) as In order to relate this polynomial with M X (x, y), we prove the following fact. Let us assume that X does not contain 0. For every C ∈ C(X), set Proof. By induction on the codimension of C. If it is 0 or 1, the statement is trivial. Otherwise, by the inductive hypothesis Proving that this sum is equal to the claimed one amounts to prove that  Hence The polynomial M X (x, y) is composed by the following summands: • (x − 1) 2 , corresponding to the empty set; • 6(x − 1), corresponding to the 4 singletons, each giving contribution (x − 1) or 2(x − 1); • 14, corresponding to the 6 pairs: indeed, the basis X = {(2, 0), (0, 2)} spans a sublattice of index 4, while the other bases span sublattices of index 2; • 8(y−1), corresponding to the 4 triples, each contributing with 2(y−1); • 2(y − 1) 2 , corresponding to the whole set X.

Notice that
as claimed in Theorem 5.6.

Poincaré polynomial
For every C ∈ C X , let us define Remark 5.8. The set X C defines a hyperplane arrangement in the vector space Let L(X C ) be its intersection poset. Let C(X, C) be the poset of the elements of C(X) that contain C. The map is an order-preserving bijection. Indeed, given L ∈ L(X C ), set Proof. By the previous remark, since X C ⊥ is the origin in V C , and hence the only element of rank 0. Thus by Theorem 4.3 and Formula (3), Let T 1 , . . . , T h be the connected components of T . We denote by C(X) i the set of layers that are contained in T i . This clearly gives a partition of the layers: We now give some formulae for the Poincaré polynomial P (q) and the Euler characteristic of R X . We start from a restatement of a result proved in [8,Theor. 4.2] (see also [9, 14.1.5]). In this paper is considered an arrangement of hypersurfaces in a torus, in which every hypersurface is obtained by translating by an element of the torus the kernel of a character. It is clear that the restriction of the arrangement T (X) on every T i is an arrangement of this kind. Then the cohomology of R X ∩T i can be expressed as a direct sum of contributions given by the layers of this arrangement, which are the elements of C(X) i . In terms of the Poincaré polynomial P i (q) of R X ∩ T i , this expression is: Thus the Poincaré polynomial of R X = i (R X ∩ T i ) is just the sum of these polynomials: Theorem 5.10.

Now we prove:
Theorem 5.11.
Proof. By definition, we have that We compare this formula with the one in the previous Theorem. We have to prove that for every k = 0, . . . , n the coefficient of (q + 1) k q n−k is the same in the two expressions. In fact by applying Formula (4) and then Lemma 4.9 we get the claim: Therefore, by comparing Theorem 5.6 and Theorem 5.11, we get the following formula, which relates the combinatorics of C(X) with the topology of R X , and is the "toric" analogue of Theorem 4.1.
We recall that the Euler characteristic of a space can be defined as the evaluation at −1 of its Poincaré polynomial. Hence by Theorem 5.11 we have: Corollary 5.13. (−1) n M X (1, 0) is equal to the Euler characteristic of R X .

Number of regions of the compact torus
In this section we consider the compact abelian group dual to Γ We assume for simplicity Γ to be a lattice: then T is a compact torus, i.e. it is isomorphic to (S 1 ) n , and every χ ∈ X defines a hypersurface in T : We denote by T (X) this arrangement. Clearly its poset of layers is the same as for the arrangement T (X) defined in the complex torus T . We denote by R X the complement The compact toric arrangement T (X) has been studied in [12]; in particular the number R(X) of regions (i.e. of connected components) of R X is proved to be a specialization of the characteristic polynomial χ(q): By comparing this result with Theorem 5.6 we get the following But by Lemma 5.4 which is the number of polynomials T Xp in which the summand (y − 1) |A|−n appears.
The previous lemma has an interesting consequence. Following [4], [6], and [7], we associate to every finite set X ⊂ V a space D(X) of functions V → C, and to every finite set X ⊂ Λ a space DM (X) of functions Λ → C. Such spaces are defined as the solutions to a system, of differential equations and of difference equations respectively, in the following way.
For every λ ∈ X, let ∂ λ be the usual directional derivative and let ∇ λ be the difference operator For every A ⊂ X, we define the differential operator The space D(X) is a space of polynomials, which was introduced in order to study the box spline. This is a piecewise-polynomial function studied in Approximation Theory; its local pieces, together with their derivatives, span D(X). On the other hand, DM (X) is a space of quasipolynomials which arises in the study of the partition function. We recall that a function f on a lattice Λ is a quasipolynomial if there exists a sublattice Λ 0 of finite index such that f restricted to every coset coincides with a polynomial. The partition function is the function that counts how many ways an element of Λ can be written as a linear combination with nonnegative integer coefficients of elements of X. This function is a piecewise-quasipolynomial, and its local pieces, together with their translates, span DM (X).
In the recent book [9] the spaces D(X) and DM (X) are shown to be deeply related with the hyperlane arrangement and with the toric arrangement defined by X respectively.
In order to compare these two spaces, we consider the elements of D(X) as functions Λ → C by restricting them to the lattice Λ. Since the elements of DM (X) are polynomial functions, they are determined by their restriction. For every p ∈ C 0 (X), let us define the following map: (see Section 2.4.2). In [7] (see also [9,Formula 16.1]) the following result is proved.
2. One may take as Λ the root lattice (or equivalently, replace the coroot lattice by the character lattice in the construction above). But then one obtains as T a maximal torus of the semisimple adjoint group G a , which is the quotient of G by its center.
Toric arrangements defined by root systems have been studied in [19]; we now show two applications to the present work. Let W be the (finite) Weyl group of Φ, and let W be the associated affine Weyl group. We denote by s 0 , . . . , s n its generators, and by W k the subgroup of W generated by all the elements s i but s k . Let Φ k ⊂ Φ be the root system of W k , and denote by X k the corresponding sublist of X. Then we have: Proof. Straightforward from [19, Theor. 1] and Lemma 6.1.
Furthermore, in [19] the following theorem is proved. Let W be the Weyl group of Φ. Theorem 7.3. The Euler characteristic of R X is equal to (−1) n |W |.
By comparing this statement with Corollary 5.13, we get the following It would be interesting to have a more direct proof of this fact.
Example 7.5. The toric arrangement described in Example 5.7 correspond to the root system of type C 2 . Notice that the order of the Weyl group of type C 2 is 8 = P (−1) = M X (1, 0) = R(X).