On Igusa zeta functions of monomial ideals

We show that the real parts of the poles of the Igusa zeta function of a monomial ideal can be computed from the torus-invariant divisors on the normalized blowing-up along the ideal. Moreover, we show that every such number is a root of the Bernstein-Sato polynomial associated to the monomial ideal.


Introduction
If f is a nonconstant polynomial in Z[x 1 , . . . , x n ] and p is a fixed prime, then the Igusa zeta function of f is defined by for every s ∈ C with Re(s) > 0. Here Z p denotes the ring of p-adic integers, endowed with the discrete valuation ord p and with the p-adic absolute value defined by .
The measure µ on (Z p ) n that is used in the above integral is the Haar measure characterized by As defined, Z f is a holomorphic function on the half plane {s | Re(s) > 0} and one can show that it admits a meromorphic extension to C. In fact, Z f is a rational function of 1 p s . A proof of rationality that gives also information on the real parts of the possible poles of Z f proceeds as follows. Let π : Y → X = A n be an embedded resolution of singularities of f defined over Q. This means that π is proper and birational, Y is smooth and the union of π * (div(f )) and of the exceptional locus of π is a divisor with simple normal crossings. For every prime divisor E in this union, denote by a E (f ) the order of E in π * (div(f )) and by k E the order of E in the relative canonical class K Y /X (this is the divisor locally defined by det(Jac(π))). Using the Change of Variable formula for p-adic integrals to express Z f as an integral over Y (Z p ), Igusa obtained the rationality of Z f as a function of 1 p s and the fact that if s is a pole of Z f , then Re(s) = − k E +1 a E (f ) for some divisor E as above. Our main reference for Igusa zeta functions is Igusa's book [Ig] (see also Denef's Bourbaki report [De]).
While every divisor on a log resolution of f gives a candidate for the real part of a pole of Z f , examples show that most of these numbers do not come actually from poles of Z f . In fact an outstanding open problem in the field is the following conjecture of Igusa, relating the poles to one of the basic invariants of the singularities of f , its Bernstein-Sato polynomial.
Conjecture 1.1. Let f be a non-constant polynomial in Z[X 1 , . . . , X n ]. For almost all primes p the following holds: if s is a pole of Z f , then the real part of s is a root of the Bernstein-Sato polynomial b f of f .
We recall that the Bernstein-Sato polynomial of f is a polynomial in one variable introduced independently in [Be] and [SS]. It is a subtle but very fundamental invariant of the singularities of f . We do not give its definition as we will not need it, but we mention that its roots are related to the eigenvalues of the monodromy of the hypersurface f −1 (0). There is, in fact, a weaker version of the above conjecture that is stated in terms of these eigenvalues and that is known as the Monodromy Conjecture (see [De] for more on these conjectures and also [Ve] for some recent work in this direction).
Our goal in this note is to prove the analogue of Conjecture 1.1 when we replace f by a monomial ideal. Though less studied, Igusa zeta functions for non-necessarily principal ideals in Z[x 1 , . . . , x n ] can be defined in a very similar way with (1). More precisely, if I is a nonzero proper ideal of Z[x 1 , . . . , x n ] and if we put for y ∈ (Z p ) n ord p I(y) = min{ord p (f (y))|f ∈ I}, then we have |dy|.
The above-mentioned results in the case of one polynomial extend in a straightforward way to the case of an arbitrary ideal. Note that in order to prove rationality, we need to consider a log resolution of I: this is a morphism π : Y → A n as before, such that π −1 (V (I)) is a Cartier divisor and its union with the exceptional locus of π is a divisor with simple normal crossings. If E is a prime divisor on Y contained in this union, then a E (I) is by definition the coefficient of E in π −1 (V (I)). As in the case of a principal ideal, one can show that given a log resolution π, for every pole s of Z I there is a divisor E on Y such that Re(s) = − k E +1 a E (I) . On the other hand, the definition of the Bernstein-Sato polynomial has been extended in [BMS3] from the case of one polynomial to that of an arbitrary ideal. This is again a polynomial in one variable and therefore the analogue of Conjecture 1.1 makes sense in this case. We will prove the monomial case, i.e. when I is generated by monomials.
Theorem 1.2. If I is a nonzero proper monomial ideal of Z[x 1 , . . . , x n ], then for every prime p and every pole s of Z I , the real part of s is a root of the Bernstein-Sato polynomial of I.
The key ingredient in the proof of the above theorem is a result on the poles of Igusa-type zeta functions associated to cones. Suppose that N ≃ Z n is a lattice and that σ is a pointed, rational, polyhedral cone in .
It is easy to see (and it will follow from our computations) that this is well-defined and holomorphic in {s ∈ C | Re(s) > 0}. We prove the following Theorem 1.3. For every σ, ℓ 1 and ℓ 2 as above, Z σ,ℓ 1 ,ℓ 2 is a rational function of 1 p s , and therefore can be meromorphically extended to C. Moreover, for every pole s of Z σ,ℓ 1 ,ℓ 2 there is a primitive generator v of a ray of σ such that Given a monomial ideal I, we give in the next section a formula for Z I in terms of suitable zeta functions for the cones in the normal fan to the Newton polyhedron of I (we refer for the relevant definition and for the precise formula to that section). Let us just mention that this fan defines the toric variety that is the normalized blowing-up of A n along the ideal I. Using this formula and Theorem 1.3, we will show in §3 that the real part of every pole of Z I corresponds to a torus-invariant divisor in the normalized blow-up of A n along I (despite the fact that the normalized blowing-up is not a log resolution of I).
Theorem 1.4. Let I be a nonzero proper monomial ideal of Z[x 1 , . . . , x n ]. For every pole s of Z I , there is a torus-invariant divisor E on the normalized blowing-up of A n along I such that On the other hand, explicit descriptions of the roots of the Bernstein-Sato polynomial of a monomial ideal have been obtained in [BMS1] and [BMS2]. We use the description in [BMS2] and Theorem 1.4 to prove Theorem 1.2 in the last section.
We mention that a description for the Igusa zeta function of a monomial ideal has also been obtained by Zúñiga-Galindo in [Zu]. Moreover, similar results appear in the work of Denef and Hoornert [DH], in which one describes the poles of the Igusa zeta functions for nondegenerate hypersurfaces with respect to their Newton polyhedron. It is shown in loc. cit. that for such f the real part of essentially any pole corresponds to a facet of the Newton polyhedron of f , as above. Moreover, Loeser [Lo] showed that under some mild extra assumptions, these numbers are roots of the Bernstein-Sato polynomial of f , thus proving Conjecture 1.1 for such nondegenerate hypersurfaces. On the other hand, note that the relations between the respective Igusa zeta functions and Bernstein-Sato polynomials of f and of the corresponding monomial ideal are not clear in general.

Igusa zeta functions of monomial ideals
Let I be a nonzero proper ideal of Z[x 1 , . . . , x n ] generated by monomials. If u = (u 1 , . . . , u n ) ∈ N n , we denote by x u the corresponding monomial x u 1 1 . . . x un n . The Newton polyhedron P I of I is the convex hull of those u in N n such that x u is in I.
We denote by ·, · the canonical pairing between M and N. If we consider in N R the cone generated by the elements of the standard basis e 1 , . . . , e n , then the corresponding toric variety is the affine space A n and the subscheme V (I) is invariant under the torus action (we refer to [Fu] for the basic notions on toric varieties). Hence the normalized blowing-up of A n along I is again a toric variety, and therefore it corresponds to a fan subdividing the above cone. This is the normal fan to the polyhedron P I , that we will denote by ∆ I . It is defined as follows: to each face Q of P I one associates the cone The fan ∆ I consists of the cones σ Q , when Q varies over the faces of P I . Note that dim(σ Q ) = n − dim(Q), so the rays of ∆ I correspond to the facets of P I , and the maximal cones of ∆ I correspond to the vertices of P I .
Let p be a fixed prime. We proceed now to the computation of Z I . For every a = (a 1 , . . . , a n ) ∈ N n we consider the set C a = n i=1 (p a i Z p p a i +1 Z p ). Since each p a i Z p p a i +1 Z p is a disjoint union of (p − 1) translates of p a i +1 Z p , we see that We denote by e the vector (1, . . . , 1), so e, a = i a i .
The function ord p I is constant on C a with value ν(a) := min{ u, a | x u ∈ I} = min{ u, a |u ∈ P I }.
Since the sets C a are disjoint and the complement of their union has measure zero, we deduce .
Note that ν is a linear function on each of the cones in ∆ I . Indeed, if w is a vertex of P I , then ν(a) = w, a whenever a is in σ w .
If σ is a cone in ∆ I , choose a vertex w of P I such that σ is contained in σ w and put ℓ σ := w. By letting the a in (4) vary inside the relative interior of each cone in ∆ I , and using the definition in the Introduction, we get the following Proposition 2.1. With the above notation, we have Z σ,ℓσ,e (s).

Igusa zeta functions for cones
Our goal now in this section is to discuss Igusa-type zeta functions associated to cones and prove Theorem 1.3. Let N be a lattice, M its dual, and σ a pointed, rational polyhedral cone in N R . We consider ℓ 1 and ℓ 2 in σ ∨ ∩ M, where σ ∨ is the dual cone of σ, such that σ ∩ {v | ℓ 2 (v) = 0} = {0}. We want to study the function Z σ,ℓ 1 ,ℓ 2 and its poles.
The definition of Z σ,ℓ 1 ,ℓ 2 was motivated by the formula in Proposition 2.1, but sometimes it is more natural to consider a version of this function in which we sum over all the integer points in σ: .
We start with the following The assertions in the lemma are direct consequences of this formula.
We can give now the proof of our result on Igusa-type zeta functions associated to cones.
Proof of Theorem 1.3. Arguing by induction on the dimension of σ, we may assume that the theorem holds for all cones of smaller dimension than dim(σ) (the case when dim(σ) is zero being trivial). In this case, we see that proving the assertions in the theorem for Z σ,ℓ 1 ,ℓ 2 is equivalent with proving them for Z σ,ℓ 1 ,ℓ 2 .
We show first that it is enough to prove the theorem when σ is a simplicial cone. Indeed, it is well-known that one can always find a fan Γ refining the cone σ such that every cone in Γ is simplicial and the one-dimensional cones in Γ are precisely the rays of σ. Since and since each ray of a cone in Γ is a ray of σ, we see that it is enough to prove the theorem for each (maximal) cone in Γ.
Therefore we may assume that σ is simplicial and our goal is to show that Z σ,ℓ 1 ,ℓ 2 satisfies the assertions in the theorem. Let v 1 , . . . , v r be the primitive generators of the rays of σ. Since σ is simplicial, the v i are linearly independent over Q. The semigroup σ ∩ N is finitely generated, so we may choose generators w 1 , . . . , w s . The v i span σ over Q, hence we can find a positive integer m such that every mw j is in the semigroup generated by the v i . It follows that after replacing {w 1 , . . . , w s } by {q 1 w 1 + . . . + q s w s | 0 ≤ q j ≤ m − 1}, we may assume that where S is the semigroup generated by the v i .
If I ⊆ {1, . . . , s}, let us put where the sum is over v in ∩ j∈I (w j + S). We claim that ∩ j∈I (w j + S) is either empty or it is equal to w + S for a suitable w in N. Indeed, by an obvious induction on |I| it is enough to show this when I has two elements, say j and k. The intersection of w j + S and w k + S is nonempty if and only if w j − w k lies in the lattice generated by the v i . If this is the case, let us write w j − w k = r i=1 a i v i for suitable integers a 1 , . . . , a r . If we put w = w j + r i=1 max{0, −a i }v i , then it is easy to check that (w j + S) ∩ (w k + S) = w + S, which proves our claim.
It follows from our claim and Lemma 3.1 that each A I is a rational function of 1 p s . Moreover, if s is a pole of A I , then there is i such that Re(s) = − ℓ 2 (v i ) ℓ 1 (v i ) . On the other hand, it follows from (7) that where the sum is over all nonempty subsets I of {1, . . . , s}. Therefore Z σ,ℓ 1 ,ℓ 2 satisfies the assertions of the theorem, which completes the proof.
Putting together Theorem 1.3 and the description of the Igusa zeta function of a monomial ideal from the previous section we can relate the poles of this zeta function with the torus-invariant divisors in the blowing-up along the ideal.
Proof of Theorem 1.4. It follows from Proposition 2.1 and Theorem 1.3 that if s is a pole of Z I , then there is a primitive generator v of a ray of the normal fan ∆ I to the Newton polyhedron P I such that Here w is a vertex of P I such that v is contained in the maximal cone σ w of ∆ I corresponding to w.
On the other hand, recall that the torus-invariant divisors on the toric variety defined by ∆ I correspond precisely to the rays of ∆ I . Moreover, if E is the divisor corresponding to the ray through v, then k E = e, v − 1. Since we also have as v lies in σ w , we deduce the assertion in the theorem.

Poles and roots of the Bernstein-Sato polynomial
We show now that the real part of any pole of Z I is a root of the Bernstein-Sato polynomial b I associated to I. In fact, we prove the following stronger statement that together with Theorem 1.4 implies Theorem 1.2.
Proposition 4.1. If I is a nonzero proper monomial ideal and if E is a prime divisor in the normalized blowing-up of the affine space along I such that a E (I) is nonzero, then − k E +1 a E (I) is a root of the Bernstein-Sato polynomial b I .
Proof. The divisor E corresponds to a ray in the normal fan ∆ I to P I . Let v be a primitive generator of this ray. If w is a vertex of P I such that the corresponding maximal cone σ w of ∆ I contains v, then we have seen in the proof of Theorem 1.4 that k E + 1 = e, v and a E (I) = w, v . Note that since w, v = 0, the facet Q of P I corresponding to v is not contained in a coordinate hyperplane: if, for example, Q is contained in the hyperplane (x i = 0), then v = e i and since w lies in Q we get w, v = 0, a contradiction.
In order to show that (k E + 1)/a E (I) is a root of the Bernstein-Sato polynomial b I associated to I, we use the description of the roots of b I from [BMS2] (in fact, the ones that we need for the theorem are "the most straightforward" of the roots of b I ). Since Q is a facet of P I that is not contained in a coordinate hyperplane, there is a unique linear function L Q on M R such that Q = P I ∩ L −1 Q (1). With this notation, it is shown in [BMS2] (see Remark 4.6) that −L Q (e) is a root of b I .
On the other hand, since the ray through v corresponds to the facet Q and since w is in Q, we have Therefore L Q is given by 1 w,v ·v and since −L Q (e) is a root of b I , we see that (k E +1)/a E (I) is, indeed, a root of b I .
Remark 4.2. We do not know whether the analogue of Proposition 4.1 holds for a nonnecessarily monomial ideal I. Note that if I = (f ) is principal, then the assertion is trivial: the divisor E is one of the irreducible components of the divisor H defined by f , k E = 0 and a E (I) is the multiplicity of E in H. The fact that − 1 a E (f ) is a root of b f follows then by restricting to an open subset where E is smooth and H = a E (f ) · E.
Remark 4.3. The arguments in the previous two sections can be used to analyze also the orders of the possible poles of the Igusa zeta function Z I . Indeed, it follows from Proposition 2.1 and from the proof of Theorem 1.3 that if s is a pole of order r of Z I , then r ≤ n and there are r invariant divisors E 1 , . . . , E r on the normalized blowing-up along I such that E 1 ∩ . . . ∩ E r = ∅ and Re(s) = −(k E i + 1)/a E i (I) for every i. We would like to deduce that in this case Re(s) is a root of order r of b I , but unfortunately, we do not understand well enough the multiplicities of the roots of b I .
Remark 4.4. While Proposition 2.1 gives in principle a formula for the Igusa zeta function of a monomial ideal, and Theorem 1.4 gives an estimate on the denominator of this function (written as a rational function of 1/p s ), getting a general explicit formula for the denominator is rather difficult. A Maple code for computing p-adic and motivic zeta functions of monomial ideals via resolution of singularities is available, upon request, from the first author.
Remark 4.5. Using motivic integration, Denef and Loeser defined in [DL] a motivic analogue of the Igusa zeta function. For concreteness, we preferred to work with p-adic integrals. However, as the reader familiar with this topic will certainly notice, all the above results have analogues in the motivic setting, "replacing p by L". For example, if σ, ℓ 1 and ℓ 2 are as in Theorem 1.3, then the series (9) v∈σ∩N L −(ℓ 1 (v)s+ℓ 2 (v)) can be written as a sum of fractions with numerator in K[L −s ] and denominator of the form where r ≤ dim(σ) and v 1 , . . . , v r are primitive generators of the rays of σ. Here K is the ring obtained from the Grothendieck ring of varieties over a base field k by inverting L = [A 1 k ]. Similarly, if I is a monomial ideal, then the motivic zeta function of I where r ≤ n and E 1 , . . . , E r are divisors on the normalized blowing-up of A n along I such that E 1 ∩ . . . ∩ E r is nonempty.