POLES OF THE TOPOLOGICAL ZETA FUNCTIONASSOCIATED TO AN IDEAL IN DIMENSION TWO
Lise Van Proeyen and Willem Veys ∗5th November 2007

Abstract
To an ideal in C[x, y] one can associate a topological zeta function.
This is an extension of the topological zeta function associated to one
polynomial. But in this case we use a principalization of the ideal instead of an embedded resolution of the curve.
In this paper we will study two questions about the poles of thiszeta function. First, we will give a criterion to determine whether or
not a candidate pole is a pole. It turns out that we can know this
immediately by looking at the intersection diagram of the principalization, together with the numerical data of the exceptional curves.
Afterwards we will completely describe the set of rational numbersthat can occur as poles of a topological zeta function associated to an
ideal in dimension two. The same results are valid for related zetafunctions, as for instance the motivic zeta function.
2000 Mathematics Subject Classification. 14E15, 14H20, 32S05.

1Introduction

We will first define the topological zeta function for one polynomial in nvariables over C and mention a number of important results about the polesof these functions. Afterwards, we will concentrate on the topological zetafunction associated to an ideal in C[x, y] and make some similar statements
about its poles.
Let f ∈ C[x1 , . . . , xn ] be a non-constant polynomial satisfying f (0) = 0.
To define the topological zeta function Ztop,f (s), we take an embedded resolution h : X → Cn of f −1 {0}. Let Ei for i ∈ S be the irreducible componentsof h−1 (f −1 {0}), then we denote by Ni and νi − 1 the multiplicities of Ei in∗
K.U.Leuven, Departement Wiskunde, Celestijnenlaan 200B, B-3001 Leuven, Belgium,
email: Lise.VanProeyen@wis.kuleuven.be, Wim.Veys@wis.kuleuven.be. The research waspartially supported by the Fund of Scientific Research - Flanders (G.0318.06).

1

the divisor on X of f ◦h and h∗ (dx1 ∧. . .∧dxn ), respectively. (Further on wegive a description of these multiplicities with local coordinates.) With thesenumerical data we can define the local topological zeta function associatedto f :X
Y
1
Ztop,f (s) :=χ(EI◦ ∩ h−1 {0}),Ni s + νi
I⊂Si∈I

where χ(·) denotes the topological Euler-Poincaré characteristic and EI◦ :=(∩i∈I Ei )\(∪j6∈I Ej ).
There is also a global topological zeta function, where we replace EI◦ ∩−1h {0} by EI◦ . Denef and Loeser proved in [DL1] that these definitions are
independent of the choice of the resolution.
In particular, the poles of the topological zeta function of f are interesting numerical invariants. For example, the monodromy conjecture relates
the poles with eigenvalues of the local monodromy of f (see e.g. [DL1]). It iseasy to see that all poles belong to the set {−νi /Ni | i ∈ S}. These elements
are called the candidate poles associated to the given resolution. They are
all negative rational numbers. It is an important question to determine
whether or not a candidate pole is a pole.
In [V3], the second author proved a fast criterion to answer this question
if we work with a curve f ∈ C[x1 , x2 ]. He showed that we can read the polesout of the minimal embedded resolution of the curve: a candidate pole s0
is a pole if and only if s0 = − Nνii for some exceptional curve Ei intersecting
at least three times other components or s0 = − N1i for some irreducible
component Ei of the strict transform of f.
There are also various results about the set
Pn := {s0 | ∃f ∈ C[x1 , . . . , xn ] : Ztop,f (s) has a pole in s0 }.
For example, in [LSV] it is shown that each rational number in the interval[−(n − 1)/2, 0) is contained in Pn . For n = 2 this means that we know P2completely, as in [SV] it is proven that P2 ∩ (−∞, −1/2 ) = {−1/2−1/i | i ∈Z>1 }.
The construction of blowing-up that is used to desingularize varieties,
can also be used to principalize an ideal. This means that after these blowups, the ideal is locally principal and monomial. This is a result of Hironaka[H].
Theorem 1.1. (Hironaka.) Let X0 be a smooth algebraic variety over a
field of characteristic zero, and I a sheaf of ideals on X0 . There exists a
principalization of I, that is a sequence
σ

σ

σ

σ

1
2
i
r
X0 ←−
X1 ←−
X2 · · · ←−Xi ←− · · · ←−Xr = Xof blow-ups σi : Xi−1 ← Xi in smooth centers Ci−1 ⊂ Xi−1 such that2

1. the exceptional divisor E i of the induced morphism σ i = σ1 ◦ . . . ◦ σi :Xi → X0 has only simple normal crossings and Ci has simple normal
crossings with E i , and2. the total transform (σ r )∗ (I) is the ideal of a simple normal crossings
divisor E. If the subscheme determined by I has no components ofcodimension one, then E is a natural combination of the irreducible
components of the divisor E r .
Remark 1.2. In order to denote the total transform (σ r )∗ (I), other authors
may use the notation IOX . If I has components of codimension one, we canwrite the total transform as a product of two (principal) ideals: the supportof the first one is the exceptional locus, where the support of the second one
is formed by the irreducible components of the total transform that are notcontained in the exceptional locus. This second ideal is the ‘weak transform’of I.
When we have a principalization σ = σ r , we can define numerical data(N, ν) for each component of the support of σ ∗ (I) such that for every b ∈ X
there exist local coordinates (y1 , . . . , yn ) which satisfy the following conditions:• if E1 , . . . , Ep are the irreducible components of the divisor E containingb, we have on some neighbourhood of b that Ei is given by yi = 0 for
i = 1, . . . , p ,Q• σ ∗ (I) is generated by ε(y) pi=1 yiNi , andQ• σ ∗ (dx1 ∧ . . . ∧ dxn ) = η(y) pi=1 yiνi −1 dy1 ∧ . . . ∧ dyn ,where ε(y) and η(y) are units in the local ring of X at b.
We can associate a topological zeta function to an ideal f = (f1 , . . . , fl ),where we suppose that 0 ∈ Supp(f). We use the numerical data that originate
from a chosen principalization to define the local topological zeta function
Ztop,f (s) :=X

χ(EI◦ ∩ σ −1 (0))

I⊂T

Y
i∈I

1
,
νi + sNi

with Ei (Ni , νi ) for i ∈ T the components of the support of the total transform of f, and again EI◦ = (∩i∈I Ei )\(∪j6∈I Ej ).
When l = 1, Denef and Loeser showed in [DL1] that the expression above
does not depend on the chosen resolution by expressing it as a limit of padic Igusa zeta functions. They introduced later in [DL2], still for l = 1, themotivic zeta function of f, which is intrinsically defined. It has however a
formula of the same kind as above in terms of a resolution. Specializing thisformula to Euler characteristics yields the topological zeta function of f.
3

One can associate more generally a motivic zeta function to an ideal
and obtain a similar formula in terms of a principalization using the same
argument as in [DL2]. Again specializing to Euler characteristics yields thedefining expression above for the topological zeta function of an ideal. This
generalization to ideals is mentioned in [VZ, (2.4)].
Alternatively, one can check that this expression is independent of thechosen principalization by verifying that it is invariant under a blow-up withallowed center (this is straightforward) and then applying the Weak Factorization Theorem of WÃlodarczyk et al. [AKMW]. Note that in dimension 2
one does not need the Weak Factorization Theorem since there is a minimal
principalization.
Remark 1.3. As in the case of one polynomial, there is also a global versionof this zeta function, where we replace EI◦ ∩ σ −1 (0) by EI◦ . However, in thispaper we will work with the local one.
Now we can ask the same questions for the topological zeta function of
an ideal in C[x, y] as we mentioned for the case of one polynomial: how
can we determine which candidate poles are poles? Which rational numbers
occur as poles of a zeta function of an ideal in dimension two?
Theorem 4.2 will answer the first question as a generalization of the result of the second author for the topological zeta function of a curve. It turns
out that you can determine which candidate poles are poles by drawing anintersection diagram of the Ei associated to the minimal principalization
together with their numerical data. In the case of one polynomial, a component of the strict transform as well as an exceptional variety that intersects
at least three times an other component, give rise to a pole. This will still
hold for the topological zeta function of an ideal in dimension two. But thistime it is not true that an exceptional variety that intersects once or twice
an other component never causes a pole. Sometimes it will, sometimes it
won’t. To solve this question, we will associate a “generic” curve to the ideal
and we will prove that a principalization of the ideal also gives an embedded
resolution of this curve with the same numerical data. Afterwards, we show
how these numerical data tell us whether or not a candidate pole is a polein this case.
Further on in this paper we will answer the second question. We willshow that the possible poles of a zeta function of an ideal in dimension two,are exactly the rational numbers in [−1, 0)∪{−1− 1i | i ∈ Z>0 } (see Theorem
5.3).
In the end, we will also draw conclusions about poles of other zeta functions of ideals in dimension two. In fact, we can say that the same results
as we prove for the topological zeta function, are also true for the Hodge
and the motivic zeta function and for most p-adic Igusa zeta functions. We
don’t need to prove these statements separately, but we can extract themout of the results for the topological zeta function.
4

2

Resolution of a generic curve

Let f = (f1 , . . . , fl ) be an ideal in C[x, y]. We suppose in this section thatl > 1. Then we can look at the linear system {λ1 f1 +. . .+λl fl | λi ∈ C for i =1, . . . , l}. A generic curve of f is a general element of this linear system. So
actually, the definition of a generic curve of an ideal is dependent on thegenerators we use to represent the ideal.
Lemma 2.1. A series of blow-ups used to principalize an ideal of C[x, y],also gives an embedded resolution of a generic curve of this ideal.
Remark 2.2. This resolution will -in general- not be minimal, but we can still
use a lot of the results about the numerical data of an embedded resolution
and use them in our context.
Proof. When we start with an ideal I = (f1 , . . . , fl ) ⊂ C[x, y], we can first
determine whether there are common components among the fi and put
them together. So we will write
I = (h)(f10 , . . . , fl0 )with (f10 , . . . , fl0 ) a finitely supported ideal.
We need two chains of blow-ups to have a principalization:(A) a composition of blow-ups σ : X̃ → C2 to transform (f10 , . . . , fl0 ) in a
locally principal ideal, and(B) a series of blow-ups τ : X → X̃ to desingularize the strict transform of
h = 0 and make it have normal crossings with all exceptional curves.
We will look now at the situation after the first series of blow-ups. The
ideal σ ∗ I = (f1∗ , . . . , fl∗ ) is locally principal. So in every point b ∈ X̃ wehave local coordinates (y1 , . . . , yn ) and a generator g(y) such thatfi∗ (y) = g(y)f˜i (y)for i = 1, . . . , l. Moreover,P we know that there exist regular
Pfunctions µi (y) onX̃ to write that g(y) = li=1 µi (y)fi∗ (y). So g(y) = g(y) li=1 µi (y)f˜i (y) andP
1 = li=1 µi (y)f˜i (y). We can conclude that the f˜i (y) don’t have a common
zero.
We study the linear system {λ1 f˜1 +. . .+λl f˜l = 0 | λi ∈ C for i = 1, . . . , l}.
This is a linear system without base points. By Bertini’s theorem (see e.g.
[J, Theorem 6.10]) we know that a general element of the system is nonsingular and connected.
We can also restrict the linear system to an exceptional curve or to acomponent of the strict transform of h = 0. (Note that there are a finite
number of such varieties.) On these curves, we get a new linear system
5without base points. We can again use the theorem of Bertini to say thata general element is non-singular. In this case, this means that every intersection point of a general element of the original linear system with acomponent of the strict transform of h = 0 or with an exceptional curve hasintersection multiplicity one.
Now we look at the following set of points: intersection points of an
exceptional curve with a component of a strict transform of h = 0 andsingular points of the strict transform of h = 0. This is a finite set. A
general element of the linear system doesn’t contain any of them.
We use all this to conclude the following: if we take a generic curve
λ1 f10 + . . . + λl fl0 = 0 with λ1 , . . . , λl ∈ C (necessarily reduced by Bertini’s
Theorem), we can suppose that the strict transform of this curve after the
first series of blow-ups (which is locally given by λ1 f˜1 +. . .+λl f˜l = 0) is nonsingular, intersects the strict transform of h = 0 and the exceptional curvestransversely, and doesn’t contain any of the points in the mentioned set.
This implies that after series (B), the components of the strict transformof the generic curve λ1 f1 + . . . + λl fl = 0 are still non-singular and thetransform (σ ◦ τ )∗ (λ1 f1 + . . . + λl fl ) = 0 is a normal crossings divisor.
So a principalization of the ideal (f1 , . . . , fl ) gives also an (in general nonminimal) embedded resolution of a generic curve λ1 f1 + . . . + λl fl = 0.
Remark 2.3. This lemma is well-known. We stated and proved it in dimension two, but one can do the same in higher dimensions. In our proof, we
made a separation in two series of blow-ups. This is not really necessary andin higher dimensions one better avoids this. However, we chose to make thisbreak to get a clearer view on the role of the common component(s) h = 0.
Example 2.4. We will study the ideal (x4 y, x7 + xy 4 ) ⊂ C[x, y]. We takethe generic curve x4 y + x7 + xy 4 of this ideal and we perform the same
blow-ups as are used to principalize the ideal.

6

Chart 1 (x, xy)
Chart 2 (xy, y)Chart 1.1 (x, xy)
Chart 1.2 (xy, y)Chart 1.1.1 (x, xy)
Chart 1.1.2 (xy, y)principalization of(x4 y, x7 + xy 4 )x5 (y, x2 + y 4 )E1 ↔ x = 05xy (x3 , x6 y 2 + 1)E↔x=0E1 ↔ y = 0
x6 (y, x + x3 y 4 )E2 ↔ x = 0
x5 y 6 (1, x2 y + y 3 )E1 ↔ x = 0
E2 ↔ y = 0
x7 (y, 1 + x6 y 4 )E3 ↔ x = 0
x6 y 7 (1, x + x3 y 6 )E2 ↔ x = 0
E3 ↔ y = 0

embedded resolution ofx4 y + x7 + xy 4 = 0
x5 (y + x2 + y 4 )E1 ↔ x = 05xy (x3 + x6 y 2 + 1)E↔x=0E1 ↔ y = 0
x6 (y + x + x3 y 4 )E2 ↔ x = 0
x5 y 6 (1 + x2 y + y 3 )E1 ↔ x = 0
E2 ↔ y = 0
x7 (y + 1 + x6 y 4 )E3 ↔ x = 0
x6 y 7 (1 + x + x3 y 6 )E2 ↔ x = 0
E3 ↔ y = 0

We can also construct the intersection diagram of this principalization andresolution, together with the numerical data (N, ν).
E3 (7, 4)
E2 (6, 3)

E 0 (1, 1)

E(1, 1)E1 (5, 2)

The curves E and E 0 are the components of the strict transform of thegeneric curve. The first one is also the support of the weak transform of theideal, the second one does not occur in the principalization.
Remark 2.5. In this example you can also see that the numerical data of
the principalization and those of the resolution are the same. This is true ingeneral. The equality of the νi is obvious, the Ni are equal since for general
λ1 , . . . , λl , the vanishing order of a divisor E along λ1 f1 + . . . + λl fl is equal
to the minimum of the vanishing orders of E along the fi .
Remark 2.6. Although the embedded resolution of the generic curve is in
general not minimal, not every blow-up is allowed in the minimal principalization. We will only blow up with center on the intersection of at least one
exceptional curve with the support of the weak transform of the ideal. Note
7

that this means that ‘superfluous’ blowing-ups in the non-minimal embedded resolution of our generic curve have center on the intersection of the
exceptional locus with the strict transform of the generic curve.

3

Relations between numerical data

For the numerical data of an embedded resolution of a generic curve of theideal (f1 , . . . , fl ) ⊂ C[x, y], we know that the following relation holds: whenE(N, ν) is an exceptional curve that intersects k times other components
Ei (Ni , νi ) and αi = νi − Nν Ni for i = 1, . . . , k, thenkX

αi = k − 2.

i=1This relation between the numerical data was proved by Loeser in [L] andgeneralized by the second author in [V2].
The intersection diagram with the numerical data of the principalization is almost the same as the one that arises from the (in general nonminimal) resolution of the generic curve h · (λ1 f10 + . . . + λl fl0 ) = 0. Here
we use again the notation of the previous section, so we suppose that
(f1 , . . . , fl ) = (h)(f10 , . . . , fl0 ), with (f10 , . . . , fl0 ) a finitely supported ideal.
The only difference between the two intersection diagrams is that the strict
transform of λ1 f10 + . . . + λl fl0 = 0 disappears in the principalization.
So we can divide the k intersections of an exceptional curve of an embedded resolution of the generic curve in two groups: there are n intersections
with the strict transform of λ1 f10 + . . . + λl fl0 = 0 and m = k − n intersections that are preserved in the intersection diagram of the principalizationof the ideal. Since we know that the first mentioned curve has numerical
data (1, 1), we can write -after renumbering the intersections- thatmX

αi + n(1 −

i=1

or

m
X

ν
) = m + n − 2,Nαi = m − 2 +

i=1

νn
.
N

(1)

Proposition 3.1. Let E(N, ν) be an exceptional curve of a principalizationof (f1 , . . . , fl ) ⊂ C[x, y], intersecting Ei (Ni , νi ) for i = 1, . . . , m, and set
αi = νi − Nν Ni for all i ∈ {1, . . . , m}. Then −1 ≤ αi < 1 for every i.
Moreover, αi = −1 only occurs when m = 1.
Proof. This proposition has been proven by Loeser in [L, Proposition II.3.1]
for the numerical data of minimal embedded resolutions. Since we already8noticed that the numerical data of the principalization and the (possibly nonminimal) embedded resolution of a generic curve are the same (see Remark
2.5), we can look at these data as if they were coming from a resolution of
the generic curve.
We can divide the exceptional curves in two groups: the ones that werefirst created are part of the minimal embedded resolution of the genericcurve. As a consequence of the mentioned theorem of Loeser, the αi thatoriginate from these will satisfy the condition −1 ≤ αi < 1. The second
group of blow-ups will have center on the intersection of one exceptional
curve and the strict transform of the generic curve. Moreover, since wesuppose that we have already an embedded resolution, we know that themultiplicity of the generic curve in the center of the blow-up is one.
So we only need to look at the following situation:(1, 1)(N + 1, ν + 1)
L99(N, ν)(N, ν)(1, 1)

We can suppose that −1 ≤ 1 − Nν < 1 (or that 0 < Nν ≤ 2) and we only
need to show that(i) −1 ≤ ν + 1 − Nν (N + 1) < 1,(ii) −1 ≤ ν − Nν+1
+1 N < 1and(iii) −1 ≤ 1 − Nν+1
+1 < 1.
This is straightforward.
Corollary 3.2. Let E(N, ν) be an exceptional curve of a principalization
σ of an ideal I ⊂ C[x, y]. Suppose that E intersects the other components
Ei (Ni , νi ) for i = 1, . . . , m of the total transform σ ∗ I. Let αi = νi − Nν Ni
for all i ∈ {1, . . . , m}. Then we have the following statements.
1. At most one Ei , 1 ≤ i ≤ m, occurs such that αi < 0.
2. If m ≥ 3, then there is at most one i such that αi ≤ 0.
3. If m = 2, we see that Nν11 < Nν ⇒ Nν < Nν22 .
This is a direct consequence of the previous proposition and equation (1). In[V3] there are almost the same results for the numerical data of an embedded
resolution of a curve. However, the analogue of the third statement in that9context is an equivalence instead of an implication. Roughly said, this isdue to the presence of the positive term νn
N in our equation (1).
The mentioned corollary in [V3] is used there to determine the ‘orderedtree’-structure of the resolution graph. The same can be done in our case.
We can draw a dual principalization graph by associating a vertex to every
exceptional curve and every (analytically irreducible) component of the support of the weak transform. For each intersection we have an edge, connecting the corresponding vertices.
By using Corollary 3.2, it is not so difficult to derive the next proposition.
For example, this can be done as in [V3, Theorem 3.3].
Proposition 3.3. The part of the dual principalization graph where Nν isminimal, is connected. Moreover, when we follow a path that moves awayfrom this minimal part, the ratio Nν will strictly increase.

4

Poles of a zeta function of an ideal

In this section we always consider ideals I ⊂ C[x, y] with 0 ∈ Supp(I). Since
we study the local topological zeta function associated to I, we need in fact
only a principalization of I in the neighbourhood of 0.
We know that the only possible poles of the topological zeta functions
are rational numbers − Nν with (N, ν) numerical data of components of theminimal principalization. We can see that the largest candidate pole plays aspecial role. The following arguments show that it is always a pole. If there
are different components with this maximal ratio − Nν , these components
need to intersect and we find a pole of order two. Moreover, this is the only
value where a pole of order two is possible. This is a consequence of the‘ordered tree’-structure of the graph (see Proposition 3.3). When there is
only one component E(N, ν) with this minimal ratio, we have a candidate
pole of order one. Its residue is then given by
Ã
!
m
X
1
1
R=
2−m+
,
N
αi
i=1

where we suppose that E intersects m times other components Ei (Ni , νi )(i = 1, . . . , m) of the principalization, and αi = νi − Nν Ni . When Nν is minimal, then 0 < αi < 1 for every i, so R > 0 and − Nν is a pole.
Not every other candidate pole gives rise to a pole. For the topologicalzeta function associated to a curve in C2 , the second author proved thefollowing theorem in [V3].
Theorem 4.1. Let f ∈ C[x, y] be a non-constant polynomial satisfying
f (0) = 0, and let h : X → C2 be the minimal embedded resolution of f −1 {0}10in a neighbourhood of 0. Let Ei (Ni , νi ) be the irreducible components of
h−1 (f −1 {0}) with their associated numerical data. We have that s0 is a poleof Ztop,f (s) if and only if s0 = − Nνii for some exceptional curve Ei intersecting at least three times other components or s0 = − N1i for some irreducible
component Ei of the strict transform of f = 0.
This gives a criterion to filter the poles out of the series of candidate
poles. The next theorem will do the same for the topological zeta function
associated to an ideal in C[x, y]. With this theorem we can easily determine
the poles of the zeta function when we have the principalization of the ideal.
Theorem 4.2. Let I ⊂ C[x, y] be an ideal satisfying 0 ∈ Supp (I) andσ : X → C2 the minimal principalization of I in a neighbourhood of 0. Let
E• (N• , ν• ) be the components of the support of the total transform σ ∗ I with
their associated numerical data.
The rational number s0 is a pole of the local topological zeta function of
I if and only if one of the following conditions is satisfied:
1. s0 = − N1 for a component E(N, ν) of the support of the weak transformof I;
2. s0 = − Nν for E(N, ν) an exceptional curve that intersects no othercomponent;
3. s0 = − Nν for E(N, ν) an exceptional curve that intersects once anothercomponent Ei (Ni , νi ) with νi − Nν Ni 6= −1;
4. s0 = − Nν for E(N, ν) an exceptional curve that intersects two times
other components Ei (Ni , νi ) and Ej (Nj , νj ) with (νi − Nν Ni ) + (νj −νN Nj ) 6= 0;
5. s0 = − Nν for E(N, ν) an exceptional curve that intersects at least threetimes other components.
Remark 4.3. In the proof we will work with the following notation. If
E(N, ν) is a curve in the support of the total transform of I that intersects once another curve Ei (Ni , νi ), we write α = νi − Nν Ni . If E(N, ν) intersects the curves Ei1 (Ni1 , νi1 ), Ei2 (Ni2 , νi2 ), . . . , Eim (Nim , νim ), we write
αj = νij − Nν Nij .
Proof. We have already said that the only possible pole of order two is thelargest candidate pole. We can see that if s0 is maximal, at least one of thefive conditions is satisfied.
Now we will calculate the contribution to the residue of s0 as a poleof order one in the various cases. (We can suppose that all αi 6= 0.) We
will see that the five situations of the theorem are the only ones wherethat contribution is non-zero. Moreover we will show that this contribution11

is negative, unless s0 is the largest candidate pole. Notice that this lastcondition corresponds with “every αi > 0”.
Suppose that s0 = − N1 for a component E(N, ν) of the support of the
weak transform. Such a component only intersects one exceptional curve
Ei (Ni , νi ) and we see that the contribution to the residue of a pole of order one is R = N1α . This is positive if s0 is the largest pole and negative
otherwise.
If we have an exceptional curve E(N, ν) that doesn’t intersect any othercomponent, we know that this is the only curve in the principalization. So
2and the value s0 = − Nν is a
the topological zeta function is given by ν+sN
pole.
Let s0 = − Nν for E(N, ν) an exceptional curve that intersects once another component. The contribution to the residue for a pole of order one, is
R = N1 (2 − 1 + α1 ). So we see immediately thatR = 0 ⇔ α = −1.
The case α = 0 is excluded, because we only look at candidate poles of orderone. If α > 0, then R > 0. If α < 0 and α 6= −1, we can use α > −1 toconclude that R < 0.
Now suppose that s0 = − Nν for E(N, ν) an exceptional curve that intersects two times other components. In this case R = N1 (2 − 2 + α11 + α12 ) =1 α1 +α2
N α1 α2 . We conclude thatR = 0 ⇔ α1 + α2 = 0.
If α1 + α2 6= 0, we can use equation (1) to know that α1 + α2 > 0. So weare only interested in the sign of α1 α2 to know the sign of R. We know that
α1 and α2 can’t be both negative. If they are both positive, then s0 is thelargest pole and R > 0. In the other case we have R < 0.
The next case is where s0 = − Nν for E(N, ν) an exceptional curve thatintersects at least three times P
an other component. Here, the contribution
1
to the residue R = N1 (2 − m + m
i=1 αi ) is always non-zero. If every αi > 0,
we can easily conclude that R > 0. When there is a αi < 0, we can use theresults for the resolutionof a curve that are written ´in [V3, Proposition 2.8]
³
P
1
1
n
0
to see that R := N 2 − (m + n) + m< 0. (Here we use theν
i=1 αi + 1− Nnotation of the previous section.) Because we know that there can exist at
most one negative α, we can deduce that 0 < 1 − Nν < 1 and
µ
¶
n
1
n−
< 0.
R = R0 +N
1 − Nν
From these calculations we can conclude that all contributions to theresidue are negative if s0 is not maximal. So if one of these contributions is
non-zero, the total residue is non-zero and s0 is a pole of order one.
12

Remark 4.4. When we work with one element f ∈ C[x, y] instead of an ideal,
only the first and the last case can occur. Moreover, a principalization of
the ideal (f ) is the same as an embedded resolution of the curve given by
f = 0. So this theorem is a generalization of Theorem 4.1.
Example 4.5. We will continue Example 2.4 and calculate explicitly the
topological zeta function of the ideal f = (x4 y, x7 + xy 4 ) ⊂ C[x, y]. With thecalculations done in the previous example, we can see that
Ztop,f (s) =

1
1
1
1
+
+
+
.
4 + 7s (3 + 6s)(4 + 7s) (2 + 5s)(3 + 6s) (1 + s)(2 + 5s)

With a little calculation, we can simplify this expression toZtop,f (s) =5s2 + 16s + 8,(4 + 7s)(2 + 5s)(1 + s)

which implies that the poles of this function are −4/7, −2/5 and −1.
We can obtain the same result by using Theorem 4.2 in the followingway:• E(1, 1) is a component of the support of the weak transform, so −1 isa pole;• E1 (5, 2) intersects twice other components, with (1− 52 )+(3 − 52 6) 6= 0,
so −2/5 is a pole;• E2 (6, 3) also has two intersections with other components, this timewith (4 − 36 7) + (2 − 36 5) = 0, hence the candidate pole −1/2 is no pole;• E3 (7, 4) intersects one other component with 3− 74 6 6= −1, so this gives
the last pole −4/7.

5

Determination of all possible poles

In this section, we will determine which numbers can occur as a pole ofa topological zeta function associated to an ideal in dimension 2. For the
topological zeta function of a curve, this question has been answered in [SV]and [LSV].
In the first article, Segers and the second author proved that the poles
smaller than − 12 are given by {− 12 − 1i | i ∈ Z>1 }. In the second article,
Lemahieu, Segers and the second author showed that every rational numberin the interval [− 12 , 0) is a pole of a zeta function of a curve. This determines
all possible poles.
We will prove an analogue of these results for the topological zeta function of an ideal in dimension 2.

13

For an exceptional variety E(N, ν) of the minimal embedded resolutionof a curve, one can show that ν ≤ N. Analogously, we prove the following
proposition.
Proposition 5.1. Let E(N, ν) be an exceptional curve of the minimal principalization of an ideal in C[x, y]. Then the numerical data satisfyν ≤ N + 1.
Proof. We will prove this proposition by induction. If E1 (N1 , ν1 ) is the firstcreated exceptional curve, then ν1 = 2 and N1 ≥ 1, so the statement isproven. Now we will suppose that the inequality is satisfied for all alreadycreated exceptional curves and we will prove it for the next one.
• First, suppose that the center of the blow-up is contained in two
exceptional curves Ei1 (Ni1 , νi1 ) and Ei2 (Ni2 , νi2 ). Then we see that
νi = νi1 + νi2 and Ni = Ni1 + Ni2 + (minimal multiplicity of the generators of the ideal in the center). If we use the induction hypothesis, we
see that νi ≤ Ni1 + Ni2 + 2, but we also know that Ni ≥ Ni1 + Ni2 + 1,
so νi ≤ Ni + 1.
• When only Ei1 exists, then νi = νi1 + 1 ≤ Ni1 + 2 and Ni = Ni1 +(minimal multiplicity of the generators of the ideal in the center) ≥Ni1 + 1, so νi ≤ Ni + 1.
Remark 5.2. This implies that all candidate poles of the topological zetafunction associated to an ideal are rational elements of [−1, 0) ∪ {−1 − 1i | i ∈Z>0 }. The next proposition will show that every rational number in thisrange really occurs as a pole of a certain topological zeta function. Hence
we have a complete description of the possible poles.
Theorem 5.3. The set of rational numbers s0 for which there exists an ideal
I ⊂ C[x, y] such that Ztop,I (s) has a pole in s0 , is given by Q ∩ ([−1, 0) ∪{−1 − 1i | i ∈ Z>0 }).
Proof. Choose a, b ∈ Z≥0 with a > b. Look at the ideal(xb y , xa + y b+1 ) ⊂ C[x, y].
After principalization, we find the following numerical data and intersection
diagram:
E1 (b + 1, 2) , E2 (b + 2, 3) , . . . , Ea−b−1 (a − 1, a − b) , Ea−b (a, a − b + 1).

14

E3

Ea−b
..

.

E1
E2

Ea−b−2Ea−b−1The last exceptional variety only intersects Ea−b−1 with α 6= −1, so
this causes a pole in − a−b+1
a . Easy calculations show that this implies thatevery element of Q ∩ ([−1, 0) ∪ {−1 − 1i | i ∈ Z>0 }) occurs as a pole of the
topological zeta function of an ideal in C[x, y].

6

Other zeta functions

There are finer variants of the topological zeta function of an ideal. For
instance, using the notation of the introduction, there is the (local) Hodge
zeta function
ZHod,f (s) =X
I⊂T

H(EI◦ ∩ σ −1 (0); u, v)Yi∈Iuv − 1∈ Q(u, v)((uv)−s )(uv)νi +sNi − 1

for the ideal f, where H( · ; u, v) ∈ Z[u, v] denotes the Hodge polynomial.
Even finer is the (local) motivic zeta function of f, which was already mentioned in the introduction. Its formula involves classes in the Grothendieckring of algebraic varieties, instead of Euler characteristics or Hodge polynomials. We refer to e.g. [DL2], [R] or [V4] for these zeta functions and theirglobal versions associated to one polynomial and to [VZ] for ideals.
The results in this paper on poles of the topological zeta function of an
ideal in dimension 2, i.e. Theorems 4.2 and 5.3, are also valid for the Hodge
and the motivic zeta function. We chose not to give the details here about
these zeta functions, since for results of the kind we proved, the version for
the topological zeta function is the strongest, and implies the same results
for the finer zeta functions.
The point is that the motivic zeta function specializes to the Hodge zetafunction, which in turn specializes to the topological zeta function. (Notefor instance that H( · ; 1, 1) = χ(·).) In particular, a pole of the topologicalzeta function will induce a pole of the other two. (The converse is not clear.)
We refer to [R] and [RV] for the precise description of the notion of a pole
for the Hodge and the motivic zeta function. Here we should note that theanalogue of Theorem 4.2 for the finer zeta functions also requires the verification of the following in the context of for instance Hodge polynomials.
Exceptional curves intersecting once or twice other components such thatα = −1 or α1 + α2 = 0, respectively, should not contribute to the residue
15

of the induced candidate pole. Now this is as straightforward as with Euler
characteristics (and well known).
Theorems 4.2 and 5.3 are also valid for (most) p-adic Igusa zeta functions.
We briefly introduce the necessary notation to introduce these zeta functions
and to state the precise result.
Let K be a finite extension of the p-adic numbers with valuation ringR, maximal ideal P, and residue field K̄ = R/P (∼= Fq ). Denote for z ∈ K
by |z| its standard absolute value, and put kzk := max1≤i≤l |zi | for z =(z1 , . . . , zl ) ∈ K l . Let f1 , . . . , fl be polynomials in K[x1 , . . . , xn ]. The (local)
p-adic Igusa zeta function associated to the mapping f = (f1 , . . . , fl ) : K n →K l isZ
ZK,f (s) :=kf(x)ks |dx|
Pnfor s ∈ C with <(s) > 0, where |dx| is the usual Haar measure on K n . A
global version consists in replacing P n by Rn . This function is analytic in s
and admits a meromorphic continuation to C as a rational function of q −s .
This was first proved by Igusa for l = 1 (see [I]). For arbitrary l there aredifferent proofs in [M], [D1] and [VZ].
Considering polynomials f1 , . . . , fl over a number field F, one can study
ZK,F (s) for all (non-archimedean) completions K of F. For all but finitely
many completions K there is a concrete formula for ZK,F (s) in terms of aprincipalization of the ideal (f1 , . . . , fl ), similar to the formulas for the otherzeta functions in this paper. This was proved for l = 1 by Denef in [D2], andcan be generalized to arbitrary l, see [VZ, (2.3)]. (In fact the motivic zetafunction specializes to ‘almost all’ p-adic zeta functions, see [DL2, (2.4)].)
When the number field F is large enough, then for all but finitely many
completions K of F we have that the analogues of Theorems 4.2 and 5.3 arevalid for ZK,F (s), replacing ‘pole’ by ‘real pole’. One can derive this from theresults for the topological zeta function, or by completely analogous proofs.
(Previous such results for l = 1 are in [V1] and [S].)References[AKMW] D. Abramovich, K. Karu, K. Matsuki, J. WÃlodarczyk, Torification and factorization of birational maps, J. Amer. Math. Soc. 15 (2002), 531-572.
[D1]

J. Denef, The rationality of the Poincaré series associated to the p-adic points
on a variety, Invent. Math. 77 (1984), 1-23.

[D2]

J. Denef, On the degree of Igusa’s local zeta function, Amer. J. Math. 109(1987), 991-1008.

[DL1]

J. Denef and F. Loeser, Caractéristiques d’ Euler-Poincaré, fonctions zêta locales, et modifications analytiques, J. Amer. Math. Soc. 5 (1992), 705-720.

[DL2]

J. Denef and F. Loeser, Motivic Igusa zeta functions, Journal of Algebraic
Geometry 7 (1998), 505-537.

16

[H]

H. Hironaka, Resolution of singularities of an algebraic variety over a field of
characteristic zero, Ann. Math. 79 (1964), 109-326.

[I]

J. Igusa, Complex powers and asymptotic expansions I, J. Reine Angew. Math.
268/269 (1974), 110-130; II, ibid. 278/279 (1975), 307-321.

[J]

J.-P. Jouanolou, Théorèmes de Bertini et applications, Progr. Math. 42,
Birkhäuser, Boston, 1983.

[L]

F. Loeser, Fonctions d’ Igusa p-adiques et polynômes de Bernstein, Amer. J.
Math. 110 (1988), 1-21.

[LSV]

A. Lemahieu, D. Segers and W. Veys, On the poles of topological zeta functions,
Proc. Amer. Math. Soc. (to appear).

[M]

D. Meuser, On the rationality of certain generating functions, Math. Ann. 256(1981), 303-310.

[R]

B. Rodrigues, On the geometric determination of the poles of Hodge and motivic
zeta functions, J. Reine Angew. Math. 578 (2005), 129-146.

[RV]

B. Rodrigues and W. Veys, Poles of zeta functions on normal surfaces, Proc.
London Math. Soc. 87 (2003), 164-196.

[S]

D. Segers, On the smallest poles of Igusa’s p-adic zeta functions, Math. Z. 252(2006), 429-455.

[SV]

D. Segers and W. Veys, On the smallest poles of topological zeta functions,Compositio Math. 140 (2004), 130-144.

[V1]

W. Veys, On the poles of Igusa zeta functions for curves, J. Lond. Math. Soc.,
41 (1990), 27-32.

[V2]

W. Veys, Relations between numerical data of an embedded resolution, Amer.
J. Math. 113 (1991), 573-592.

[V3]

W. Veys, Determination of the poles of the topological zeta function for curves,
Manuscripta Math. 87 (1995), 435-448.

[V4]

W. Veys, Arc spaces, motivic integration and stringy invariants, Advanced
Studies in Pure Mathematics, Proceedings of “Singularity Theory and its applications, Sapporo (Japan), 16-25 September 2003” (to appear), 43p.

[VZ]

W. Veys and W.A. Zuniga-Galindo, Zeta functions for analytic mappings, logprincipalization of ideals, and Newton polyhedra, preprint (2006), 23p.

17