Compositio Math. 140 (2004) 667–682DOI: 10.1112/S0010437X03000630

The zeta function of a quasi-ordinary singularity

Lee J. McEwan and Andras Nemethi

Abstract

We prove that the zeta function of an irreducible hypersurface quasi-ordinary singular-ity f equals the zeta function of a plane curve singularity g. If the local coordinates(x1, . . . , xd+1) of f are ‘nice’, then g = f(x1, 0, . . . , 0, xd+1). Moreover, the Puiseux pairsof g can also be recovered from (any set of) distinguished tuples of f .

1. Introduction

We consider the germ of an irreducible quasi-ordinary singularity f : (Cd+1, 0) → (C, 0). Thequasi-ordinary assumption means that, in some local coordinates, the reduced discriminant of theprojection pr : (F, 0) := ({f = 0}, 0) → (Cd, 0), induced by (x, xd+1) �→ x (x = (x1, . . . , xd) ∈ Cd),is included in ({x1 · · · xd = 0}, 0).

The main goal of this note is the computation of the zeta function of f . This is defined as follows.For any germ f : (Cd+1, 0) → (C, 0), we fix a sufficiently small closed ball Br in Cd+1 of radius r,

then Fε := f−1(ε) ∩ Br (0 < ε � r) is called the Milnor fiber of f . In fact, by a theorem of Milnor[Mil68], f−1({|w| = ε}) ∩ Br → {|w| = ε} is a fibration with fiber Fε. Let mq ∈ AutHq(Fε, R)(q � 0) be the algebraic monodromy operators of this fibration. Then the zeta function of f isdefined by the following rational function:

ζ(f)(t) :=∏q�0

det(I − tmq)(−1)q.

In general, ζ(f) is not computed by this expression. The most efficient way to determine ζ(f) isby A’Campo’s formula ([A’C75], see also § 4) via the embedded resolution of f . Hence, in the case ofthose families of singularities whose embedded resolution is well understood, one gets ζ(f). This isthe case for plane curve singularities (see e.g. [EN85]) and isolated singularities with non-degenerateNewton boundary ([Var76], see also [MO70]). For non-isolated singularities, the methods of series ofsingularities, provide partial results (see e.g. [Sie90, Sch90, Nem93]) provided that the singular locusis one-dimensional. But, in general, for non-isolated singularities there is no nice, explicit formulaof ζ(f).

We recall that in general the singular locus of quasi-ordinary singularities is large. Our mainresult for these singularities is the following.

Theorem A. Assume that f : (Cd+1, 0) → (C, 0) (d � 2) is an irreducible quasi-ordinary singularity.For convenience, we assume that the coordinates are ordered (cf. § 2.2). Then ζ(f) = ζ(f |xd=0).In fact, by induction

ζ(f) = ζ(f |xd=···=x2=0),

Received 17 December 2001, accepted in final form 17 September 2002.2000 Mathematics Subject Classification 14B05, 32Sxx.Keywords: quasi-ordinary singularities, Milnor fiber, monodromy, zeta function, Newton polyhedron, toric resolution.

The second author is partially supported by NSF grant DMS-0088950.This journal is c© Foundation Compositio Mathematica 2004.

L. J. McEwan and A. Nemethi

hence ζ(f) can be computed as the zeta function of the plane curve singularity (x1, xd+1) �→f(x1, 0, . . . , 0, xd+1).

Here some comments are in order. First, this type of result is rather surprising. In general,for germs f with one-dimensional singular locus and generic hyperplane section xd, an identityζ(f) = ζ(f |xd=0) holds if the corresponding polar curve is empty. Our germ and hyperplane sectionare more complicated, but still the above theorem suggests the vanishing of some polar obstruction.This is a remarkable property of quasi-ordinary germs; the details will appear elsewhere.

We recall also that the quasi-ordinary singularities are generalizations of the plane curve singular-ities. For any (d-dimensional) irreducible quasi-ordinary singularity f , one can define the general-ization of the Puiseux pairs: they are called the normalized distinguished tuples {λi}g

i=1 of f .(For details, see [Lip65, Lip83, Lip88].) Using Zariski’s result on saturation of local rings, one canprove that these pairs determine the embedded topological type of f (cf. [Zar68] and [Lip88], or[Oh93] for a different proof). They suffice as well to determine an embedded resolution (see [BMc00]and [GP00]). In addition, by [BMcN02] (cf. also with [LDT73] and [Tei74]), one has that, for anyreduced hypersurface singularity, the zeta function depends only on the embedded topological typeof the singularity. Therefore, it is natural to ask for explicit formulae for ζ(f) in terms of thedistinguished tuples of f .

In the next section we reformulate Theorem A. This second version provides this explicit formulaas well. It is remarkable that not all the data of the distinguished pairs are needed.

Theorem A is the consequence of Theorem B.

Theorem B. Let λ1 be the first distinguished tuple of f . For any α ∈ Zd�0 set I(α) := {i | αi > 0}.

Assume that 0 �= β ∈ Zd�0 satisfies #I(β + λ1) � 2. Then

ζ(xβf)(t) = 1.

The precise definition of the distinguished tuples will be given in § 2. As a general reference, see[Lip65, Lip83, Lip88]. The general properties of the Milnor fibration and smoothing invariants, inparticular, of the zeta function, can be found in the books [Mil68], [EN85] and [AGV88].

The proof of Theorem B occupies all of §§§ 3, 4 and 5.In § 3 we recall some results from the thesis of Gonzalez Perez [GP00], namely the construction

of a toric modification π(Σ) associated with the first distinguished tuple λ1 of f . Moreover, we addsome new facts which will be important in our proof. For general facts about toric geometry, werecommend e.g. the books [Oda88] and [Ful93]. For the properties of the toric modification π(Σ)see also [GT00].

In § 4 we will consider a generalization of the classical A’Campo theorem [A’C75]. Using thiswe prove a ‘splitting property’ for the invariant ζ, which is one of the key arguments in the proofof Theorem B; moreover, Theorem A is the consequence exactly of this property. At the end of thissection we also recall the main result of [GLM97], which provides a formula for the zeta function inthe presence of a partial embedded resolution. We will apply this result in the case of π(Σ).

Section 5 contains the proof of the theorems collecting all the information from the previoussections.

Theorems A and B were conjectured in the article [BMcN02], where the authors proved thecorresponding results for d = 2 and the invariant χ(Fε) = deg(ζ(f)) (instead of ζ(f)).

Finally, we note that in the literature the notation for the parametrization of a quasi-ordinarysingularity is ζ, and for the zeta function of a hypersurface singularity it is ζ(f). Even if thesenotations are almost identical, we do not modify them: the meaning of the corresponding notationwill also be clear from the context.

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2. Quasi-ordinary singularities and their topological type

In this section we recall the definition of the distinguished tuples associated with an irreduciblequasi-ordinary singularity (F, 0) ⊂ (Cd+1, 0), its connection with the embedded topological type of(F, 0) ⊂ (Cd+1, 0), and we reformulate our Theorem A.

As a general reference for quasi-ordinary singularities, see [Lip65, Lip83, Lip88].

2.1 We will use the notation x = (x, xd+1) = (x1, . . . , xd, xd+1) ∈ Cd × C = Cd+1 for coordinatesin Cd+1 (or for local coordinates in (Cd+1, 0)).

By the very definition of an irreducible hypersurface quasi-ordinary singularity f : (Cd+1, 0) →(C, 0), there exist local coordinates (x, xd+1) such that f can be expressed as a (Weierstrass) pseudo-polynomial in the variable xd+1:

f(x, xd+1) = xnd+1 + g1(x)xn−1

d+1 + · · · + gn(x),

where gi ∈ C{x} (= the ring of convergent power series in x), and the reduced discriminant ofpr : ({f = 0}, 0) → (Cd, 0) induced by (x, xd+1) �→ x is contained in ({x1 · · · xd = 0}, 0). Moreover,by the Jung–Abhyankar Theorem [Abh55], there exists a parametrization of (F, 0) = ({f = 0}, 0)by a fractional power series ζ = H(x1/m

1 , . . . , x1/md ), where H(s1, . . . , sd) is a power series and m is a

suitable natural number depending on f . (This means that there exists a finite map (Cd, 0) → (F, 0)given by xd+1 = H(s1, . . . , sd), xi = sm

i for i = 1, . . . , d.)

The conjugates of ζ are obtained by multiplying any of x1/mi (i = 1, . . . , d) by mth roots of

unity; the number of different conjugates {ζi}i of ζ is precisely the degree n of the covering pr, and

f(x, xd+1) =n∏

i=1

(xd+1 − ζi).

There is an important finite subset {λi}gi=1 of the set of exponents λ = (λ1, . . . , λd) ∈ Qd

�0 ofthe terms xλ := xλ1

1 · · · xλdd appearing in ζ with non-zero coefficient, called the distinguished d-tuples

associated with ζ (in some articles they are called characteristic exponents, see [GP00]). They playa role similar to Puiseux pairs for plane curve singularities.

By unique factorization of the discriminant one has

ζi − ζj = xλij · εij(x1/m1 , . . . , x

1/md ) with εij(0) �= 0,

and λij,k ∈ (1/m)Z�0. The set {λij}ij = {(λij,1, . . . , λij,d)}ij constitutes the set of distinguishedtuples.

2.2 In the rest of this paper we use the following partial ordering. For any λ, µ ∈ Qd, we saythat λ � µ if λi � µi for all i = 1, . . . , d. If λ � µ, but λ �= µ, then we write λ < µ.

The distinguished tuples {λi}gi=1 are totally ordered:

0 < λ1 < λ2 < · · · < λg. (1)

Let αλ be the coefficient of the term xλ in ζ having exponent λ. Then the distinguished tuples{λi}g

i=1 generate all the other exponents in the following sense:

If αλ �= 0 then λ ∈ Zd +∑λi�λ

Z(λi); and λj �∈ Zd +∑

λi<λj

Z(λi) for 1 � j � g. (2)

Sometimes it is convenient to order the coordinates x1, . . . , xd: after permuting (x1, . . . , xd) we canassume

(λ1,1, . . . , λg,1) > (λ1,2, . . . , λg,2) > · · · > (λ1,d, . . . , λg,d) (lexicographically in Qg). (3)

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The parametrization ζ of F , in general, is not unique. Consider, for example, an arbitraryparametrization ζ as above. ζ can be written in a unique way as ζ(0) + ζ(>0), so that ζ(0) ∈ C{x},and for any non-zero term αλxλ of ζ(>0) one has λ �∈ Zd. We say that the local coordinates (x, xd+1)(of ζ) are ‘good’ (cf. [GP00, p. 71], cf. also with [BMcN02, property (4)]) if

ζ(0) = 0. (4)

Obviously, if the coordinates of ζ are not ‘good’, then the local change of variables x′d+1 = xd+1−

ζ(0), x′i = xi for i = 1, . . . , d, transforms ζ into ζ ′ = ζ − ζ(0) with (ζ ′)(0) = 0. This transformation

preserves the distinguished tuples of the parametrizations.In fact, different parametrizations could produce different sets of distinguished tuples. But if

we consider only the ‘normalized’ parametrizations, then the corresponding tuples are independentof the choice of parametrization. A parametrization ζ is normalized if in addition to (1)–(3) we havethe following:

If λ1 = (λ1,1, 0, . . . , 0) then λ1,1 > 1. (5)We recall the following facts about distinguished tuples.

2.3 By a result of Lipman [Lip65, Lip83, Lip88], by a suitable change of variables, any paramet-rization ζ can be transformed into a normalized parametrization ζ ′. The normalized distinguishedtuples of ζ ′ (or of f) can be determined explicitly from the distinguished tuples of ζ.

From [Lip88, Gau88] the embedded topological type of f determines, and is determined by, thenormalized distinguished tuples of f . In particular, via the above statement, any set of distinguishedtuples (normalized or not) contains the complete information about the embedded topological typeof (F, 0) ⊂ (Cd+1, 0).

We add to this list the following general fact about hypersurface singularities.Let f : (Cd+1, 0) → (C, 0) be a reduced hypersurface singularity with Milnor fiber Fε and

geometric monodromy mgeom : Fε → Fε (defined up to an isotopy). Then the homotopy type of thepair (Fε,mgeom) can be recovered from the embedded topological type of f . (For isolated singularitiesit was proved in [LDT73], cf. also [Yau89] and [Tei74]; in the general case by [BMcN02].)

2.4 The above results show that there should exist an explicit formula of ζ(f) of any irreduciblequasi-ordinary singularity f in terms of the distinguished tuples {λi}g

i=1 of any parametrization ζ(normalized or not) of (F, 0). Our main theorem provides such a formula.

Theorem A (reformulated). Assume that f : (Cd+1, 0) → (C, 0) (d � 2) is an irreducible quasi-ordinary singularity represented in a local coordinate system (x, xd+1) as in § 2.1. For convenience,we assume that the coordinates are ordered, i.e. (3) is satisfied. Then ζ(f) = ζ(f |xd=0), hence (byinduction):

ζ(f) = ζ(f |xd=···=x2=0).Moreover, if the first distinguished tuple λ1 = (λ1,1, . . . , λ1,d) satisfies λ1,2 �= 0, then

f |xd=···=x2=0 = (xd+1 − ζ(0)(x1, 0, . . . , 0))n,

hence ζ(f)(t) = 1 − tn.

If λ1,2 = 0 (hence λ1 = (λ1,1, 0, . . . , 0)), then let i0 � 1 be that index which satisfies λi0,2 = 0but λi0+1,2 �= 0 (if λi,2 = 0 for all i = 1, . . . , g, then i0 = g). Then

f |xd=···=x2=0 = (f(x1, xd+1))n/degf ,

where f is an irreducible plane curve singularity, which has the equisingularity type of the singular

germ defined by the Puiseux parametrization xd+1 =∑i0

i=1 xλi,1

1 , and deg f is the xd+1-degree of f .

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Remark. It is well known that, for any germ f and integer s � 1, one has

ζ(f s)(t) = ζ(f)(ts).

On the other hand, for any irreducible plane curve singularity f , ζ(f) has an explicit expressionin terms of the Puiseux pairs, see e.g. [EN85]. Moreover, deg f can also be determined from theexponents {λi,1}i0

i=1: it is the number of conjugates of∑

1�i�i0x

λi,1

1 .

3. The partial toric resolution: results of Gonzalez Perez

In this section we will assume that f is a quasi-ordinary singularity as in § 2.1, satisfying property (4).(Note that properties (1) and (2) are automatically satisfied.) We will consider a toric modificationπ(Σ) : Z(Σ) → Cd+1 whose restriction above a small ball Br provides a partial resolution of(F, 0) := (f−1(0), 0) ⊂ (Cd+1, 0). In order to be uniform in the next discussions, we will use Cd+1

instead of Br, but this will not affect the proof of our theorems (since in the formula given inTheorem 4.7 only π(Σ)−1(0) is used; see also Remark 4.6).

Some results of this section are proved in the thesis of Gonzalez Perez [GP00]. We will indicatethe corresponding page numbers at the corresponding places.

3.1 The function f1

Let f be as above. Properties (2) and (4) guarantee that, if the coefficient αλ of xλ in theparametrization ζ is non-zero, then λ1 � λ, i.e.

ζ = αλ1xλ1 +

∑λ1<λ

αλxλ (α1 := αλ1 �= 0). (6)

Define ζ ′ := α1xλ1 . Obviously, ζ ′ determines a quasi-ordinary singularity, which has only one

distinguished tuple (namely λ1), and degree n1 = the number of conjugates of xλ1 . Its monic minimalpolynomial is

f1 =n1∏i=1

(xd+1 − ωiα1xλ1) = xn1

d+1 − αn11 xn1λ1 , (7)

where {ωi}i are the n1 roots of unity. If L denotes the fraction field of C{x}, then n = [L(ζ) : L] =[L(xλ1 , . . . , xλg ) : L], and n1 = [L(xλ1) : L]. Therefore, e1 := n/n1 ∈ Z>0.

3.2 The Newton polyhedron of f

Given f =∑

v cvxv ∈ C{x}, its Newton polyhedron N (f) is the convex hull of the set

∑cv �=0 v +

Rd+1�0 . Any vector u ∈ (Rd+1)∗�0 defines the face Fu of N (f) by

Fu :={

v ∈ N (f) : 〈u, v〉 = infv′∈N (f)

〈u, v′〉}

.

If f is a quasi-ordinary singularity as above, then N (f) has only two vertices, namely (0, . . . , 0, n)and (nλ1,1, . . . , nλ1,d, 0). Let Fco be the segment determined by these two vertices. Notice thatFco and its vertices are the only compact faces of N (f). By (6) one has

f |Fco = (f1)e1. (8)

Above f |F denotes the (symbolic) restriction of f to the face F , namely f |F =∑

v∈F cvxv.

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3.3 The fan Σ(N (f)) in (Rd+1)∗�0

We say that two vectors in (Rd+1)∗�0 are related if they define the same face in N (f). The fanΣ(N (f)) is defined in such a way that the classes of the above relation are the relative interiorsof the cones of Σ(N (f)). In general, Σ(N (f)) is not regular. We fix a regular fan Σ, supported by(Rd+1)∗�0, which subdivides Σ(N (f)). This, in particular, means that the intersection of any coneσ ∈ Σ with the linear subspace l given by the equation

l(v) :=d∑

j=1

λ1,jvj − vd+1 = 0 (9)

is a cone of Σ. For any σ ∈ Σ, we denote by Fσ the face Fu of any vector u from the relative interiorof σ.

3.4 The regular fan Σ supported by (Rd+1)∗�0 defines a modification π(Σ) : Z(Σ) → Cd+1. Thevariety Z(Σ) is smooth, and it is covered by affine spaces {Z(σ)}dim σ=d+1 (i.e. each Z(σ) ≈ Cd+1).The inclusion σ ⊂ (Rd+1)∗�0 provides a morphism π(σ) : Z(σ) → Cd+1. If σ = 〈a1, . . . , ad+1〉 and ai

has coordinates (ai1, . . . , a

id+1), then π(σ) has the form:

x1 = ua11

1 · · · uad+11

d+1 , . . . , xd+1 = ua1

d+1

1 · · · uad+1d+1

d+1 . (10)

The total transform of f =∑

v cvxv by π(σ) is

f ◦ π(σ) =∑

v

cvu〈a1,v〉1 · · · u〈ad+1,v〉

d+1 = um(a1)1 · · · um(ad+1)

d+1 · fσ, (11)

where m(a) := infv∈N (f)〈a, v〉 and fσ defines the equation of the strict transform of f in the chartZ(σ) ⊂ Z(Σ).

If l(a) � 0 (cf. Equation (9)), then (0, . . . , 0, n) ∈ Fa, hence

m(a) = nad+1. (12)

3.5 The divisors D(a) in Z(Σ)To any primitive vector a of Σ(1) (:= 1-skeleton of Σ), we associate a divisor D(a) in Z(Σ). Inany chart Z(σ), associated with a cone σ = 〈a1, . . . , ad+1〉, D(ai) is given by {ui = 0}. If b �∈{a1, . . . , ad+1}, then D(b)∩Z(σ) = ∅. By (10) one gets that π(Σ)(D(ai)) is the coordinate subspacedefined by xj = 0 for all j with ai

j �= 0. If e1, . . . , ed+1 denote the canonical basis on Zd+1, thenπ(Σ)(D(ei)) = {xi = 0}, but for any a ∈ Σ(1) \ {e1, . . . , ed+1} one has dim π(Σ)(D(a)) � d− 1. Thecritical locus (exceptional divisor) of π(Σ) is exactly the union of these divisors:⋃

a∈Σ(1)\{ei}i

D(a).

Moreover, D(a) is compact (i.e. D(a) ⊂ π(Σ)−1(0)) if and only if a is an interior point of (Rd+1)∗�0.More generally, one has the following fact.

Lemma 3.1. Let A = {a1, . . . , as} be a non-empty subset of Σ(1). Denote DA :=⋂

a∈A D(a). Then:

a) if DA �= ∅, then {a1, . . . , as} forms a cone σA in Σ. Moreover, LA := π(Σ)(DA) is a linearcoordinate subspace of Cd+1;

b) assume that DA �= ∅, then

DA compact ⇔ σA ∩ (Rd+1)∗>0 �= ∅ ⇔ LA = 0.

The proof follows from standard facts of toric geometry and the previous discussion.

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3.6 The divisor D :=⋃

a∈Σ(1) D(a) in the smooth variety Z(Σ) is a normal crossing divisor withsmooth irreducible components. Gonzalez Perez proves [GP00, Theorem 3.6] that π(Σ) : Z(Σ) →Cd+1 provides an embedded resolution of f1. If F1 denotes the strict transform of F1 = {f1 = 0},then F1 is smooth and D ∪ F1 is a normal crossing divisor. Moreover, the restriction π(Σ)|F1

:F1 → F1 is an isomorphism above F1 \ Sing F1 (cf. [GP00, Lemma 3.6]). But, for a general Σ (e.g.if Σ(1) ∩ {vd+1 = 0} is ‘too rich’), it is not true that π(Σ) : Z(Σ) → Cd+1 is an isomorphismabove Cd+1 \ Sing F1. On the other hand, in the proof of Theorem B, the property that π(Σ) is anisomorphism (at least) above Cd+1 \ F1 (respectively above Cd+1 \ F ) is crucial (cf. Theorem 4.7).(We notice that a similar property is used in [Var76] as well.)

Proposition 3.2. There exists a regular fan Σ which subdivides Σ(N (f)), supported by (Rd+1)∗�0,

so that π(Σ) is an isomorphism above Cd+1 \ (F1 ∩ F ). This means that the image

∆Σ := π(Σ)( ⋃

a∈Σ(1)\{ei}i

D(a))

of the exceptional divisor of π(Σ) is included in F1 ∩ F .

Proof. A) First we assume that λ1,i �= 0 for all i; and we construct a regular fan Σ with ∆Σ ⊂ F1.Define H := {v ∈ (Rd+1)∗�0 : vd+1 = 0}. By assumption about λ1 one has

l ∩ H ∩ (Rd+1)∗�0 = 0. (13)

In [Ful93, pp. 47–48], Fulton gives the following algorithm to find a regular refinement Σ. Thisalgorithm provides our wanted fan as well.

Given a fan Σ0 and any lattice point v, one can subdivide Σ0 to a fan Σ1 as follows: each conethat contains v is replaced by the joins of its faces with the ray through v; each cone not containingv is left unchanged.

Now, the construction of the regular refinement is done by adding successively vectors v asabove. In this procedure there are two steps.

Step 1. On can subdivide any fan by successively adding vectors in larger and larger cones, until itbecomes simplicial.

Step 2. Any additional new vector v has the following form: for some simplicial k-cone σ generatedby the primitive vectors v1, . . . , vk, the new v is a lattice point of the form v =

∑ki=1 tivi (0 � ti < 1).

We have to show that, at any time in this procedure, one can make the choice of the vector vso that v �∈ H. In this case, for any a ∈ Σ(1) \ {ei}i, we will have ad+1 �= 0, hence π(Σ)(D(a)) ⊂ F1

(cf. § 3.5).Since H ∩ Σ(N (f)) is simplicial (cf. (13)), we can assume that all the vectors v introduced in

Step 1 have vd+1 > 0. For Step 2, notice that any vector in H ∩ {v1, . . . , vk} is automatically oftype {ei}i�d, and the new (lattice) vector v cannot be of the form

∑i�d tie

i (0 � ti < 1). Hencevd+1 > 0.

B) Assume that λ1,j0 �= 0, but λ1,j0+1 = 0 for some j0 < d. First construct a refinement Σj0

of Σ(N (f)) ∩ (Rj0+1)∗�0 in (Rj0+1)∗�0 (with coordinates v1, . . . , vj0 , vd+1), then define Σ as the fangenerated by Σj0 and the vectors {ei}j0<i�d. Then again ∆Σ ⊂ F1.

C) Consider the linear coordinate subspace La := π(Σ)(D(a)). Then it is easy to verify (using (6))that if La ⊂ F1 then La ⊂ F as well.

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3.7 The strict transforms F 1 and F

Let g denote one of the functions f or f1, and set S for the strict transform of {g = 0} via π(Σ).In this section we will analyze the intersections D ∩ S.

By [GP00, (3.2)], a divisor D(a) (a ∈ Σ(1)) has a non-empty intersection with S if and only ifa ∈ l. In fact, the intersection D ∩ S can be covered by charts {Z(σ)}σ , where we consider onlythose (d + 1)-dimensional cones σ = 〈a1, . . . , ad+1〉 of Σ which satisfy ai ∈ l for all i = 1, . . . , d, andl(ad+1) > 0 (cf. [GP00, page 110]). Therefore, all the properties of the intersections D∩F1 and D∩Fcan be verified in these charts.

First we analyze the intersection D ∩ F1.Fix a chart Z(σ) as above. Using N (f1) = (1/e1)N (f) and (12), the analog of (11) written for

f1 (and in Z(σ)) reads as

f1 ◦ π(σ) = (ua1

d+1

1 · · · uad+1d+1

d+1 )n1 (1 − αn11 u

n1l(ad+1)d+1 ),

where n1l(ad+1) = 1 (cf. [GP00, p. 111]). Therefore:

(f1)∼σ = 1 − αn11 ud+1.

This means that F1 ∩ Z(σ) has equation w := 1 − αn11 ud+1 = 0. Notice that this equation is

independent of the choice of σ. (Indeed, ud+1, as a vector of σˇ, is a primitive vector generatingl⊥, cf. [Ful93, ch. 1].) This has the following important consequence. We say that the ‘naturalstratification’ of D ∩ F1 is given by (non-empty) strata of type:( ⋂

a∈A

D(a)∖ ⋃

a�∈A

D(a))∩ F1 (A ⊂ Σ(1)).

Then the above discussion guarantees that the natural stratification of D ∩ F1 is equivalent to thestratification of the toroidal embedding whose combinatorics corresponds to the cones of Σ ∩ l (cf.[GP00, 3.2.3 and 3.3.2]). In particular,

for any stratum Ξ of D ∩ F1 with dimΞ > 0, one has χ(Ξ) = 0. (14)

This can be explained in the following way as well. If a stratum Ξ of D ∩ F1 has a non-emptyintersection with the chart Z(σ), then it is completely contained in Z(σ), and in this chart is givenby the intersection ΞD ∩ {w = 0}, where ΞD is a toric stratum (orbit) of D contained in Z(σ).Hence Ξ itself is a torus.

By similar argument as in Lemma 3.1, and using the above discussion, one obtains the followinglemma.

Lemma 3.3. For any A = {a1, . . . , as} ⊂ Σ(1), one has the following.

a) If DA ∩ F1 �= ∅, then {a1, . . . , as} is a cone σA in Σ ∩ l.

b) Assume that DA ∩ F1 is not empty. Then

DA ∩ F1 is compact ⇔ σA ∩ l ∩ (Rd+1)∗>0 �= ∅.In particular, we obtain via Lemma 3.1, that

c) DA ∩ F1 compact ⇔ DA compact.

The point is that these facts are true for the strict transform F of {f = 0} as well.

Lemma 3.4. The statement of Lemma 3.3 is also true for F instead of F1.

Proof. a) is clear from the above discussion. By (8), f is a perturbation of f e11 . Using the local

equations of the corresponding strict transforms, one obtains that, if DA ∩ F1 is not compact, thenDA ∩ F cannot be compact either. Hence the lemma follows from Lemmas 3.1 and 3.3.

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The final goal of the above discussions is the next result.

Proposition 3.5. Assume that for a non-empty A ⊂ Σ(1) the intersection DA is non-empty andcompact. Then

DA ∩ F1 = DA ∩ F .

This (via Lemmas 3.3 and 3.4) gives that:

π(Σ)−1(0) ∩ F1 = π(Σ)−1(0) ∩ F .

Proof. By Lemmas 3.1, 3.3 and 3.4 we can assume that A defines a cone σA in l in such a way thatσA∩ l∩ (Rd+1)∗>0 �= ∅. This implies that the face FσA

⊂ N (f) associated with σA is compact, and infact it is exactly Fco. Then Proposition 3.5 follows from (8) and from the next technical lemma.

Lemma 3.6 [GP00, p. 105]. Set f =∑

v cvxv ∈ C{x}, and fix a compact face Fσ associated with

the cone σ = 〈a1, . . . , as〉 ∈ Σ. Then in Z(σ), the intersection of the strict transform of {f = 0}with

⋂1�i�s D(ai) has equation (f |Fσ)∼ = 0.

3.8 The strict and total transforms of f along π(Σ)−1(0) ∩ F

The divisor D ∩ F (also) has a ‘natural stratification’ given by strata of type( ⋂a∈A

D(a)∖ ⋃

a�∈A

D(a))∩ F (A ⊂ Σ(1)).

We are interested only in those strata which are situated in π(Σ)−1(0). They are situated incompact intersections of type DA by Lemmas 3.3 and 3.4. Consider such a (connected) strata Ξwith dimΞ > 0. We claim that both the strict and the total transforms of f along Ξ determinean equisingular family of singularities. This follows from [GP00], which provides a simultaneousresolution of the strict transform of f along Ξ. This shows in particular that the zeta function ofthe strict transform along Ξ is constant.

3.9 Now we concentrate our discussion on the zero-dimensional strata {Ξ}dim Ξ=0 of D ∩ F .Obviously, all of these points are in π(Σ)−1(0) ∩ F .

For this, we fix an arbitrary d-cone σ′ = 〈a1, . . . , ad〉 of l∩Σ, and we consider the unique (d+1)-cone σ = 〈a1, . . . , ad, ad+1〉 of Σ with a primitive vector ad+1 satisfying l(ad+1) > 0. Then, in thechart Z(σ) one can find a zero-dimensional stratum Oσ′ := F ∩ D(a1) ∩ · · · ∩ D(ad) of D ∩ F .

In fact, there is a one-to-one correspondence between the zero-dimensional strata of D ∩ F andthe points Oσ′ where σ′ runs over the d-cones of Σ ∩ l.

By § 3.7, Oσ′ is given by the equations u1 = · · · = ud = 0 and w := 1 − αn11 ud+1 = 0.

Gonzalez Perez in his thesis [GP00] shows that the strict transform fσ of f in the local coordinates(u1, . . . , ud, w) is a quasi-ordinary singularity, but this coordinate system is not ‘good’ (cf. (4)).However, we can construct a local coordinate system (u1, . . . , ud, w1) (i.e. only w is modified, cf.(4)) in such a way that the projection (u,w1) �→ u induces a quasi-ordinary projection of (fσ, Oσ′),which satisfies (1), (2) and (4). (In fact, by [GP00], w1 = 0 is the strict transform of the hypersurfacegiven by the vanishing of the first approximate root of f .) The germ (fσ, Oσ′) has smaller complexitythan the original germ f : it has only g − 1 distinguished tuples. (They can be computed from theoriginal tuples of f , but this formula is not essential here.)

3.10 The total transform of xβf

Fix β ∈ Zd�0, and consider the germ xβf . Notice that π(Σ)−1({xi = 0}) = D(ei) (for i = 1, . . . , d),

hence D ∪ F = D ∪ {the total transform of xβf}, and both of them have the same ‘naturalstratification’ (i.e. by introducing the factor xβ , we do not create new strata).

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In the proof of Theorem B, we will use an inductive step, which replaces the germ xβf by itstotal transform at the points {Oσ′}σ′ . Obviously, the total transform of xβf at the point Oσ′ hassimilar form, namely uγ fσ for some γ (cf. (11)).

In the next lemma we will use the following notation: For any α = (α1, . . . , αd) ∈ Zd�0, we define

I(α) := {i | αi > 0}.Proposition 3.7. Let σ′ be a cone as in § 3.9. Let uγ fσ be the total transform of xβf at thepoint Oσ′ . Then if #I(β) � 1 and #I(β + λ1) � 2, then #I(γ) � 2. (In particular, uγ fσ satisfiesthe inductive property #I(γ) � 1 and #I(γ + λ1(fσ, Oσ′)) � 2.)

Proof. For simplicity we write 〈ai, β〉 =∑d

j=1 aijβj . Moreover, assume that the coordinates are

ordered, i.e. (3) is satisfied as well.Then by (10)–(12) one gets that

γi = naid+1 + 〈ai, β〉, i = 1, . . . , d.

If aid+1 = 0 for all i = 1, . . . , d, then ai ∈ l ∩ {ud+1 = 0} for all i. However, l ∩ {ud+1 = 0} is

(d − 1)-dimensional, and σ′ is d-dimensional, hence this inclusion is not possible.Assume that for some i0 one has ai0

d+1 �= 0, but aid+1 = 0 for all i �= i0 (1 � i � d). This

means that {ai}i�=i0 is a basis for l ∩ {ud+1 = 0}. Notice that each ai ∈ (Rd+1)∗�0, and 0 = l(ai) =〈ai, λ1〉 − ai

d+1 = 〈ai, λ1〉 for i �= i0. This can happen only if #I(λ1) = 1, i.e. λ1,1 �= 1, but λ1,i = 0for i � 2. In this case l ∩ {ud+1 = 0} = Z(e2, . . . , ed). Since #I(β + λ1) � 2, there exists an indexi′ � 2 such that 〈ei′ , β〉 = βi′ > 0. Therefore, there exists at least one index i′0 �= i0 such that〈ai′0 , β〉 > 0. Therefore, γi0 > 0 and γi′0 > 0 for some i′0 �= i0, hence #I(γ) � 2.

4. Generalities about the zeta function

4.1 Definitions/notations

Let f : (Cd+1, 0) → (C, 0) be the germ of an analytic function, Fε = f−1(ε)∩Br the Milnor fiber of f ,mgeom the geometric monodromy acting on Fε (defined up to an isotopy), and mq ∈ AutHq(Fε, R)the algebraic monodromy induced by mgeom (q � 0). Then the rational function

ζ(f)(t) :=∏q�0

det(I − tmq)(−1)q

is called the zeta function of f .If, additionally, one has a local analytic divisor (V, 0) ⊂ (Cd+1, 0), then one can take a geometric

monodromy mgeom compatible with (V, 0). This means that mgeom acts on (Fε, Fε ∩ V, Fε \ V ).This gives rise to two additional fibrations (over the circle) with fibers Fε ∩ V , respectively Fε \ V .The corresponding zeta functions are denoted as follows:

ζ(f |V )(t) =∏q�0

det(I − tmgeom,q|Hq(Fε∩V,R))(−1)q

,

ζ({f = ε} \ V )(t) =∏q�0

det(I − tmgeom,q|Hq(Fε\V,R))(−1)q

.

By a Mayer–Vietoris argument one has:

ζ(f) = ζ(f |V ) · ζ({f = ε} \ V ).

In particular, if f, g : (Cd+1, 0) → (C, 0) are germs of hypersurface singularities, then taking V ={g = 0}, ζ(f |g=0) means ζ(f |V ).

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Notice that the ‘degree’ of the rational function ζ(f)(t) is exactly the Euler characteristic χ(Fε)of Fε. Sometimes we denote this number by χ(f). Similarly, the degree of ζ(f |g=0) will be denotedby χ(f |g=0) (which is the Euler characteristic of the Milnor fiber {f = ε} ∩ {g = 0} ∩ Br of therestriction f |g=0). The following example will be a building block in the proof of Theorems A and B.

Example 4.1. Let f : (Cd+1, 0) → (C, 0) be defined by f(x, xd+1) = xβ(xn1d+1 + cxλ), where x =

(x1, . . . , xd), xλ = xλ11 · · · xλd

d (similarly for xβ), n1 � 1, and c ∈ C. Assume that #I(β) � 1 and#I(β + λ) � 2. (Here, as above, I(α) = {i | αi > 0}.) Then ζ(f)(t) = 1.

Indeed, since f is given by a weighted homogeneous polynomial, we can replace the local situationby the affine one. Then Fε = {f = ε} ⊂ Cd+1. Assume that c �= 0, and consider the projectionpr : Cd+1 → Cd given by (x, xd+1) �→ x. Then for ε �= 0, pr(Fε) = Cd \ ⋃

βj �=0{xj = 0}. Thediscriminant locus ∆ of the projection pr : Fε → pr(Fε) is {cxβ+λ = ε}. Since I(β) � 1, one getsχ(pr(Fε)) = 0; similarly, since I(β + λ) � 2, χ(∆) = 0. Therefore, χ(Fε) = 0. Notice that the abovegeometric picture is compatible with the monodromy action, hence ζ(f) = 1 by similar argument(cf. [Nem91, Lemma 3.3.9]). (If c = 0 then the result is trivial.)

4.2 One has the following generalization of A’Campo’s formula [A’C75], which provides the zetafunction in terms of an embedded resolution.

Fix the following data:

i) an arbitrary analytic germ h : (Cd+1, 0) → (C, 0);ii) a local analytic divisor (V, 0) ⊂ (Cd+1, 0);iii) an analytic subset (S, 0) ⊂ (V, 0) ∪ ({h = 0}, 0).

Let Br be a sufficiently small ball in Cd+1 centered at the origin. We write {h = ε} for the Milnorfiber h−1(ε) ∩ Br (0 < ε � r).

Assume that φ : X → Br is a birational modification such that:

1) φ−1({h = 0} ∪ V ) is a normal crossing divisor;2) φ is an isomorphism above Br \ S.

Let E be the total transform of {h = 0}∪V , {Ei}i the irreducible components of E, and mEi(h)the vanishing order of h ◦ φ along Ei.

Proposition 4.2. With the above notations, one has:

ζ({h = ε} \ V ) =∏

i

(1 − tmEi(h))χ(Ei\

⋃j �=i Ej).

The proof is similar to the proof of the classical case [A’C75] (see also [AGV88, Theorem 3.10]),and it is left to the reader. The classical case corresponds to V = ∅ (cf. also with Remark 4.6.)

Corollary 4.3 (Splitting property for χ) [BMcN02]. Let f , g : (Cd+1, 0) → (C, 0) be two germsof analytic functions. Then

χ(fg) = χ(f) + χ(g) − χ(f |g=0) − χ(g|f=0).

In order to emphasize the similarities and the differences between the splitting property of χand ζ, we present a short proof of Corollary 4.3.

Proof. Fix an embedded resolution φ of the divisor ({fg = 0}, 0) ⊂ (Cd+1, 0) which has the propertythat it is an isomorphism above Cd+1 \S for some S ⊂ Sing{fg = 0}. Let E be the total transformof {fg = 0}. By applying Proposition 4.2 for h = f and V = {g = 0}, we obtain χ(f)− χ(f |g=0) =∑

i mEi(f) · χ(Ei \⋃

j �=i Ej).

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L. J. McEwan and A. Nemethi

One can write a similar identity for χ(g) − χ(g|f=0), and for χ(fg) (with h = fg and V = ∅).The corollary then follows from mEi(fg) = mEi(f) + mEi(g).

4.3 Notice that Corollary 4.3 has no analog for ζ, since, in general, (1−tmEi(f))χi ·(1−tmEi

(g))χi �=(1 − tmEi

(fg))χi . In fact, the reader can easily find examples of germs f, g such that ζ(fg) �=ζ(f)ζ(g)/ζ(f |g=0)ζ(g|f=0).

Nevertheless, we prove the following weak version of the splitting property for ζ.

Proposition 4.4 (Weak splitting property for ζ). Let f , g : (Cd+1, 0) → (C, 0) be two germs ofanalytic functions. Assume that for any k � 1 we have ζ(fkg)(t) = 1. Then

ζ(f) = ζ(f |g=0) and ζ(g) = ζ(g|f=0).

Proof. Consider the geometric set-up of the proof of Corollary 4.3. By Proposition 4.2 one gets

ζ(f)ζ(f |g=0)

=∏

i

(1 − tmEi(f))χi ,

where χi := χ(Ei \⋃

j �=i Ej); and there is a similar formula for ζ(g)/ζ(g|f=0). Moreover,

ζ(fkg) =∏

i

(1 − tkmEi(f)+mEi

(g))χi .

Now the result follows from the next lemma (with s = 1). If the pairs {mEi(f),mEi(g)}i arenot distinct, regroup them replacing {χi}i by {χj :=

∑i∈Ij

χi}j for convenient index sets {Ij}j

corresponding to equal pairs.

Lemma 4.5. Let J be a finite set, s ∈ Z>0, {χj}j∈J a set of integers, and finally mjl > 0 (j ∈

J, 0 � l � s) positive integers such that the vectors mj = (mj0, . . . ,m

js) ∈ Zs+1

>0 are distinct (j ∈ J).Assume that for any (k1, . . . , ks) ∈ Zs

>0 we have:∏j∈J

(1 − tmj0+∑

l�1 mjl kl)χj = 1. (15)

Then χj = 0 for all j ∈ J .

Proof. If mj0 is maximal in the lexicographic order, then

M := mj00 +

∑l�1

mj0l kl > mj

0 +∑l�1

mjl kl

for any j �= j0 and some well-chosen kl’s. Fix such kl’s, take a primitive M -root ξ of unity, and seethe multiplicity of t − ξ in the left- and right-hand sides of (15). This gives χj0 = 0. Then proceedby induction.

Remark 4.6. In Proposition 4.2 it was convenient to formulate the A’Campo type theorem in termsof the total transform. However, when V = ∅, it is more natural to formulate it in terms of thecompact space φ−1(0). In fact, if we stratify the spaces considered in Proposition 4.2 with respectto φ−1(0) and its complement φ−1(Br \ {0}), the contribution to the zeta function over Br \ {0}is 1 by vanishing of some Euler characteristics. Therefore, it is enough to consider only the compactspace φ−1(0). In fact, this will be crucial in the proof of the main theorems. On the other hand, insome cases, we need a similar result formulated in the context of partial resolutions. This type ofstatement was proved in [GLM97].

Theorem 4.7. Let φ : X → Br be an arbitrary birational modification such that φ is an isomor-phism above the complement of {f = 0}. Let S be a semi-analytic stratification of φ−1(0) such that

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along each stratum Ξ of S the zeta function of the germ (f ◦ φ, x) at x ∈ Ξ does not depend onx ∈ Ξ. Denote this rational function by ζΞ(t). Then:

ζ(f)(t) =∏Ξ∈S

ζΞ(t)χ(Ξ).

(In this paper stratification means a pre-stratification, without any regularity assumption.)

Remark 4.8. In the setting of an embedded resolution when the total transform is a normal crossingdivisor (and the stratification is a ‘natural’ one), then all the contributions from the non-top-dimensional strata Ξ vanish, since along them ζΞ(t) = 1. (In fact this follows from Proposition 4.2,case c = 0. Compare also with Proposition 4.2 or with the original case of A’Campo [A’C75].)

5. The proofs of Theorems A and B

5.1 Proof of Theorem BTheorem B (reformulated). Let f be a quasi-ordinary singularity as in § 2.1. We fix β ∈ Zd

�0 with#I(β) � 1. Assume that g > 0 and the first distinguished tuple λ1 satisfies #I(β + λ1) � 2. Then

ζ(xβf) = 1.

Proof. First notice that we can assume that f satisfies property (4) of § 2.2 ((1) and (2) areautomatically satisfied, cf. § 2.2). Then consider a regular fan Σ as in Proposition 3.2, and thepartial resolution π(Σ) : Z(Σ) → Cd+1 as described in § 3. The critical locus of π(Σ) is included inboth F and F1 (cf. Proposition 3.2). Therefore, Theorem 4.7 can be applied for both xβf and xαf1

(for some α, see below). First we apply for the germ xβf .Define the index set

Σ(1)(β) := Σ(1) \ {ei}i�∈I(β).

The corresponding divisor is

D(β) :=⋃

a∈Σ(1)(β)

D(a).

Obviously, D(β) ⊂ D, and D(β) ∪ F is exactly the total transform of xβf .The divisor D has two natural stratifications provided by intersections of type {DA}A, where in

the first case A ⊂ Σ(1), and in the second case A ⊂ Σ(1)(β). Obviously, the second one is coarser.In fact, in this proof we will need the following ‘mixed stratification’ of D:{( ⋂

a∈A

D(a)∖ ⋃

a�∈A

D(a))∖

F

}A⊂Σ(1)(β)

,

{( ⋂a∈A

D(a)∖ ⋃

a�∈A

D(a))∩ F

}A⊂Σ(1)

.

(The first group of strata has the advantage that it is compatible with A’Campo’s theorem inthe sense that ζΞ(t) = 1 for all non-top-dimensional strata Ξ; cf. Remark 4.8; the second grouphas the advantage that it corresponds to a toroidal embedding, hence χ(Ξ) = 0 for all non-zero-dimensional strata Ξ; cf. (14). See the details below.)

This stratification of D induces a stratification on π(Σ)−1(0) which will be denoted by S.By Lemma 3.3, each stratum Ξ of S is included in some compact intersection of type

⋂a∈A D(a).

By the very definition, the stratification is compatible with F . Denote by S ∩ F (respectively byS \ F ) the collection of those strata which are in F (respectively are in D \ F ). The stratificationis compatible with the assumptions of Theorem 4.7 by § 3.8. Then

ζ(xβf)(t) =∏

Ξ∈S\F(ζΞ(t))χ(Ξ) ·

∏Ξ∈S∩F

(ζΞ(t))χ(Ξ). (16)

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L. J. McEwan and A. Nemethi

The first factor on the right-hand side of (16), which we denote ζS\F , corresponds to the productover Ξ ∈ S \ F . It can be computed as follows. Since D(β) is the total transform of xβf , and thedivisor D(β) \ F is a normal crossing divisor, one gets that for any Ξ ∈ S \ F the correspondingcontribution is

ζΞ(t)χ(Ξ) = (1 − tmΞ(xβf))χ(Ξ),

provided that dim Ξ = d, and equals 1 otherwise (cf. Remark 4.8). If dim Ξ = d, then Ξ ⊂ D(a) forsome a ∈ Σ(1), and by (11) we obtain

mΞ(xβf) = mD(a)(xβf) = m(a) + 〈a, β〉 = infv∈N (f)〈a, v〉 + 〈a, β〉.

ThereforeζS\F =

∏(1 − tm(a)+

∑l alβl)χ(Ξ), (17)

where the product is over Ξ ∈ S \ F with Ξ ⊂ D(a) (and 1 � l � d).Next, for any k = (k1, . . . , kd) ∈ Zd

>0, consider the germ

xk·βf1 := xk1β11 · · · xkdβd

d · f1.

We apply Theorem 4.7 for this germ and modification π(Σ).The main point is that the same stratification S is good in this case as well (cf. Proposition 3.5).

Moreover, D ∪ {total transform of xk·βf1} is a normal crossing divisor (cf. § 3.6), hence in theζ(xk·βf1) we have contributions only from the top-dimensional strata, and all of these are situatedin S \ F .

On the other hand, we know from Example 4.1 that ζ(xk·βf1) = 1. Therefore:∏(1 − t(1/e1)m(a)+

∑l alklβl)χ(Ξ) = 1 (18)

for any (k1, . . . , kd) ∈ Zd>0, and the product is over Ξ ∈ S \ F with Ξ ⊂ D(a) (and 1 � l � d).

In this formula we used for a stratum Ξ ⊂ D(a) that

mΞ(f1) = infv∈N (f1)

〈a, v〉 = infv∈(1/e1)N (f)

〈a, v〉 =1e1

mΞ(f).

But (17), (18) and Lemma 4.5 imply that in (16) the first factor ζS\F = 1.

Now, the second factor on the right-hand side of (16) is a product over Ξ ∈ S ∩ F . By Proposi-tion 3.5 and (14), each positive dimensional stratum has Euler characteristic χ(Ξ) = 0. Hence, (16)reads as

ζ(xβf)(t) =∏

dimΞ=0,Ξ∈S∩F

ζΞ(t)

=∏

σ′⊂Σ∩l,dimσ′=d

ζ (total transform of xβf at Oσ′)(t).

But each total transform at Oσ′ has the form uγ fσ satisfying all the inductive properties: fσ is quasi-ordinary in local coordinates (u,w1) with properties (1), (2) and (4), it has less complexity (i.e. withg − 1 distinguished tuples), and #I(γ) � 1, #I(γ + λ1(fσ)) � 2 (cf. § 3.10 and Proposition 3.7).

Notice that, in the case g = 1, uγ fσ has the form uγw in some local coordinates (u,w), henceby Example 4.1 its zeta function is 1. Hence Theorem B follows by induction (over g).

5.2 Proof of Theorem ATheorem A is a consequence of Theorem B and the splitting property. Indeed, we can assume thatf satisfies properties (1)–(4) (cf. § 2.2). Set β := (0, . . . , 0, k). Then #I(β) � 1 and #I(β +λ1) � 2.Then by Theorem B one has ζ(xk

df) = 1 for any k � 1. Then ζ(f) = ζ(f |xd=0) by Proposition 4.4.

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The zeta function of a quasi-ordinary singularity

5.3 Final remarks

1) In the case of an irreducible plane curve singularity (i.e. in the case of d = 1), ζ(f) is acomplete topological invariant. If d � 2, then this is not true any more: ζ(f) forgets almost all theinformation about the (normalized) distinguished tuples {λi}i of f . For example, if λ1,2 �= 0, thenζ(f)(t) = 1 − tn, hence from ζ(f)(t) only the degree n can be recovered. On the other hand, thisfact can have interesting consequences (e.g. in the Jung program).

2) It is not true that the Milnor fiber Fε(f) of f has the same homotopy type as the Milnorfiber of Fε(f |xd=0) of f |xd=0. For example, if d = 2 and f = xn

3 − x1x2, then Fε(f) is∨

n−1 S2,but Fε(f |x2=0) ≈ disjoint union of n discs. Notice that Fε(f) can be obtained from Fε(f |x2=0) byidentifying the boundaries of the discs (i.e. by an S1-equivariant surgery).

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Lee J. McEwan mcewan@math.ohio-state.eduDepartment of Mathematics, Ohio State University, 231 West 18th Avenue, Columbus, OH 43210,USA

Andras Nemethi nemethi@math.ohio-state.eduDepartment of Mathematics, Ohio State University, 231 West 18th Avenue, Columbus, OH 43210,USA

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