ON THE MILNOR FIBER OF A QUASI-ORDINARY SURFACE SINGULARITY

CHUNSHENG BAN, LEE J. MCEWAN, AND ANDRÁS NÉMETHI

Abstract. We verify a generalization of (3.3) from [Lê73] proving that the homotopy type of the Milnor fiber of a reduced hypersurface singularity depends only on the embedded topological type of the singularity. In particular, using [Zariski68, Lipman83, Oh93, Gau88] for irreducible quasi-ordinary germs, it depends only on the normalized distinguished pairs of the singularity. The main result of the paper provides an explicit formula for the Euler-characteristic of the Milnor fiber in the surface case.

1. Introduction

We consider the germ of an irreducible quasi-ordinary surface singularity at the origin in C^{3}, defined by the equation f(x, y, z) = 0. Thus, we suppose that the projection π : (F, 0) → (C^{3}, 0) on the first two coordinates (x, y) restricted to (F, 0) = ({f = 0}, 0) is a finite map onto (C^{2}, 0) with (reduced) discriminant (Δ, 0) contained in ({xy = 0}, 0). The main goal of this note is the computation of the Euler-characteristic χ(F_{ϵ}) of the Milnor fiber F_{ϵ} of f. We recommend as general references for hypersurface singularities the books [Milnor68] and [AGV88]. The Milnor fiber F_{ϵ} is defined as follows. For any germ f : (C^{d+1}, 0) → (C, 0), we fix a sufficiently small closed ball B_{r} in C^{d+1} of radius r, then F_{ϵ} := f^{−1}(ϵ) ∩ B_{r} (0 < ϵ ≪ r). For χ(F_{ϵ}) we will use the simplified notation χ(f). Our main result is the following

Theorem. Assume that f : (C^{3}, 0) → (C, 0) is an irreducible quasi-ordinary singularity represented in a “normalized” coordinate system (cf. 2.7). Then χ(f) = χ(f|_{y=0}). More precisely, the Euler-characteristic of the Milnor fiber of f is exactly the Euler-characteristic of the Milnor fiber of the plane curve singularity (x, z) → f(x, 0, z).

The importance of the above theorem can be illuminated by the following qualitative results as well. First we recall that the embedded topological type of an irreducible plane curve singularity is completely determined by its Puiseux pairs. Moreover, these pairs determine χ(F_{ϵ}) as well.

The point is that for any higher dimensional irreducible quasi-ordinary singularity f, one can define the generalization of the Puiseux pairs: they are called the normalized distinguished tuples of f. (In fact, they are defined via a parametrization ζ of F, but it turns out that they are independent of ζ and depend only on the analytic type of (F, 0).) For details, see [Lipman65, Lipman83, Lipman88].

Using Zariski’s result on saturation of local rings, one can prove that these pairs determine the embedded topological type of f (i.e., the homeomorphism type of the pair (F, 0) ⊂ (C^{d+1}, 0), cf. [Zariski68] and [Lipman88], or [Oh93] for a different proof.) Moreover, using Lipman’s results [Lipman88], Gau in [Gau88] proves that from the embedded topological type one can recover the normalized distinguished tuples of f. In addition, a generalization of Theorem 3.3 of [Lê73] (cf. also [Teissier74], and see (2.4) in our Section 2) says that for any reduced hypersurface singularity, the homotopy type of the Milnor fiber depends only on the embedded topological type of the singularity. Therefore, it is natural to ask for explicit formulae for the homotopical invariants of F_{ϵ} in terms of the distinguished tuples of f. In the next section we will present a more precise version of our theorem which provides χ(F_{ϵ}) in the surface case in terms of the normalized distinguished pairs (cf. 2.8).

The precise definition of the distinguished pairs will be given in Section 2. Since the main result is formulated for surface singularities, in order to avoid an unnecessary flood of notations, we give the notations in this case only. The general situation can be found e.g. in [Lipman88].

The proof starts in Section 3, where we generalize A’Campo’s formula for χ(F_{ϵ}) [A’Campo75] (in terms of the embedded resolution). In fact, the proof of the main theorem is based on the following fact: the expression χ(f) − χ(f|_{y=0}) is “stable” with respect to all the types of blow ups which appear in the resolution process of f. In order to verify this, we need the “splitting property” (3.3) (which itself is proved by our generalized A’Campo-type theorem). The verification of the “stability” is in Section 4.

We end the introduction with the following remark. Upon inspecting the literature of quasi-ordinary singularities, one is very surprised to discover that the amount of results about the smoothing invariants is incredibly small. The point is that although the strong assumption about F (namely, the existence of the projection π with normal crossing discriminant) provides a nice list of analytic and geometric properties of F, it is surprisingly difficult to relate these properties to any kind of property of the level sets F_{ϵ} (ϵ ≠ 0). The present note is the first in a series of articles in which the authors plan to analyze the smoothing invariants (like the zeta function of the monodromy of f, etc.). We remark that, while a central purpose of the definition of quasi-ordinary singularities is to generalize features of plane curve singularities to higher dimensions, our main theorem applies only to surfaces (though see note added at the end of the paper). Also the result has no analog for curves.

2. Quasi-ordinary singularities and their topological type

In this section we recall the definition of the normalized distinguished pairs associated with an irreducible quasi-ordinary singularity (F, 0) ⊂ (C^{3}, 0), and we reformulate our theorem in terms of these pairs. (2.1) By the very definition of irreducible hypersurface quasi-ordinary singularities, there exists coordinates such that f can be expressed as a pseudo-polynomial in the third variable z: f(x, y, z) = z^{m} + g_{1}(x, y)z^{m−1} + ⋯ + g_{m}(x, y), where g_{i} are power series, and the (reduced) discriminant of π : ({f = 0}, 0) → (C^{2}, 0) induced by (x, y, z) ↦ (x, y) is contained in ({xy = 0}, 0). Moreover, there exists a parametrization of F = {f = 0} by a fractional power series ζ = H(x^{1/n}, y^{1/n}), where H(s, t) is a power series and n is a suitable natural number depending on f. (This means that the parametrization (C^{2}, 0) → (F, 0) is given by x = s^{n}, y = t^{n}, z = H(s, t).)

The conjugates of ζ are obtained by multiplying x^{1/n} and y^{1/n} by n^{th}-roots of unity; the number of different conjugates {ζ_{i}}_{i} of ζ is precisely the degree m of the covering π, and

f(x, y, z) = ∏_{i=1}^{m}(z − ζ_{i}).

The roots ζ_{i} are called the branches of f. There is a finite subset {(λ_{i}, μ_{i})}_{i=1}^{s} of the set of exponents {(λ_{i}, μ_{i})}_{i} of the monomials appearing in ζ with non-zero coefficient, called the distinguished pairs, which play a role similar to Puiseux pairs for plane curve singularities. Namely, by unique factorization of the discriminant one has:

ζ_{i} − ζ_{j} = x^{u_{ij}/n}y^{v_{ij}/n}ϵ_{ij}(x^{1/n}, y^{1/n}) with ϵ_{ij}(0, 0) ≠ 0,

and the set of exponents {(u_{ij}/n, v_{ij}/n)}_{i,j} constitute the set of distinguished pairs.

Say that (λ, μ) ≤ (λ′, μ′) if λ ≤ λ′ and μ ≤ μ′. Then the distinguished pairs are ordered:

(1) (0, 0) < (λ_{1}, μ_{1}) < (λ_{2}, μ_{2}) < ⋯ < (λ_{s}, μ_{s});

and (after permuting x and y if necessary) we can assume:

(2) (λ_{1}, … λ_{s}) > (μ_{1}, … μ_{s}) (lexicographically).

Let c_{λ,μ} be the coefficent of the term in ζ having exponent pair (λ, μ). The distinguished pairs generate all the other exponent pairs in the following sense:

(3) If c_{λ,μ} ≠ 0 then (λ, μ) ∈ Z × Z + ∑_{(λ_{i},μ_{i})≤(λ,μ)} Z(λ_{i}, μ_{i}) (and (λ_{j}, μ_{j}) ∉ Z × Z + ∑_{(λ_{i},μ_{i})<(λ_{j},μ_{j})} Z(λ_{i}, μ_{i})).

The parametrization ζ of F, in general, is not unique. Moreover, different parametrizations could produce different sets of distinguished pairs. But if we consider only the “normalized” parametrizations, then the corresponding pairs are independent of the choice of parametrization. A parametrization is said to be normalized if in addition to (1)-(3) we have

(4) If c_{λ,μ} ≠ 0, then λ and μ are not both integers. (In the literature this condition is not usually included, and as we see in Example (4.3) is not really necessary.)

(5) If μ_{1} = 0 then λ_{1} > 1.

Given a parametrization

ζ = cx^{a}y^{b} + higher order terms ∈ C{x^{1/n}, y^{1/n}} (c ≠ 0)

then if ζ is normalized, it follows from (3) and (4) that (λ_{1}, μ_{1}) = (a, b). If ζ is not normalized and (λ_{1}, μ_{1}) ≰ (λ, μ) with c_{λ,μ} ≠ 0 then (λ, μ) ∈ Z × Z.

(2.2) If a parametrization is not normalized, then by a change of variables ϕ : (x, y, z) → (x′, y′, z′) one can transform ζ into a normalized parametrization ζ′ = ζ ∘ ϕ. Basically, there are two cases to consider.

(a) If c_{λ,μ} ≠ 0 with (λ, μ) ∈ Z × Z, then we make a change of variables (x′, y′, z′) = (x, y, z − c_{λ,μ}x^{λ}y^{μ}). In this way ζ′ will have no term with exponent (λ, μ). This applied repeatedly assures the validity of (4). Moreover, this type of transformation preserves the distinguished pairs of the parametrization.

(b) If (5) is not satisfied then first one has to make a transformation of type (x, y, z) → (z, y, x) (and express f as a pseudo-polynomial in z′ = x), then, if it is necessary, one has to perform a finite number of times the transformation described in (a). (This sequence of transformations is called “inversion”, for details see [Lipman65].) In this case the distinguished pairs will be changed (cf. also Lipman’s Table presented in paragraph (4.2)). In fact the transformation (x, y, z) → (z, y, x) (and f rewritten as a pseudo-polynomial in z′ = x) changes the distinguished pairs {(λ_{i}, μ_{i})}_{i=1}^{s} into the new pairs (the first of which may be integral, hence non-distinguished):

{((λ_{i} + 1 − λ_{1})/λ_{1}, μ_{i})}_{i=1}^{s}.

If the first new pair (1/λ_{1}, μ_{1}) is in Z × Z, then we eliminate this pair (and any other previously non-distinguished pair which now becomes integral) by a sequence of transformations of type (a). All the other pairs will survive as distinguished pairs; for this notice that (λ_{2} + 1 − λ_{1})/λ_{1}, μ_{2}) cannot become integral unless (λ_{2}, μ_{2}) depends integrally on (λ_{1}, μ_{1}), which is forbidden by (3) above.

(2.3) As we already mentioned in the introduction, in any dimension, for any irreducible hypersurface quasi-ordinary singularity f : (C^{d+1}, 0) → (C, 0) the information codified in the set of normalized distinguished tuples is the same as the information codified in the embedded topological type of f, i.e., in the homeomorphism type of the pair L ⊂ S^{2d+1}, where S^{2d+1} = ∂B_{r}, and L = f^{−1}(0) ∩ S^{2d+1} is the link of f.

On the other hand, we can verify the following general result. For isolated singularities it was proved in [Lê73] (cf. also [Yau89]); the topological invariance of the Milnor number for isolated singularities was first noticed in [Teissier74].

(2.4) Proposition. Denote by f : (C^{d+1}, 0) → (C, 0) a reduced hypersurface singularity, and let F_{ϵ} be its Milnor fiber, and m_{geom} the geometric monodromy acting on F_{ϵ} (defined up to an isotopy). Then:

(a) The homotopy type of (F_{ϵ}, m_{geom}) can be recovered from the embedded topological type of f.

(b) F_{ϵ} is connected.

(c) If f is irreducible then π_{1}(F_{ϵ}) = [G, G], where G = π_{1}(S^{2d+1} \ L, ∗). Proof. The proof is similar to the proof of Theorem (3.3) in [Lê73]. For the convenience of the reader, we give the details as well. F_{ϵ} is a manifold with boundary ∂F_{ϵ} = f^{−1}(ϵ) ∩ S^{2d+1}. Let F_{ϵ}^{0} be the open Milnor fiber F_{ϵ} \ ∂F_{ϵ}. Then there is a (Milnor) fibration arg := f/|f| : S^{2d+1} \ L → S^{1} with fiber (diffeomorphic to) F_{ϵ}^{0} [Milnor68]. The fibration is completely characterized (up to a homotopy) by the induced map arg_{1} : π_{1}(S^{2d+1} \ L, ∗) → π_{1}(S^{1}) = Z. Indeed, since arg_{1} is onto (see below), F_{ϵ}^{0} can be homotopically identified with the covering space of S^{2d+1} \ L associated with arg_{1} (or with the subgroup ker(arg_{1})). Moreover, the corresponding Galois action is exactly Z, and its generator 1_{Z} can be identified with m_{geom}.

On the other hand, arg_{1} can be completely determined from (S^{2d+1}, L). Indeed by Alexander duality H_{1}(S^{2d+1} \L, Z) = H^{2d−1}(L, Z). Consider the irreducible decomposition f = f_{1}f_{2} … f_{t} of f. Let L_{i} = f_{i}^{−1}(0)∩S^{2d−1}, then obviously L = ∪_{i}L_{i}. Moreover, H^{2d−1}(L_{i}, Z) = Z, and by a Mayer-Vietoris argument H^{2d−1}(L, Z) = Z^{t}. The isomorphism H_{1}(S^{2d+1} \L, Z) = Z^{t} is realized as follows: for any smooth point p of L_{i} take a local transversal slice of L_{i} in S^{2d+1} and fix a small oriented circle in it going around L_{i} in S^{2d+1}. Its homology class is the i^{th} base element of Z^{t}. This shows that arg_{1} is the composed map π_{1}(S^{2d+1}\L, ∗) →^{h} H_{1}(S^{2d+1}\L, Z) = Z^{t} →^{s} Z where h is the Hurewicz map and s((α_{1}, …, α_{t})) = ∑_{i} α_{i}. This shows that arg_{1} is onto, hence F_{ϵ}^{0} is connected. If f is irreducible, then arg_{1} = h, and π_{1}(F_{ϵ}^{0}) = [G, G] by the long homotopy exact sequence.

(2.5) If we start with f = f_{1}^{m_{1}} … f_{t}^{m_{t}}, then in the above proof s((α_{1}, …, α_{t})) = ∑_{i} α_{i}m_{i}, hence F_{ϵ}^{0} has g.c.d.(m_{1}, … m_{t}) connected components, and the homotopy type of F_{ϵ}^{0} can be recovered from the type of (S^{2d+1}, L) and the set of integers (m_{1}, … m_{t}) (i.e., from the “multilink” L ⊂ S^{2d+1}) (see also [Dimca92], page 76].)

(2.6) The discussion (2.3) and proposition (2.4) show that, for any irreducible quasiordinary singularity, there should be an explicit formula for χ(F_{ϵ}) in terms of the distinguished pairs of f. The goal of the next part is the presentation of this formula.

(2.7) Now we return to an irreducible quasi-ordinary surface singularity f : (C^{3}, 0) → (C, 0). We assume that it is represented in some coordinate system (x, y, z) which admits a normalized parametrization ζ.

First assume that SingF = ∅. Then f has no normalized distinguished pairs, and f = z. Hence m = deg_{z}(f) = 1. We also define g(x, z) := z, and m̃ = deg_{z}(g) = 1.

If SingF ≠ ∅, then denote the normalized distinguished pairs by {(λ_{i}, μ_{i})}_{i=1}^{s}. In this paragraph we make more precise the connection between the curve sin-

gularity f(x, 0, z) and the parametrization ζ.

First assume that μ_{1} = 0. Then by (5) λ_{1} > 1, and by (1) all λ_{i} are non-zero. Suppose μ_{1} = μ_{2} = ⋯ = μ_{k} = 0, and μ_{k+1} > 0. Then, again by (1) μ_{k+i} > 0 for all i ≥ 1. In this case H(s, t) (cf. 2.1) has a decomposition H(s, t) = K(s)+L(s, t), where K and L are power series and L(s, 0) = 0. Recall that ζ = H(x^{1/n}, y^{1/n}); then we define the fractional power series φ = K(x^{1/n}) and ψ = L(x^{1/n}, y^{1/n}). Let m̃ be the number of conjugates {φ_{j}}m̃ ; j=1 of φ. We emphasize that even if ζ (respectively φ) has some terms whose exponents are not distinguished pairs, the number of conjugates m (respectively m̃) of ζ (respectively of φ) depends only on the distinguished pairs {(λ_{i}, μ_{i})}_{i=1}^{s} (respectively {(λ_{i}, 0)}_{i=1}^{k}). (Indeed, m, respectively m̃, is the number of different conjugates of the “simplified parametrization” ζ̃ = ∑_{i=1}^{s} x^{λ_{i}}y^{μ_{i}}, respectively of φ̃ = ∑_{i=1}^{k} x^{λ_{i}}.)

Since each conjugate of ζ may be obtained as a conjugate of φ plus a conjugate of ψ, it is clear that m̃ divides m. Let’s consider the curve singularity given by the conjugates of φ:

g(x, z) = ∏_{j=1}^{m̃}(z − φ_{j}).

If μ_{1} ≠ 0, then by definition, φ ≡ 0, m̃ = 1 and g(x, z) ≡ z.

With these notations one has: g(x, z) = f(x, 0, z)_{red}, f(x, 0, z) = g(x, z)^{m/m̃} (and φ is a parametrization of {g = 0}). Moreover, the set {λ_{i}}_{i=1}^{k} provides exactly the set of Puiseux pairs of g.

If μ_{1} ≠ 0 then π^{−1}({xy = 0}) is the union of the x and the y-axis of C^{3}. If μ_{1} = 0, then π^{−1}({x = 0}) is the y-axis of C^{3}, but π^{−1}({y = 0}) is the curve {g = 0}. The integer m̃ also has the following interpretation: for a given point (x_{0}, 0) with x_{0} ≠ 0, the fiber π^{−1}(x_{0}, 0) consists of m̃ points (counted without multiplicities).

Now we formulate our main result in terms of the normalized distinguished pairs of f: (2.8) Theorem. Consider f and g as in (2.7). Then

χ(f) = χ(g) · \frac{m}{m̃}; i.e. χ(f) = χ(f|_{y=0}).

In particular, χ(f) depends only on the integer m and the subset {(λ_{i}, 0)}_{i=1}^{k} of the set of normalized distinguished pairs.

Moreover, in the “symmetric case”, i.e. if either SingF = ∅, or μ_{1} ≠ 0, one has:

f|_{y=0} = f|_{x=0} = z^{m},

hence χ(f) = m.

For the computation of χ(g) in terms of {λ_{i}}_{i=1}^{k}, see e.g. [BK86]. In a normalized coordinate system as in (2.7)-(2.8), we sometimes say that the y-axis is “distinguished.”

The proof of (2.8) is given in the next two sections.

3. The first part of the proof. General facts about χ(F_{ϵ}).

(3.1) We start with the following generalization of A’Campo’s formula [A’Campo75]. Fix the following data:

• an arbitrary analytic germ h : (C^{d+1}, 0) → (C, 0).

• a local analytic divisor (V, 0) ⊂ (C^{d+1}, 0).

• an analytic subset (S, 0) ⊂ (V, 0) ∪ ({h = 0}, 0). Let B_{r} be a sufficiently small ball in C^{d+1} centered at the origin. We write {h = ϵ} for the Milnor fiber h^{−1}(ϵ) ∩ B_{r} (0 < ϵ ≪ r).

Assume that ϕ : X → B_{r} is a birational modification such that

(1) ϕ^{−1}({h = 0} ∪ V) is a normal crossing divisor.

(2) ϕ is an isomorphism above B_{r} \ S.

Let E be the total transform of {h = 0} ∪ V, {E_{i}}_{i} the irreducible components of E, and, m_{E_{i}}(h) the vanishing order of h ∘ ϕ along E_{i}.

(3.2) Proposition. With the above notations, one has:

χ({h = ϵ} \ V) = ∑_{i} m_{E_{i}}(h) · χ(E_{i} \ ⋃_{j≠i} E_{j}).

The proof is similar to the proof of the classical case [A’Campo75] (see also [AGV88] Theorem 3.10), and it is left to the reader.

In the next corollary we will use the following notation. If f, g are hypersurface singularities (C^{d+1}, 0) → (C, 0) then f|_{g=0} : ({g = 0}, 0) → (C, 0) denotes the restriction of f, and χ(f|_{g=0}) is the Euler-characteristic of its Milnor fiber (i.e., χ({f = ϵ} ∩ {g = 0} ∩ B_{r}) with 0 < ϵ ≪ r).

(3.3) Corollary. The splitting property. Let f, g : (C^{d+1}, 0) → (C, 0) be two germs of analytic functions. Then

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

Proof. Fix an embedded resolution ϕ of the divisor ({fg = 0}, 0) ⊂ (C^{d+1}, 0) which has the property that it is an isomorphism above C^{d+1} \S for some S ⊂ Sing{fg = 0}. Let E be the total transform of {fg = 0}. By applying (3.2) for h = f and V = {g = 0}, we obtain χ(f) − χ(f|_{g=0}) = ∑_{i} m_{E_{i}}(f) · χ(E_{i} \ ∪_{j≠i}E_{j}).

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 m_{E_{i}}(fg) = m_{E_{i}}(f) + m_{E_{i}}(g).

(3.4) Remarks.

(a) Notice that (3.3) has no analogue for the zeta function associated with the monodromy action of f.

(b) We want to emphasize the following fact regarding A’Campo-type theorems. In order to be able to apply the classical formula [A’Campo75], or (3.2), we need to know the structure and topology of the exceptional divisors in an embedded resolution. But typically what we know is only an algorithm (and even this can be very involved) which gives the steps of the resolution (i.e., tells what to blow up in the next step), without giving a global overview of the result of the resolution process. In such cases it is extremely helpful to have instead a formula for χ(F_{ϵ}) in terms of a partial resolution.

(3.5) Theorem. [GLM97] Let ϕ : X → B_{r} be an arbitrary birational modification such that ϕ is an isomorphism above the complement of {f = 0}. Let S be an analytic stratification of the total transform of {f = 0} such that along each stratum Ξ of S the Euler-characteristic of the Milnor fiber of f ∘ ϕ at x ∈ Ξ does not depend on x ∈ Ξ. Denote this number by χ_{Ξ}. Then:

χ(f) = ∑_{Ξ∈S} χ_{Ξ} · χ(Ξ).

This theorem will be applied in the resolution process of a quasi-ordinary singularity. The computation is in the next section.

We end this section with the following list of Euler-characteristic computations associated with the different plane sections of {f = 0}.

(3.6) Lemma. Let f : (C^{3}, 0) → (C, 0), f(x, y, z) = z^{m} + g_{1}(x, y)z^{m−1} + … be an irreducible quasi-ordinary singularity, such that the discriminant of the projection π : (F, 0) → (C^{2}, 0) is contained in {xy = 0}. Assume that (F, 0) admits a parametrization z = H(s, t) = cs^{a}t^{b} + …, x = s^{n}, y = t^{n} (c ≠ 0) having the form H(s, t) = cs^{a}t^{b}(unit), but which is possibly not normalized. Assume that a ≠ 0. Then (1) χ(f|_{x=0}) = χ(f|_{x=y=0}) = m.

(2) χ(x|_{f=0}) = χ(x|_{f=y=0}) (= m̃ = ((f|_{y=0})_{red}, x)_{0}). In fact {x = ϵ} ∩ {f = y = 0} is a strong deformation retract of {x = ϵ} ∩ {f = 0}

(3) χ(y|_{f=0}) = χ(y|_{f=x=0}) = χ(y|_{f=z=0}) = 1. In fact {y = ϵ} ∩ {f = 0} is contractible and {y = ϵ} ∩ {f = x = 0} is a strong deformation retract of {y = ϵ} ∩ {f = 0}.

(4) If b ≠ 0, then χ(z|_{f=0}) = 0; in fact there is a topological covering F_{ϵ}(H) → {z = ϵ} ∩ {f = 0}, where F_{ϵ}(H) is the Milnor fiber of H. In this case, {z = ϵ} ∩ {f = y = 0} = ∅, hence χ(z|_{f=y=0}) = 0 as well.

If b = 0, then χ(z|_{f=0}) = χ(z|_{f=y=0}) = ((f|_{y=0})_{red}, z)_{0}. In fact {z = ϵ} ∩ {f = y = 0} is a strong deformation retract of {z = ϵ} ∩ {f = 0}.

(5) If b ≠ 0, then χ(f|_{z=0}) = 0; and {f = ϵ} ∩ {z = y = 0} = ∅, hence χ(f|_{z=y=0}) = 0 as well.

If b = 0, then χ(f|_{z=0}) = χ(f|_{z=y=0}) = (f, z, y)_{0}. In fact {f = ϵ} ∩ {z = y = 0} is a strong deformation retract of {f = ϵ} ∩ {z = 0}. (In the above, in all cases ϵ ≠ 0 and all spaces are considered in the ball B_{r}; and (,)_{0} denotes the intersection multiplicity.) Proof.

For (1), notice that f|_{x=0} = f|_{x=y=0} = z^{m} because of the condition a ≠ 0. For (2), observe that the Milnor fiber of x|_{f=0} is just the inverse image under the projection π of the disk Δ = {(ϵ, y)}. Since π is a topological covering over the set {xy ≠ 0} and the punctured disk Δ^{∗} = Δ \ (0, 0) lies in this region, π^{−1}(Δ^{∗}) must be a union of disjoint puntured disks E_{i}. The closure of each of these punctured disks must be a disk, whence π^{−1}(Δ) is a finite union of disks, any two of which intersect at most at a single point of the (finite) set π^{−1}(ϵ, 0). Clearly each point in π^{−1}(ϵ, 0) belongs to at least one E_{i}. Thus each point of π^{−1}(ϵ, 0) is the center of a non-empty collection of disks which intersect only at that point, which shows that π^{−1}(ϵ, 0) is a strong deformation retract of π^{−1}(Δ). The cardinality of π^{−1}(ϵ, 0) is precisely the z degree of (f|_{y=0})_{red}. Part (3) of the lemma is the same as part (2) except that (f|_{x=0})_{red} has degree one, i.e. (f|_{x=0})_{red} = z. For part (4) and b ≠ 0, the set {z = ϵ} ∩ {f = 0} must lie above {xy ≠ 0}, i.e. on the portion of (F, 0) for which the parametrization (x, y, z) = (s^{n}, t^{n}, z) is a covering space. But the pull-back is F_{ϵ}(H), which has the form {cs^{a}t^{b}(unit) = ϵ}, a collection of disjoint punctured disks; hence χ(F_{ϵ}(H)) = 0. On the other hand if b = 0, the situation and the argument are the same as in part (3) with the roles of x and z reversed. Finally, for part (5), {f = ϵ} ∩ {z = 0} is given by {cx^{a}y^{b}(unit) = ϵ}, and this has Euler characteristic zero if b ≠ 0. If b = 0 then it is clear that the Euler characteristic is unchanged when y is restricted.

The splitting property (3.3), part (3) of lemma (3.6), and the main theorem have the following consequence.

(3.7) Corollary. Assume that f : (C^{3}, 0) → (C, 0) is an irreducible quasi-ordinary singularity, and the y axis is distinguished (cf. 2.8). Then χ(yf) = 0.

Remark. The main theorem (2.8) is actually equivalent to the statement χ(yf) = 0. This shows that in some sense it may be more natural to consider the object yf rather than just f.

4. The second part of the proof. The resolution of f.

We will apply the following fundamental facts:

(4.1) Fact. Every surface (F, 0) ⊂ (B_{r}, 0) (or F ⊂ C^{3}) has an embedded resolution obtained by blowing up smooth centers within the singular locus, i.e we may assume that we have a finite sequence

(∗) \begin{matrix}M_{n} & →^{r_{n}} ⋯ → & M_{2} →^{r_{2}} & M_{1} →^{r_{1}} & C^{3} \\ ∪ & & ∪ & ∪ & ∪ \\ F_{n} & →^{r_{n}} ⋯ → & F_{2} →^{r_{2}} & F_{1} →^{r_{1}} & F_{0} = F \end{matrix}

such that r_{i} blows up a smooth center in F_{i−1}, F_{n} is smooth, and F_{n} has normal crossing with the exceptional divisors of r_{1} ∘ ⋯ ∘ r_{n}. Moreover we may assume that the center of each blow up is a point or an equimultiple curve.

There are many proofs of this fact, see e.g. [Hironaka64, Abhyankar98, BM91, Villamayor89].

In addition we have the following result of Lipman [Lipman65].

(4.2) Fact. If an irreducible quasi-ordinary surface F is blown up at a point or a smooth equimultiple curve inside the singular locus, then the strict transform F′ of F is again quasi-ordinary. (Note that this is not the case for quasi-ordinary varieties of higher dimension.)

Lipman also provides rules of transformation by which the (possibly not normalized) distinguished pairs of F′ may be read from those of F. We summarize those rules in the following table.

Transformation  Pairs of resulting branch

Inversion  \frac{λ_{i} + 1 − λ_{1}}{λ_{1}} , μ_{i}

Monoidal Transformation

Center (x, z)  λ_{i} − 1, μ_{i}
Center (y, z)  λ_{i}, μ_{i} − 1

Quadratic Transformation

“Transversal Case” (λ_{1} + μ_{1} ≥ 1)

Direction (1 : 0 : 0)  λ_{i} + μ_{i} − 1, μ_{i}
Direction (0 : 1 : 0)  λ_{i}, λ_{i} + μ_{i} − 1

“Non-Transversal Case” (λ_{1} + μ_{1} < 1)

Direction (1 : 0 : 0)  λ_{i} + \frac{(1 + μ_{i})(1 − λ_{1})}{μ_{1}} − 2, \frac{1 + μ_{i}}{μ_{1}} − 1

Direction (0 : 1 : 0)  μ_{i} + \frac{(1 + λ_{i})(1 − μ_{1})}{λ_{1}} − 2, \frac{1 + λ_{i}}{λ_{1}} − 1

Direction (0 : 0 : 1)  \frac{λ_{i}(1 − μ_{1}) + μ_{i}λ_{1}}{1 − λ_{1} − μ_{1}} , \frac{λ_{i}μ_{1} + μ_{i}(1 − λ_{1})}{1 − λ_{1} − μ_{1}}

In the resolution of a surface quasi-ordinary singularity, in (∗) each morphism r_{i} is either a quadratic transformation (blow up of a point) or a monoidal transformation (blow up of a smooth equimultiple line, locally a disc) with center inside Sing(F_{i−1}).

It is proved in [BMc00] that there exists a resolution process (∗) which depends only on the normalized distinguished pairs of (the original) F, but at each step i the type of modification r_{i} required depends on the history of the process (i.e., on the steps r_{i−1}, r_{i−2}, …, r_{1}). We will not use this complicated algorithm.

Instead, we will examine the effect of each possible type of transformation on the local equation for F_{i−1}. For convenience, we drop the subscript i − 1 and we write f for the local equation of F_{i−1}. The blow up of F_{i−1} (in the case of a blow up of a point) is covered by charts U_{x}, U_{y}, U_{z} (defined by the local equations x, y, and z); we denote the strict transform of f in each chart by f_{x}, f_{y} and f_{z}.

We will assume that f is written in some coordinate system (x, y, z) in which f admits a normalized parametrization. By the above notation, f_{x}, f_{y} and f_{z} are the germs of the strict transform of f considered at the points [1 : 0 : 0], [0 : 1 : 0] and [0 : 0 : 1] respectively. In order to avoid burdensome notation, we will continue to denote by (x, y, z) the new local coordinates: e.g. at the point [1 : 0 : 0] these are obviously equal to (x, xy, xz) in terms of the coordinates of f. We call this new coordinate system at the point [1 : 0 : 0] the “natural coordinate system” of f_{x}. For f_{y} and f_{z} the discussion is similar. If we blow up an axis, then we use similar notations.

We notice that if ζ = cx^{λ_{1}}y^{μ_{1}} + ⋯ is a normalized parametrization of f, and we blow up a center, then the leading term of a (possibly not normalized) parametrization of the strict transform of f in the “natural coordinate system” can be read from the above table by taking i = 1 (i.e. the leading term is exactly the first pair provided by the table). In the non-transversal case, x and y-directions, a permutation of the variables is needed to normalize the new parametrization (see Table 2 below).

Then we will apply theorem (3.5) for the modification r_{i}. In particular there are only two components of the 2-dimensional stratum. One of them is E^{o} = E \{strict transform of {f = 0}}, where E is the exceptional divisor created at this step r_{i}. The contribution on the left hand side of the formula (3.5) is χ(E^{o})m_{0}(f), where m_{0}(f) is the multiplicity of f along E. (In general m_{0}(f) ≠ m; where m is the degree of f.)

The other component is supported by the strict transform St of f, but (as we will now show) χ(St \ E) (=χ(F \ Center)) = 0 hence it gives no contribution. The center is either the origin or the w- axis (w = x or y); if it’s the origin, let w be the smooth component of the pre-image of the discriminant (which must exist). Decompose F \ Center as the disjoint union of A = (F \ Center) ∩ {w ≠ 0} and B = (F \ Center) ∩ {w = 0}. Then A is a fiber bundle over a punctured disk (namely the punctured local w-axis), hence χ(A) = 0. Similarly, B is a punctured local curve, hence χ(B) = 0 as well.

One can verify that in all cases (see the list of cases below) the one dimensional strata are either punctured discs D^{∗}’s or C^{∗}’s, hence their Euler characteristics vanish. Therefore, the theorem reads as

χ(f) = m_{0}(f)χ(E^{o}) + a sum corresponding to the zero-dimensional strata.

At each point of the zero-dimensional strata the local equation is the total transform of f. We split these local equations using the splitting property (cf. 3.3). Then, for any f expressed in a normalized coordinate system, we define D(f) := χ(f) − χ(f|_{y=0}) and we verify that D(f) is stable, i.e.

(S) D(f) = ∑_{dimΞ=0} D(f_{Ξ}).

We emphasize that the expression D(f_{Ξ}) must also be computed in the normalized coordinate system for the germ f_{Ξ} and that the corresponding local coordinate “y” is distinguished in the sense of (2.8).

On the other hand, D(f_{Ξ}) can often be computed in the natural local coordinate system.

(4.3) Examples. (a) Given the germ ζ = c_{λ}x^{λ} + …, not normalized, we claim that D(f) = χ(f) − χ(f|_{y=0}). Indeed, it is easy to check that the normalization transformations preserve the germ {f|_{y=0}}, i.e. {f|_{y=0}} = {f|_{y′=0}} where (x′, y′, z′) are the normalized coordinates.

(b) If f can be normalized by a sequence of transformations of type 2.2 (a) alone, i.e. (x′, y′, z′) = (x, y, z − c_{a,b}x^{a}y^{b})), where at each stage ζ = c_{a,b}x^{a}y^{b}u(x, y) and u(x, y) is a unit, then again D(f) can be computed in the non-normalized coordinates (x, y, z). As in (a), the germs {f|_{y=0}} and {f|_{y′=0}} are equal if ab ≠ 0, and isomorphic otherwise.

Finally, notice that if f is smooth, then D(f) = χ(f) − χ(f|_{y=0}) = 1 − 1 = 0, hence (S) by induction will imply that D(f) = 0 for the original singularity F = F_{0}. The stability formula (S) will be verified for all the possible transformations.

Now we verify the stability (S). Hence we start with a quasi-ordinary singularity f represented in a coordinate system (x, y, z) with a normalized parametrization (satisfying 2.1 (1)-(5) ) ζ = cx^{λ_{1}}y^{μ_{1}} + ⋯, where λ_{1} ≥ μ_{1} ≥ 0.

Case I. Blowing up an axis. First assume that the center C is the y-axis. Since (by our assumption, cf. 4.1) the center C is an equi-multiple curve, we must have λ_{1} ≥ 1.

We distinguish three cases:

(A) μ_{1} = 0; (B) μ_{1} ≠ 0, λ_{1} > 1; (C) μ_{1} ≠ 0, λ_{1} = 1.

(A) Since μ_{1} = 0 we have λ_{1} > 1 by the normalization property 2.1(5). The blow up of C is constructed by gluing two charts U_{x} and U_{y} together. The strict transform St of f is contained in U_{x} (this follows from λ_{1} > 1). Then E ≈ D × P^{1}, where D is the disc given by the new y-axis (i.e. D ≈ C), and the strict transform St intersects E along D. Since λ_{1} > 1, m_{0}(f) = m. There is only one zero-dimensional strata (namely the origin of D); the strict transform of f will be denoted by f_{x}. Then:

χ(f) = mχ(E^{o}) + χ(x^{m}f_{x}).

Here E^{o} = E \ St ≈ D × C, hence χ(E^{o}) = 1. We write a similar formula for f|_{y=0}: set Ẽ := E ∩ {y = 0} ≈ P^{1} and Ẽ^{0} = Ẽ \ St ≈ C. Then:

χ(f|_{y=0}) = mχ(Ẽ^{0}) + χ(x^{m}f_{x}|_{y=0}).

Now, by the splitting property (3.3) (and from the fact that for any germ h one has χ(h^{m}) = mχ(h)):

χ(x^{m}f_{x}) = m + χ(f_{x}) − mχ(x|_{f_{x}=0}) − χ(f_{x}|_{x=0}).

There is a similar relation for χ(x^{m}f_{x}|_{y=0}). Subtracting them one has:

(4.4) D(f) = χ(f_{x}) − χ(f_{x}|_{y=0}) − m [ χ(x|_{f_{x}=0}) − χ(x|_{f_{x}=y=0}) ]

− [ χ(f_{x}|_{x=0}) − χ(f_{x}|_{x=y=0}) ].

We emphasize that in general at this step it can happen that f_{x}, expressed in the new variables (x, y, z) is not normalized. Moreover, we have to identify the distinguished axis for f_{x} as well. In order to do this, we will find the parametrization of f_{x}. From Table (4.2) we see that if ζ = cx^{λ_{1}} + … is the (normalized) parametrization of f then ζ_{x} = cx^{λ_{1}−1} + … is the parametrization of f_{x} in the new natural local coordinates. If λ_{1} − 1 > 1 then ζ_{x} is already normalized (and the y axis is distinguished), thus D(f_{x}) = χ(f_{x}) − χ(f_{x}|_{y=0}). On the other hand if λ_{1} − 1 < 1, we apply Example (4.3) to conclude again D(f_{x}) = χ(f_{x}) − χ(f_{x}|_{y=0}). Moreover, Lemma (3.6) works as well, because a = λ_{1} − 1 ≠ 0 and ζ_{x} = x^{−1}ζ guarantees that the first term of ζ_{x} divides all the others.

In particular, χ(x|_{f_{x}=0}) = χ(x|_{f_{x}=y=0}) (by 3.6 (2)), and χ(f_{x}|_{x=0}) = χ(f_{x}|_{x=y=0}) (by 3.6 (1)).

Hence (4.4) gives D(f) = D(f_{x}).

(B) If μ_{1} ≠ 0 but λ_{1} > 1, the above argument works with a small modification. In this case ζ_{x} = cx^{λ_{1}−1}y^{μ_{1}} + ⋯, which is already normalized up to a permutation of x and y. On the other hand, this is the “symmetric case” (cf. 2.8), hence either the x- or the y-axis is distinguished. Moreover, in (3.6) we can freely permute the x and y axes. In particular χ(x|_{f_{x}=0}) = χ(x|_{f_{x}=y=0}), and χ(f_{x}|_{y=0}) = χ(f_{x}|_{x=0}) = χ(f_{x}|_{x=y=0}). Hence (4.4) can be reduced to

D(f) = χ(f_{x}) − χ(f_{x}|_{y=0}) − χ(f_{x}|_{x=0}) + χ(f_{x}|_{x=y=0}).

where the last three Euler characteristics are equal. Since either x or y is distinguished, D(f) = χ(f_{x}) − χ(f_{x}|_{w=0}) = D(f_{x}) (where w is x or y).

(C) Assume that μ_{1} ≠ 0 and λ_{1} = 1. Then E ≈ D × P^{1} as above, but E ∩ St is a curve singularity containing the center O of D (and whose intersection with Ẽ = {O}×P^{1} is exactly O). By additivity χ(E \St) = χ(D^{∗} ×P^{1} \St)+χ(Ẽ \St). But χ(D^{∗} × P^{1} \ St) = 0 because the first projection D^{∗} × P^{1} \ St → D^{∗} is a topological fiber bundle over D^{∗}. Since Ẽ \ St ≈ C one obtains that χ(E^{o}) = χ(Ẽ^{o}) = 1. Moreover, the argument (A) works again as far as (4.4).

In this case ζ = cx^{1}y^{μ_{1}} + …, therefore ζ_{x} = cy^{μ_{1}} + . . .. After permuting x and y, we apply the same argument as in (A) (for the case λ_{1} − 1 < 1). Therefore D(f_{x}) = χ(f_{x}) − χ(f_{x}|_{x=0}).

Moreover, (permuting the x axis and the y axis in (3.6)) we obtain: χ(x|_{f_{x}}) = χ(x|_{f_{x}=y=0}) (3.6(3)), and χ(f_{x}|_{y=0}) = χ(f_{x}|_{x=y=0}) (3.6(1)). Therefore D(f) = D(f_{x}) again.

Finally, suppose we blow up the x axis instead of the y axis. Since the center must be equimultiple, μ_{1} ≥ 1. If μ_{1} > 1 then ζ_{y} = cx^{λ_{1}}y^{μ_{1}−1} +… is already normalized. This is also the symmetric case (cf. 2.8) so we have χ(f|_{y=0}) = χ(f|_{x=0}) and χ(f_{y}|_{y=0}) = χ(f_{y}|_{x=0}). Therefore D(f) = χ(f) − χ(f|_{x=0}). We repeat argument (A) as far as (4.4), treating x as the distinguished variable; no normalization is required and the last two expressions vanish by (3.6) exactly as before. Thus D(f) = D(f_{y}).

If μ_{1} = 1 then λ_{1} > 1 (since (λ_{1}, μ_{1}) ∉ Z × Z) and once again ζ_{y} = cx^{λ_{1}} + … is already normalized. Here we (as usual) take y as the distinguished axis for f_{y}. We repeat the argument of case (A) (with no normalization) but use the argument of (C) to show that χ(E^{o}) = χ(Ẽ^{o}) = 1.

Case II. Blowing up a point. Again there are a few cases to consider, depending on the “size” of (λ_{1}, μ_{1}), the first characteristic pair. If λ_{1} +μ_{1} ≥ 1 (the “transverse case” in Lipman’s language), then the strict transform lives in only two of the three special directions. If λ_{1} + μ_{1} < 1 then the strict transform lives in all three special directions. These possibilities are then multiplied by the condition that φ (cf. 2.7) and/or φ_{x}, φ_{y}, φ_{z} are identically zero or not (i.e. if μ_{1}(f) and/or μ_{1}(f_{x}), etc. vanish or not)). We summarize all the cases along with the forms of the parametrizations ζ_{∗} of f and the strict transforms f_{x}, f_{y}, f_{z} in the next table. To obtain the first characteristic pair of each ζ_{∗} we use the table from paragraph (4.2); each ζ_{∗} is expressed in the natural local coordinates. For simplicity we drop the subscripts and coefficients, e.g. (λ, μ) = (λ_{1}, μ_{1}).

(A) λ + μ > 1 ζ = x^{λ} + …  ζ_{x} = x^{λ−1} + …  ζ_{y} = x^{λ}y^{λ−1} + …
μ = 0
(B) λ + μ > 1 ζ = x^{λ}y^{μ} + … ζ_{x} = x^{λ+μ−1}y^{μ} + …  ζ_{y} = x^{λ}y^{λ+μ−1} + …
λμ ≠ 0
(C) λ + μ = 1 ζ = x^{λ}y^{μ} + … ζ_{x} = y^{μ} + …  ζ_{y} = x^{λ} + …
λμ ≠ 0
(D) λ + μ < 1 ζ = x^{λ}y^{μ} + … ζ_{x} = x^{1−λ−μ/μ}z^{1/μ} + … ζ_{y} = y^{1−λ−μ/λ}z^{1/λ} + …
λμ ≠ 0  ζ_{z} = x^{λ/1−λ−μ}y^{μ/1−λ−μ} + …

(TABLE 2)

Cases (A)-(B)-(C) (λ_{1} +μ_{1} ≥ 1). Obviously E = P^{2}. In cases (A)-(B), E ∩St = E ∩ {strict transform of {z = 0}}, i.e. E^{o} ≈ P^{2} \ C^{1} ≈ C^{2}. Therefore χ(E^{o}) = 1. In case (C), E^{o} = P^{2} \ {z^{m} = x^{λm}y^{μm}} (recall λ + μ = 1). We verify that in this case χ(E^{o}) = 1 as well. Indeed, letting h(x, y, z) be the homogeneous polynomial z^{m}−x^{λm}y^{μm} we have χ(P^{2}\{h = 0}) = χ(P^{2})−χ({h = 0}). Since h is a projective curve of degree m, its Euler characteristic is −m(m − 3) + ∑ μ_{i}, where μ_{i} are the Milnor numbers of h at each of its singular points. It is easy to check (using λμ ≠ 0) that ∑ μ_{i} = (m − 1)(m − 2). Thus χ(E^{o}) = 3 + m(m − 3) − (m − 1)(m − 2) = 1.

Since in all cases (A)-(C) λ_{1} + μ_{1} ≥ 1 we see that m_{0}(f) = m. Therefore

χ(f) = m + χ(x^{m}f_{x}) + χ(y^{m}f_{y}).

On the other hand, Ẽ^{o} = E^{o} ∩ {y = 0} ≈ C^{1} in all these cases, hence χ(f|_{y=0}) = m + χ(x^{m}f_{x}|_{y=0}). Using the splitting property for χ, one gets:

D(f) = m − mχ(x|_{f_{x}=0}) − m · χ(y|_{f_{y}=0}) + m · χ(x|_{f_{x}=y=0})
+χ(f_{x}) − χ(f_{x}|_{x=0}) + χ(f_{y}) − χ(f_{y}|_{y=0}) − χ(f_{x}|_{y=0}) + χ(f_{x}|_{x=y=0}).

We claim that D(f_{y}) = χ(f_{y}) − χ(f_{y}|_{y=0}) in all three cases. In case (A), f_{y} is already normalized, so this is true by definition. In case (B), f_{y} is only normalized up to a permutation of x and y, but χ(f_{y}|_{y=0}) = χ(f_{y}|_{x=0}) since this is the “symmetric case” (cf. 2.8). Finally, in case (C), example (4.3) gives D(f_{y}) = χ(f_{y}) − χ(f_{y}|_{y=0}) as well. Moreover, in all three cases χ(y|_{f_{y}=0}) = 1 by lemma 3.6 (3). Now for each case we collect several identities.

(A) Applying (4.3) and (3.6) directly gives the following:

D(f_{x}) = χ(f_{x}) − χ(f_{x}|_{y=0}), χ(f_{x}|_{x=0}) = χ(f_{x}|_{x=y=0}), χ(x|_{f_{x}=0}) = χ(x|_{f_{x}=y=0}).

(C) Similarly, applying (4.3) and (3.6) with the roles of x and y reversed, we have:

D(f_{x}) = χ(f_{x}) − χ(f_{x}|_{x=0}), χ(f_{x}|_{y=0}) = χ(f_{x}|_{x=y=0}), χ(x|_{f_{x}=0}) = χ(x|_{f_{x}=y=0}).

(B) Since ζ_{x} is normalized up to a permutation of x and y, we obtain the identities of (A) or (C), depending on which of x or y is distinguished for f_{x}.

Applying these to the expression for D(f), in all three cases D(f) = D(f_{x}) + D(f_{y}).

Case (D) (λ_{1} +μ_{1} < 1). Note that since f is assumed to be normalized, λ_{1}μ_{1} ≠ 0. As before, E = P^{2}, but E^{0} = P^{2}\{strict transform of xy = 0} ≈ C×C^{∗}. Therefore χ(E^{o}) = 0. The multiplicity m_{0}(f) of f now is \underline{not} m, but ν = (λ_{1} + μ_{1})m. Therefore:

χ(f) = χ(x^{ν}f_{x}) + χ(y^{ν}f_{y}) + χ(z^{ν}f_{z}).

Moreover, the exceptional curve of the induced resolution of f|_{y=0} is P^{1} with two special points situated on it. Hence:

χ(f|_{y=0}) = χ(x^{ν}f_{x}|_{y=0}) + χ(z^{ν}f_{z}|_{y=0}).

The contribution of f_{x} to D(f) is:

χ(f_{x}) − νχ(x|_{f_{x}=0}) − χ(f_{x}|_{x=0}) − χ(f_{x}|_{y=0}) + νχ(x|_{f_{x}=y=0}) + χ(f_{x}|_{x=y=0}).

By Table 2, the parametrization of f_{x} is ζ_{x} = x^{1−λ_{1}−μ_{1}/μ_{1}}z^{1/μ_{1}} +. . .. To normalize we need only make the permutations z → x, x → y, y → z and transformations of type 2.2 (a). Making the substitutions and applying 3.6 (3), (5) and example 4.3 (b), the total f_{x} contribution to D(f) is exactly D(f_{x}).

The contribution of f_{y} to D(f) is

χ(y^{ν}f_{y}) = ν + χ(f_{y}) − νχ(y|_{f_{y}=0}) − χ(f_{y}|_{y=0}).

Again by Table 2, we see that f_{y} is normalized by permuting x and z and making transformations 2.2 (a). In particular the set {y = 0} is invariant under this change. Then by 3.6 (3) and 4.3 (b) we have χ(y|_{f_{y}}) = 1 and D(f_{y}) = χ(f_{y}) − χ(f_{y}|_{y=0}), i.e. the contribution of f_{y} to D(f) is exactly D(f_{y}). Finally, the contribution of f_{z} to D(f) is

χ(f_{z}) − νχ(z|_{f_{z}=0}) − χ(f_{z}|_{z=0}) − χ(f_{z}|_{y=0}) + νχ(z|_{f_{z}=y=0}) + χ(f_{z}|_{z=y=0}).

By Table 2, f_{z} is normalized by transformations 2.2 (a) alone. Then D(f_{z}) = χ(f_{z}) − χ(f_{z}|_{y=0}) by 4.3 (b), and the remaining contributions cancel out by 3.6 (4) and (5). Combining these calculations, we have shown that D(f) = D(f_{x}) + D(f_{y}) + D(f_{z}).

This ends the proof of the main theorem.

Note added on August 1, 2001 Using different techniques (based on the thesis of P. González Pérez) the authors have since been able to extend the results of this paper to calculate the analogous formula for the zeta function. The new result holds in all dimensions and for reducible quasi-ordinary singularities as well.

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   E-mail address: cban@math.ohio-state.edu
   E-mail address: mcewan@math.ohio-state.edu
   E-mail address: nemethi@math.ohio-state.edu

Dept. of Math., Ohio State University, 231 West 18th Ave., Columbus, OH, USA