Contemporary Mathematics

Some conjectures about quasi-ordinary singularities

L. J. McEwan and A. Némethi

Abstract. We formulate some open problems and conjectures about quasiordinary singularities.

1. Introduction

In a series of papers [BMcN00, McN01, GMcN01] the authors tried to understand the properties of the Milnor fiber of a quasi-ordinary hypersurface singularity f : (ℂ^{d+1}, 0) → (ℂ, 0). During this period some questions, problems and conjectures were formulated, some of them in a very precise, some of them in a slightly vague form. This note reflects some of them. We hope that by publishing these pages, we invite and stimulate more people to work on this area, contributing to the renaissance of quasi-ordinary singularities.

The reader is invited to consult “The zeta-function of a quasi-ordinary singularity II” [GMcN01] (these same proceedings) for details of the notations and terminology. For the convenience of the reader, we have selected some of them, together with two theorems, in the appendix.

Above, the quasi-ordinary assumption means, that in some local coordinates, the projection pr : (F, 0) := ({f = 0}, 0) → (ℂ^{d}, 0) induced by (x, x_{d+1}) ↦ x (where x = (x_{1}, …, x_{d}) ∈ ℂ^{d}) is finite with reduced discriminant included in the union of hyperplanes ({x_{1} ⋯ x_{d} = 0}, 0). (Here (F, 0) will always be assumed to be reduced, but f not necessarily so.)

The quasi-ordinary singularities are generalizations of the plane curve singularities. There is a natural generalization of Puiseux pairs, called the distinguished tuples, associated to any quasi-ordinary singularity f and the projection pr (see [Zariski67, Lipman65, Lipman83, Lipman88, González01]).

There is a rather large literature of the properties of these distinguished tuples and of the topological, geometrical and analytical properties of the zero set (F, 0) (see e.g. Lipman’s papers). Moreover, even about the embedded topological type of (F, 0) ⊂ (ℂ^{d+1}, 0) there are deep results. E.g., in the irreducible case, one can prove that the distinguished tuples determine the embedded topological type

\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

2000 Mathematics Subject Classification. 14B05, 32S**.

Key words and phrases. Quasi-ordinary singularities, Milnor fiber, monodromy, zetafunction, resolution of singularities, polar curves, discriminants.

The second author is partially supported by the NSF Grant DMS-0088950.

of f (cf. [Zariski68] and [Lipman88], or [Oh93]). (In the general case this statement is still a conjecture.) Conversely, using Lipman’s results [Lipman88], Gau in [Gau88] proves that from the embedded topological type one can recover the normalized distinguished tuples of f.

On the other hand, “classical” results about the smoothing invariants are almost non-existent. One of the explanations for this might be that it is rather difficult to relate the properties of the central fiber (F, 0) (e.g. the fact that a quasi-ordinary projection pr : (F, 0) → (ℂ^{2}, 0) exists) to any kind of property of the nearby Milnor fibers F_{ϵ} = f^{−1}(ϵ) (ϵ ≠ 0). The aim of our papers [BMcN00, McN01, GMcN01] was to reduce this gap. But the following list clearly suggests that we are definitely at the very beginning of our investigation.

The theory of quasi-ordinary singularities has revived recently with a large intensity by the work of Teissier and his students. For the convenience of the reader we also list at the end of the article some recent publications about this subject (even if we do not refer to all of them in the body of the paper).

2. Some problems and conjectures

In the next discussion we will assume that the quasi-ordinary germ f : (ℂ^{d+1}, 0) → (ℂ, 0) is represented in a “normalized” coordinate system (cf. appendix). For any germ g, we denote by ζ(g), respectively by χ(g) := χ(F_{ϵ}(g)) = deg(ζ(g)), the zeta function associated with the monodromy action, respectively the Euler characteristic of the Milnor fiber of g (cf. appendix).

By Theorem A of [GMcN01] (cf. 3.1 in the appendix), one has ζ(f) = ζ(f|_{x_{d}=0}). This result (together with its proofs at different levels in [BMcN00, McN01, GMcN01]), and concrete tests for various families of examples, suggest that any quasi-ordinary singularity f has a series of properties which are even stronger than the above mentioned homological property. One can think about these properties in two different ways: in terms of an embedded resolution or in terms of polar curves. In both cases it is sometimes more convenient to concentrate on the germ x_{d} · f, or on the pair (x_{d}, f), instead of f (cf. also with Theorems B in [McN01, GMcN01], or 3.2 here). In this way, one can reformulate Theorem A in terms of the vanishing of some Euler characteristics or in terms of the vanishing of some specific polar curve components.

I. A property of the embedded resolution.

Fix an embedded resolution ϕ : X → B_{r} of the germ x_{d} · f, i.e. of the pair {x_{d} ·f = 0} ⊂ B_{r}, where B_{r} is a sufficiently small (Milnor) ball. Let {E_{i}}_{i∈I} be the set of irreducible exceptional divisors, and S the strict transform of {x_{d} · f = 0}. For each i ∈ I we set E_{i}^{o} = E_{i} \ (S ∪ ∪_{j≠i}E_{i}), and denote by m_{E_{i}}(f) the vanishing order of f ∘ ϕ along E_{i} (and we use similar notation for x_{d} or x_{d}^{m}f as well). Recall that by A’Campo’s theorem [A’Campo75], for any m ≥ 1 one has:

χ(x_{d}^{m} · f) = ∑_{i} m_{E_{i}}(x_{d}^{m} · f)χ(E_{i}^{o}) = ∑_{i} (m · m_{E_{i}}(x_{d}) + m_{E_{i}}(f))χ(E_{i}^{o}).

Then Theorems A and B suggest the following.

2.1. Conjecture A. For any embedded resolution ϕ there is a partition {I_{1}, …, I_{t}} of I so that for any fixed k ∈ {1, 2, …, t} one has:

(a) if i, j ∈ I_{k} then m_{E_{i}}(x_{d}) = m_{E_{j}}(x_{d}) and m_{E_{i}}(f) = m_{E_{j}}(f),

(b) ∑_{i∈I_{k}} χ(E_{i}^{o}) = 0.

In fact, even if it is not explicitly stated, the statement above is valid for the specific (iterated toric) resolutions used in [McN01, GMcN01]. Nevertheless, we would like to have a natural argument for this phenomenon which is valid for an arbitrary resolution ϕ.

Notice also that we cannot expect, for any resolution ϕ, the stronger property, namely that χ(E_{i}^{o}) = 0 for all i ∈ I. Indeed, assume that this stronger property is satisfied for ϕ; then modify ϕ by a blow up in a generic point of E; then for the new modification only the weaker property is satisfied.

II. A property of the polar variety associated with (x_{d}, f).

From the polar curve point of view, one needs to understand the fiber-structure of the pair (x_{d}, f) : (ℂ^{d+1}, 0) → (ℂ^{2}, 0). The reader is invited to consult [Lê73, Teissier73, LT83] (and the references listed in them) for the theory of polar varieties.

We wish to emphasize that in our situation x_{d} is not a generic hyperplane section. This makes this geometrical situation non-standard: one cannot just apply the general facts valid for generic hyperplane sections (or for their polar curves). But still, one can try to employ the method of polar curves for this special hyperplane section as well. The reason for this choice of the section is the following: in general, it is hard to connect the geometry of a generic section with the quasi-ordinary projection. On the other hand, x_{d} is part of this projection.

One more warning: the pair (x_{d}, f) is not an isolated complete intersection singularity (icis). In particular, the general mathematical machinery which describes the fiber structure of an icis (see e.g. [Looijenga84]), cannot be automatically used in this case. Nevertheless, the reader can consult the paper [LT83] of Lê and Teissier, which contains (among many other things) the theory of “descriptible” morphisms which have a more complex fiber structure.

Even if the pair (x_{d}, f) is not an icis, still, the first obstruction to understanding the fiber-structure of (x_{d}, f) is the critical locus

Σ := \{\frac{∂f}{∂x_{i}} = 0 for all i ≠ d\},

and its image D := (x_{d}, f)(Σ).

Notice that in the case of an icis, the nontrivial polar curve components (of a similarly defined discriminant space D) constitute the obstructions for the validity of statements of type Theorems A and B. This motivates the following.

2.2. Conjecture B. Σ ⊂ {x_{d} · f = 0}, i.e. D ⊂ {union of coordinate axes}.

We also conjecture the following geometric analog. Let (u, v) be the coordinates in (ℂ^{2}, 0), and let Δ_{η}^{∗} be the punctured disc of radius η.

2.3. Conjecture C. a) There exist 0 < r ≪ 1 and 0 < η ≪ r such that the restriction

(x_{d}, f) : B_{r} ∩ (x_{d}, f)^{−1}(Δ_{η}^{∗} × Δ_{η}^{∗}) → Δ_{η}^{∗} × Δ_{η}^{∗}

is a locally trivial topological fiber bundle.

(b) In particular, by taking the restriction of this fiber bundle over {u^{m}v = ϵ} ⊂ Δ_{η}^{∗} × Δ_{η}^{∗} (0 < ϵ ≪ η), we obtain that the Milnor fiber of x_{d}^{m}f (m ≥ 1) is the total space M of a topological fiber bundle q : M → ℂ^{∗}, where ℂ^{∗} is identified with the Milnor fiber {u^{m}v = ϵ} of the germ u^{m}v : (ℂ^{2}, 0) → (ℂ, 0), and the fiber of q is the generic fiber of the pair (x_{d}, f).

Since (x_{d}, f) is not an isolated complete intersection, even if 2.2 is true (i.e. the polar curve has no components outside of the coordinate lines), in order to understand the fiber structure, one needs additionally to analyze the behaviour of (x_{d}, f) near the boundary of the Milnor ball. We conjecture that this behaviour will not have any influence on the Euler-characteristic computations. In fact, we formulate the following more precise conjecture.

2.4. Conjecture D. The Milnor fiber F_{ϵ}(f) of f can be obtained (at least homotopically) by gluing to the Milnor fiber F_{ϵ}(f|_{x_{2}=···=x_{d}=0}) of the plane curve singularity f|_{x_{2}=···=x_{d}=0} some spaces which fiber over S^{1}.

E.g., if (F, 0) ⊂ (ℂ^{3}, 0) is given by {x_{3}^{n} − x_{1}x_{2} = 0}, then the homotopy type of F_{ϵ}(f) is ∨_{n−1}S^{2}, and F_{ϵ}(f|_{x_{2}=0}) consists of n disjoint copies of the real 2-disc D^{2}. Obviously, if we glue these discs along their boundaries we get (homotopically) ∨_{n−1}S^{2}.

Note that the homotopy decomposition of the Milnor fiber predicted by Conjecture D does not correspond to the usual “bouquet decompositions” of the Milnor fiber valid in some different situations. For such homotopy decompositions and related questions, the reader is invited to consult some of the articles of Siersma and Tibăr (see e.g. [Tibăr96]).

III. A property of the polar variety associated with (x_{1}, …, x_{d}, f).

More generally, instead of the map (x_{d}, f), one can consider the map ψ : (ℂ^{d+1}, 0) → (ℂ^{d+1}, 0) given by (x, x_{d+1}) ↦ (x, f). Denote by (u_{1}, ⋯, u_{d}, u_{d+1}) the coordinates of the target space. Obviously, ψ is finite (in particular, an icis) with critical locus Σ = {∂f/∂x_{d+1} = 0} and discriminant D = ψ(Σ). It is clear that D is given by the equation Res(f − u_{d+1}, ∂f/∂x_{d+1}) = 0, where Res denotes the resultant with respect to the variable x_{d+1} (and we identify u_{i} = x_{i} for i ≤ d).

2.5. Example. Assume that d = 2 and f = x_{3}^{3}+(x_{1}+x_{2})x_{1}x_{2}x_{3}+x_{1}x_{2}. Then ∂f/∂x_{3} = 3x_{3}^{2} +(x_{1} +x_{2})x_{1}x_{2}, hence D = {(u_{3} −u_{1}u_{2})^{2} +4u_{1}^{3}u_{2}^{3}(u_{1} +u_{2})^{3}/27 = 0}.

This example shows that, in general, we cannot expect that {∂f/∂x_{d+1} = 0} is quasi-ordinary (with respect to the projection (x, x_{d+1}) ↦ x), nor can we expect it of D (with respect to the projection (u, u_{d+1}) ↦ u). Nevertheless, we believe that they are “not far” from being quasi-ordinary.

E.g., assume that f is an irreducible quasi-ordinary singularity with distinguished tuples {λ_{i}}_{i=1}^{g} (which are totally ordered: 0 < λ_{1} < ⋯ < λ_{g}). We write f_{top} for the germ parametrized by

ζ = ∑_{i=1}^{g} x^{λ_{i}}.

Then the embedded topological types of f and f_{top} are the same (cf. introduction), hence in any topological discussion f can freely be replaced by f_{top}. The following conjecture was verified for many particular families of examples by the authors.

2.6. Conjecture E. The discriminant D associated with f_{top} is quasi-ordinary (with respect to the projection (u, u_{d+1}) ↦ u). Moreover, we expect that D and its irreducible components satisfy properties similar to those of the polar curve associated with an irreducible plane curve singularity established in [Merle77] (cf. also with the discussion in Popescu-Pampu’s Thesis [PopescuP01]).

We explain in a few words what is the advantage of the fact that D is quasi-ordinary. Notice that the restriction of ψ to the Milnor fiber F_{ϵ} = {f = ϵ} of f is a finite map ψ : F_{ϵ} → {u_{d+1} = ϵ}, with target ≈ ℂ^{d} and branch locus D ∩ {u_{d+1} = ϵ}. Now, if D is quasi-ordinary with projection (u, u_{d+1}) ↦ u, then this branch locus has nice properties, and the above construction provides immediately results like those of Theorem A and B.

On the other hand, even for “bad” germs of type 2.5 (i.e. for germs f for which D is not quasi-ordinary), the statements of Theorems A and B are true, and a possible proof of them can be carried out using similar arguments about the discriminant spaces D. This leads us to expect that, for any quasi-ordinary singularity, the space D is not very far from being a quasi-ordinary germ. More precisely:

2.7. Conjecture F. There exists a class of (some kind of) “weakly quasiordinary” germs as a slight extension of the class of quasi-ordinary ones (characterized by a sheaf-theoretical property, which is omitted here), which has the property that for any quasi-ordinary germ f its discriminant space is “weakly quasiordinary”. Even more, we expect a stability property of this class, in the sense that for any “weakly quasi-ordinary” germ its discriminant is “weakly quasi-ordinary”.

IV. Intrinsic characterization of quasi-ordinary germs.

When we discuss and list different properties of quasi-ordinary singularities (hypersurfaces or not), we can also ask the following, rather natural question: do these properties characterize the quasi-ordinary germs? Here some comments are necessary. Recall that the quasi-ordinary germs are defined via the existence of a quasi-ordinary projection (cf. introduction). But, in general, for an arbitrary germ, it is extremely difficult (if not impossible) to verify the existence of such a projection (unless in the description of the germ a natural projection is built in). This motivates the necessity of a better characterization of these germs. Here we have in mind, e.g., the case of normal surface quasi-ordinary germs (here normality is in the sense of [Laufer71]). They are exactly the Hirzebruch-Jung surface singularities. They can be characterized not only by the existence of a quasi-ordinary projection, but also by the (topological) property that their minimal resolution graph is a string of rational curves.

2.8. Problem G.

• Find an intrinsic characterization of the quasi-ordinary germs which is independent of their embedding in some (ℂ^{N}, 0). First try the case of hypersurfaces or surfaces.

• Find a topological characterization of quasi-ordinary germs. (In the hypersurface case, one can use their embedded topological type as well).

In problem G, our primary interest is the understanding of hypersurface surface singularities (F, 0) ⊂ (ℂ^{3}, 0). In fact, for such germs, we found an intrinsic topological characterization [McN02].

V. The zeta-function of a quasi-ordinary singularity.

In the case of irreducible plane curve singularities (i.e. the case d = 1), ζ(f) is a complete topological invariant. Moreover, ζ(f) contains the same amount of information as the semigroup associated with the germ, cf. [GZDC99]. If d ≥ 2 the semigroup is still a complete topological invariant (for details see [González00] or [PopescuP01]). On the other hand, ζ(f) forgets almost all the information about the distinguished tuples.

2.9. Question H. What is the right analog for d ≥ 2 of the Gusein-Zade - Delgado - Campillo result [GZDC99] ?

This question was also asked by González Pérez (private communication).

3. Appendix. Quasi-ordinary singularities.

Here we recall the definition of the distinguished tuples associated with a hypersurface quasi-ordinary singularity (F, 0) ⊂ (ℂ^{d+1}, 0) (see [Lipman65, Lipman83, Lipman88] for the irreducible case, and [González01] for the non-irreducible case); the definition of a “normalized” coordinate system; and the main results of [McN01, GMcN01] about the zeta function associated with the Milnor fibration. For more details, the interested reader can consult [McN01, GMcN01], or the references provided at the corresponding places.

We write (F, 0) = ({f = 0}, 0) for some (not necessarily reduced) quasi-ordinary analytic germ f : (ℂ^{d+1}, 0) → (ℂ, 0) (see introduction). Let {f^{(i)}}_{i∈I} be the set of irreducible components of f, and set (F^{(i)}, 0) := ({f^{(i)} = 0}, 0) for any i ∈ I. Then each (F^{(i)}, 0) is an irreducible quasi-ordinary singularity. By Jung– Abhyankar Theorem, for each i ∈ I, there exists a parametrization of (F^{(i)}, 0) by a fractional power series ζ^{(i)} = H^{(i)}(x_{1}^{1/m}, …, x_{d}^{1/m}), where H^{(i)}(s_{1}, …, s_{d}) is a power series and m is a suitable natural number independent of the index i ∈ I (see [Abhyankar55, Jung08]). This means that there exists a finite map (ℂ^{d}, 0) → (F^{(i)}, 0) given by x_{d+1} = H^{(i)}(s_{1}, …, s_{d}), x_{t} = s_{t}^{m} for t = 1, …, d.

The conjugates of ζ^{(i)} are obtained by multiplying any of x_{t}^{1/m} (t = 1, …, d) by m^{th}-roots of unity; the number of different conjugates {ζ_{k}^{(i)}}_{k} of ζ^{(i)} is precisely the degree n^{(i)} of the covering pr|_{F^{(i)}} : (F^{(i)}, 0) → (ℂ^{d}, 0), and

f^{(i)}(x, x_{d+1}) = ∏_{k=1}^{n^{(i)}}(z − ζ_{k}^{(i)}).

Obviously, the equation of the discriminant of the projection pr is a product of this type:

∏(ζ_{k}^{(i)} − ζ_{l}^{(j)}).

There is an important finite subset of the set of exponents λ = (λ_{1}, …, λ_{d}) ∈ ℚ_{≥0}^{d} of the terms x^{λ} := x_{1}^{λ_{1}} ⋯ x_{d}^{λ_{d}} appearing in {ζ^{(i)}}_{i} with non-zero coefficient, called the distinguished tuples (in some articles they are called characteristic exponents). By unique factorization of the discriminant one has:

ζ_{k}^{(i)} − ζ_{l}^{(j)} = x^{λ_{kl}^{ij}} · ϵ_{kl}^{ij}(x_{1}^{1/m}, …, x_{d}^{1/m}) with ϵ_{kl}^{ij}(0) ≠ 0,

and λ_{kl,t}^{ij} ∈ \frac{1}{m}ℤ_{≥0}. The set Λ := \{λ_{kl}^{ij}\}_{ij,kl} = \{(λ_{kl,1}^{ij}, …, λ_{kl,d}^{ij})\}_{ij,kl} ⊂ \frac{1}{m}ℤ_{≥0}^{d} constitute the set of distinguished tuples. For ordering properties of the distinguished tuples, see [Lipman65, Lipman83, Lipman88, González00, González01, McN01, GMcN01]; in particular there is a minimal element min Λ.

Notice that Λ = ∅ if and only if f is irreducible and smooth. If Λ ≠ ∅, we reorder the variables (x_{1}, …, x_{d}) in such a way that the first entry of min Λ is non-zero. This coordinate system is called “normalized.”

For any index set K ⊂ {1, …, d} we define f|_{x_{K}=0} by the restriction of f to the set ∩_{k∈K}{x_{k} = 0}. The next result is proved in [GMcN01]. (For the irreducible case, also see [McN01]. The result was conjectured in [BMcN00], where the case d = 2 and χ(F_{ϵ}(f)) is treated.)

3.1. Theorem A. Fix a normalized coordinate system. Then for any index set K ⊂ {2, …, d} one has ζ(f) = ζ(f|_{x_{K}=0}). In particular, for K = {2, …, d}, one gets that ζ(f) = ζ(g), where g is the plane curve singularity (x_{1}, x_{d+1}) ↦ f(x_{1}, 0, …, 0, x_{d+1}). Taking K = {d} as another particular case, one gets that ζ(f) = ζ(f|_{x_{d}=0}). In fact, if min Λ has at least two non-zero entries, then ζ(f)(t) = 1 − t^{deg f}.

Here the zeta function of a hypersurface germ f : (ℂ^{d+1}, 0) → (ℂ, 0) is defined as follows. Fix a sufficiently small closed ball B_{r} in ℂ^{d+1} of radius r, and consider the Milnor fiber F_{ϵ}(f) := f^{−1}(ϵ)∩B_{r} (0 < ϵ ≪ r). By [Milnor68], f^{−1}(S_{ϵ}^{1})∩B_{r} → S_{ϵ}^{1} is a fibration with fiber F_{ϵ}(f), where S_{ϵ}^{1} := {w ∈ ℂ : |w| = ϵ}. Let m_{q}(f) ∈ AutH_{q}(F_{ϵ}(f), ℝ) (q ≥ 0) be the algebraic monodromy operators of this fibration. Then the zeta function of f is defined by the following rational function:

ζ(f)(t) := ∏_{q≥0} det(I − tm_{q}(f))^{(−1)^{q}}.

Obviously, the topological Euler characteristic of the Milnor fiber F_{ϵ}(f) of f can be recovered from ζ(f) by χ(F_{ϵ}(f)) = deg_{t}(ζ(f)(t)). In particular, by the above theorem one has χ(F_{ϵ}(f)) = χ(F_{ϵ}(f|_{x_{d}=0})).

Theorem 3.1 is the consequence of another theorem (cf. (3.7) in [BMcN00], and Theorem B in [McN01, GMcN01]):

3.2. Theorem B. If β = (β_{1}, …, β_{d}) ∈ ℤ_{>0}^{d} then ζ(x^{β}f)(t) = 1. If f is irreducible the same is true for any β ∈ ℤ_{≥0}^{d} provided that min Λ + β has at least two non-zero entries.

References

[A’Campo75] A’Campo, N.: La fonction zeta d’une monodromie, Comment. Math. Helvetici, 50

(1975), 233-248. [Abhyankar55] Abhyankar, S.S.: On the ramification of algebraic functions, Amer. J. Math., 77

(1955), 575-592.

[Ban93] Ban, C.: A Whitney stratification and equisingular family of quasi-ordinary singularities, Proceedings of the American Mathematical Society, 117 (2) (1993), 305-311.

[Ban94] Ban, C.: Auréole of a quasi-ordinary singularity, Proceedings of the American Mathematical Society, 120 (2) (1994), 393-404.

[BMc00] Ban, C. and McEwan, L. J.: Canonical resolution of a quasi-ordinary surface singularity, Canadian Journal of Mathematics, Vol. 52(6) (2000), 1149-1163.

[BMcN00] Ban, C, McEwan, L. J. and Némethi, A.: On the Milnor fiber of a quasi-ordinary surface singularity, Canadian Journal of Mathematics, 54 (1) (2000), 55-70.

[Costa87] Costa, A.F.: On the link of quasi-ordinary singularities, Géométrie algébrique et applications II. Singularités et géométrie complexe, J.M. Aroca, T. Sanchez Giralda, J.-L. Vicente eds., Travaux en cours 23, Hermann (1987).

[Gau88] Gau, Y.-N.: Embedded topological classification of quasi-ordinary singularities, Memoirs of the AMS, 388 (1988).

[González00] González Pérez, P. D.: Quasi-ordinary singularities via toric geometry, Thesis, Uni-

versity of La Laguna, 2000. [González01] González Pérez, P. D.: Toric embedded resolutions of quasi-ordinary singularities,

preprint 2001 June.

[GMcN01] González Pérez, P. D., McEwan, L. J. and Némethi, A.: The zeta-function of a quasiordinary singularity II; preprint June 2001; this proceedings.

[GZDC99] Gusein-Zade, S. M., Delgado, F, and Campillo, A.: On the monodromy of a Plane Curve Singularity and the Poincaré Series of the Ring of Functions on the Curve, Funct. Anal. and its Appl., 33 (1) (1999), 56-57.

[Jung08] Jung, H.W.E.: Darstellung der Funktionen eines algebraischen Körpers zweier unabhaängigen Veränderlichen x, y in der Umgebung einer stelle x = a, y = b, J. Reine Angew. Math., 133 (1908), 289-314.

[Laufer71] Laufer, H. B.: Normal two-dimensional singularities, Annals of Math. Studies 71, Princeton university Press 1971.

[Lê73] Lê D. T.: Calcul de nombre de cycles évanouissants d’une hypersurface complexe, Ann.

Inst. Fourier (Grenoble) 23 (1973), 261-270.

[LT83] Lê D. T. and Tessier, B.: Cycles evanescents, sections planes et conditions de Whitney. II, Proc. of Symp. in Pure Math., 40, Part 2 (1983), 65-103.

[Lipman65] Lipman, J.: Quasi-ordinary singularities of embedded surfaces, Thesis, Harvard University (1965).

[Lipman83] Lipman, J.: Quasi-ordinary singularities of surfaces in C^{3}, Proc. of Symp. in Pure Math., 40, Part 2 (1983), 161-172.

[Lipman88] Lipman, J.: Topological invariants of quasi-ordinary singularities, Memoirs of the AMS, 388, 1988.

[Looijenga84] Looijenga, E.J.N.: Isolated singular points on complete intersections, LMS Lecture

Note Series 77, Cambridge University Press, Cambridge 1984.

[Luengo83] Luengo Velasco, I.: On the Structure of Embedded Algebroid Surfaces, Proceedings of Symposia in Pure Mathematics, vol. 40 Part 2, (1983), 185-192.

[McN01] McEwan, L. J. and Némethi, A.: The zeta-function of a quasi-ordinary singularity, Ohio State Math. Research Inst. Preprints, 01-25, May 2001; to appear in Compositio Math.

[McN02] McEwan, L. J. and Némethi, A.: A topological intrinsic characterization of hypersurface surface quasi-ordinary singularities, manuscript in preparation.

[Merle77] Merle, M.: Invariants polaires des courbes planes, Invent. math., 41 (1977), 103-111.

[Milnor68] Milnor, J.: Singular points of complex hypersurfaces, Annals of Math. Studies, 61, Princeton Univ. Press, 1968.

[Oh93] Oh, K.: Topological types of quasi-ordinary singularities, Proc. Amer. Math. Soc., 117, Number 1 (1993), 53-59.

[Oh93] Oh, K.: The homology groups of the links of quasi-ordinary singularities, Math.Z., 213 (1993), 595-616.

[Oh95] Oh, K.: The branching orders of a quasi-ordinary projection, Math. Z., 220 (1995), 307-315.

[PopescuP01] Popescu-Pampu, P.: Arbres de contact des singularités quasi-ordinaires et graphes

d’adjacence pour les 3-variétés réelles, Thesis, University of Paris 7, 2001. [Popescu02] Popescu-Pampu, P.: On the invariance of the semigroup of a quasi-ordinary surface

singularity, C. R. Acad. Sci. Paris, Ser. I 334 (2002), 1101-1106.

[Teissier73] Teissier, B.: Cycles evanescents, sections planes et conditions de Whitney, Astérisque

7-8 (1973), 285-362. [Tibăr96] Tibăr, M.: Bouquet decomposition of the Milnor fiber, Topology, 35 (1) (1996), 227-241. [Zariski67] Zariski, O.: Exceptional Singularities of an Algebroid Surface and their Reduction,

Atti. Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur., 8 43 (1967), 135-146; reprinted in Collected papers, vol. 1 (1979).

[Zariski68] Zariski, O.: Studies in equisingularity, III. Saturation of local rings and equisingularity, Amer. J. Math., 90 (1968), 961-1023; reprinted in Collected papers, vol. 4, 96-158.

Department of Math., The Ohio State University, 231 W 18th Avenue, Columbus, OH 43210, USA

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