Contemporary Mathematics

The zeta-function of a quasi-ordinary singularity II

P. D. González Pérez, L. J. McEwan, and A. Némethi

Abstract. We prove that the zeta-function of a hypersurface quasi-ordinary singularity f equals the zeta-function of a plane curve singularity g. If we reorder the coordinates, then g has the form g = f(x_{1}, 0, …, 0, x_{d+1}). Moreover, the topological type of g can also be recovered from the set of distinguished tuples of f.

1. Introduction

The main goal of this note is the computation of the zeta-function of a hypersurface quasi-ordinary singularity f : (C^{d+1}, 0) → (C, 0). The paper generalizes [McN01], where the result is proved for an irreducible germ f. The zeta-function formula was conjectured in [BMcN00].

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 (x = (x_{1}, …, x_{d}) ∈ C^{d}), is finite and its (reduced) discriminant is included in ({x_{1} ⋯ x_{d} = 0}, 0).

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^{−1}(ϵ) ∩ B_{r} (0 < ϵ ≪ r). By [Milnor68], f^{−1}({|w| = ϵ}) ∩ B_{r} → {|w| = ϵ} is a fibration with fiber F_{ϵ}. Let m_{q} ∈ AutH_{q}(F_{ϵ}, 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})^{(−1)^{q}}.

The most efficient way to determine ζ(f) is by A’Campo’s formula [A’Campo75] via the embedded resolution of f. Hence, in the case of those families of singularities whose embedded resolution is well understood, one gets ζ(f). This is the case for plane curve singularities (see e.g. [EN85]) and isolated singularities with non-degenerate Newton boundary ([Varchenko76], see also [MO70]). For nonisolated singularities the methods of series of singularities provide partial results

\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, Newton polyhedron, toric resolution.

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

(see e.g. [Siersma90, Schrauwen90, Némethi93]) provided that the singular locus is one-dimensional. But, in general, for non-isolated singularities there is no nice, explicit formula of ζ(f).

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

Theorem A. Assume that f : (C^{d+1}, 0) → (C, 0) (d ≥ 2) is a quasi-ordinary singularity (which does not need to be reduced). Then there is a (precisely described) reordering of the coordinates (x_{1}, …, x_{d}) such that ζ(f) = ζ(f|_{x_{2}=···=x_{d}=0}).

This theorem can also be reinterpreted as follows. The quasi-ordinary singularities are generalizations of the plane curve singularities. There is a generalization of Puiseux pairs in general coordinates, called the distinguished tuples, associated to the quasi-ordinary singularity f and the projection pr (see [Zariski67, Lipman65, Lipman83, Lipman88]). The distinguished tuples can be organized in a tree generalizing the Egger-Wall weighted tree encoding the topological type of a plane curve singularity (see [Eggers83, GarcíaB00, PopescuP01]).

In the irreducible case, using Zariski’s result on saturation of local rings, one can prove that the distinguished tuples determine the embedded topological type of f (cf. [Zariski68] and [Lipman88], or [Oh93]). In the general case this fact is still a conjecture. On the other hand, by [BMcN00] (cf. also with [Lê73]), the zetafunction of any reduced hypersurface singularity depends only on the embedded topological type of the singularity. This shows that in the irreducible case ζ(f) depends only on the set of distinguished tuples (fact explicitly verified in [McN01]). In the general case, a similar result would strongly support the above conjecture.

Theorem A provides exactly this result; for its revised version, see §4.

Theorem A is the consequence of Theorem B:

Theorem B. Let f be as in Theorem A. Then ζ(x^{β}f)(t) = 1 for any β = (β_{1}, …, β_{d}) ∈ Z_{>0}^{d}, where x^{β} := x_{1}^{β_{1}} ⋯ x_{d}^{β_{d}}.

The proof of Theorem B occupies all of sections 2, 3, and the first part of 4. The definition of the distinguished tuples, and some of their properties is given

in Section 2. As a general reference, see [Lipman65, Lipman83, Lipman88] for the irreducible case, and [González01, González00a] in the general case.

In section 3 we recall the construction of a toric modification π(Σ) associated with the distinguished tuple of f from [González01], nevertheless our presentation is different from [loc. cit.] (where even a more general case is treated). Here, we insert the needed new results of [González01] in the package used in [McN01] (which was based on the thesis of the first author [González00a]). For general facts about toric geometry, we recommend e.g. the books [KKMS73, Oda88]. For the properties of the toric modification π(Σ) see also [Varchenko76] or [GT00].

In the first part of section 4 we will prove Theorem B. The proof uses a result of [GLM97], which provides a formula for the zeta-function in the presence of a partial embedded resolution. We will apply this result in the case of π(Σ). Then we prove Theorem A, as a consequence of Theorem B and the “weak splitting property” of the zeta-function proved in [McN01].

Some general properties of the Milnor fibration and smoothing invariants, in particular, of the zeta-function, can be found in the books [Milnor68] or [EN85].

Finally (and similarly as in [McN01]) we notice that in the literature the notation for the parametrization of an irreducible quasi-ordinary singularity is ζ, and for the zeta-function of a hypersurface singularity is ζ(f). Even if these notations are almost identical, we do not modify them: the meaning of the corresponding notation will also be clear from the context.

2. Quasi-ordinary singularities and their distinguished tuples

In this section 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, and [González01] for the non-irreducible case).

2.1. We will use the notation x̄ = (x, x_{d+1}) = (x_{1}, …, x_{d}, x_{d+1}) ∈ C^{d} × C =

C^{d+1} for coordinates in C^{d+1} or for local coordinates in (C^{d+1}, 0). Let C{x} denote the ring of convergent power series in x.

We assume that (F, 0) = ({f = 0}, 0) for some (not necessarily reduced) analytic germ f : (C^{d+1}, 0) → (C, 0) (i.e. F will always be considered reduced, but f not). Then, by the very definition of the hypersurface quasi-ordinary singularities, (F, 0) is quasi-ordinary, if there exist local coordinates (x, x_{d+1}) such that f can be expressed as a pseudo-polynomial f(x, x_{d+1}) = x_{d+1}^{n} + g_{1}(x)x_{d+1}^{n−1} + ⋯ + g_{n}(x) with g_{i} ∈ C{x}, and the reduced discriminant of pr : (F, 0) → (C^{d}, 0) induced by (x, x_{d+1}) ↦ x is contained in ({x_{1} ⋯ x_{d} = 0}, 0). 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 (see [Abhyankar55] Th. 3 and [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 number m can be chosen independently of the index i ∈ I, but, obviously, it depends on f. (E.g. m = n! is suitable; for an explicit construction of ζ^{(i)} one can consult [González00b] and [ALR89]).

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)}).

We recall that f is not necessarily reduced; write f_{red} = ∏_{i} f^{(i)} for its reduced germ.

Obviously, the equation of the discriminant of the projection pr is a product of 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). In the irreducible case they play a role similar to Puiseux pairs for irreducible plane curve singularities. In the case of non-irreducible germs, besides the tuples of the irreducible components there are some additional tuples as well: they measure the order of coincidence of the parametrizations of different components.

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.

Fix an irreducible component (F^{(i)}, 0) for some i ∈ I, and let ζ^{(i)} be its parametrization. Then (by similar argument as above) the distinguished tuples of (F^{(i)}, 0) are Λ^{(i)} := {λ_{kl}^{ii}}_{kl}. Set g^{(i)} := #Λ^{(i)}.

2.2. In the sequel we will use the following partial ordering. For any λ, μ ∈ Q^{d} we say that λ ≤ μ if λ_{t} ≤ μ_{t} for all t = 1, …, d. If λ ≤ μ, but λ ≠ μ, then we write λ < μ.

The point is that for any fixed irreducible component (F^{(i)}, 0), the distinguished tuples Λ^{(i)} are totally ordered (cf. [Lipman65, Zariski67, Lipman83, Lipman88]). For simplicity, we denote the tuples of Λ^{(i)} by {λ_{k}^{(i)}}_{k=1}^{g^{(i)}}. Hence:

(1) 0 < λ_{1}^{(i)} < λ_{2}^{(i)} < ⋯ < λ_{g^{(i)}}^{(i)}.

Λ^{(i)} is empty if and only if F^{(i)} is smooth.

Moreover, let α_{λ}^{(i)} be the coefficient of the term x^{λ} in ζ^{(i)} having exponent λ. Then the distinguished tuples Λ^{(i)} generate all the exponents appearing in ζ^{(i)} in the following sense:

(2) If α_{λ}^{(i)} ≠ 0 then λ ∈ Z^{d} + ∑_{λ_{k}^{(i)}≤λ} Z(λ_{k}^{(i)}).

In general, the set Λ is not totally ordered. This is a crucial difference between the irreducible and non-irreducible cases, therefore we explain it briefly.

Fix an index i ∈ I as above. Assume that ζ_{k_{1}}^{(i)} is obtained from ζ_{1}^{(i)} by multiplying any x_{t}^{1/m} by an m^{th}-root of unity. Assume that the same Galois action transforms ζ_{l_{1}}^{(i)} into ζ_{l′_{1}}^{(i)}. Then λ_{k_{1}l_{1}}^{ii} = λ_{1l′_{1}}^{ii}. Similarly, for another pair (k_{2}, l_{2}), one

has λ_{k_{2}l_{2}}^{ii} = λ_{1l′_{2}}^{ii} for some l′_{2}. Since ζ_{l′_{1}}^{(i)} − ζ_{l′_{2}}^{(i)} = (ζ_{l′_{1}}^{(i)} − ζ_{1}^{(i)}) − (ζ_{l′_{2}}^{(i)} − ζ_{1}^{(i)}) divides the discriminant of f, one must have λ_{1l′_{1}}^{ii} ≤ λ_{1l′_{2}}^{ii} or λ_{1l′_{2}}^{ii} < λ_{1l′_{1}}^{ii}. We call this line of argument “the Galois trick”.

On the other hand, in the case of Λ, the same argument does not work, since it is impossible to compare λ_{kl}^{ij} and λ_{k′l′}^{i′j′} if {i, j} ∩ {i′, j′} = ∅. If this intersection is not empty, then a similar argument as above gives (cf. [Zariski67]):

(3) Fix an index i ∈ I. Then {λ_{kl}^{ij}}_{j,kl} is totally ordered.

2.3. Definition. Fix two indices i, j ∈ I, i ≠ j, and set Λ^{(i,j)} := {λ_{kl}^{ij}}_{kl}. Since Λ^{(i,j)} ⊂ {λ_{kl}^{ij}}_{j,kl}, by (3) one obtains that Λ^{(i,j)} is totally ordered. Let λ^{(i,j)} := max Λ^{(i,j)} be its maximal element. λ^{(i,j)} is called the order of coincidence of the components f^{(i)} and f^{(j)}. Obviously, Λ^{(i,j)} = Λ^{(j,i)}, hence λ^{(i,j)} = λ^{(j,i)} as well.

In fact, λ^{(i,j)} carries all the information about the mutual positions of (F^{(i)}, 0) and (F^{(j)}, 0) (cf. the next 2.4 (a)). Moreover, one has the following “valuation type properties”. (For the proof use (3) and the “Galois trick”, cf. also with [González01], Lemma 16.)

2.4. Lemma. Assume that i ≠ a ≠ j ≠ i.

(a) Λ^{(i,j)} = {λ ∈ Λ^{(i)} ∩ Λ^{(j)}; λ < λ^{(i,j)}} ∪ {λ^{(i,j)}}.

(b) min{λ^{(i,a)}, λ^{(i,j)}} ≤ λ^{(j,a)}.

(c) If λ^{(i,a)} < λ^{(i,j)} then λ^{(i,a)} = λ^{(j,a)}.

2.5. Definition. Fix an index i ∈ I, and consider the subset A(i) := ({λ^{(i,j)}}_{j}∩ Z^{d}) ∪ {λ_{1}^{(i)}} of {λ_{kl}^{ij}}_{j,kl}. In special cases this subset can be empty; define I′ := {i|A(i) ≠ ∅}. If i ∈ I′ then A(i) is totally ordered by (3). Following [González01], we define λ_{κ(i)} := min A(i) for any i ∈ I′. For i ∈ I \ I′ we write λ_{κ(i)} = +∞. (By convention: λ < +∞ for any λ ∈ Q^{d}.)

Notice that using (1), property (2) (cf. 2.2 ) reads as:

(2*) Assume that λ ∉ Z^{d} satisfies λ_{κ(i)} ≰ λ for some i ∈ I. Then α_{λ}^{(i)} = 0.

The above facts imply the next result.

2.6. Proposition. ([González01], Lemma 17)

(a) #(I \ I′) ≤ 1. If #(I \ I′) = 1 then the number of smooth irreducible component of F is exactly one. In particular, Λ = ∅ if and only if F is irreducible and smooth.

(b) Assume that λ_{κ(i)} ∈ Z^{d} for some i ∈ I′. Then λ_{κ(j)} ≤ λ_{κ(i)} for any j ∈ I′. In particular, the set {λ_{κ(i)}}_{i∈I′} contains at most one element from Z^{d} (which is its greatest element).

(c) The set {λ_{κ(i)}}_{i∈I′} is totally ordered.

2.7. This proposition together with the next technical result will allow us to modify the coefficients of f by a change of variables in such a way that in the new situation the Newton polyhedron of f_{red} will reflect the set {λ_{κ(i)}}_{i}. The change of variable has the form x′_{d+1} = x_{d+1} − φ(x), x′_{t} = x_{t} for t = 1, …, d, with φ(x) of type ∑_{λ∈Z_{≥0}^{d}} a_{λ}x^{λ}. This provides automatically a set of new parametrizations

ζ^{′(i)} := ζ^{(i)} − φ(x) (i ∈ I). Such a change of variables and parametrizations does not modify the set of distinguished tuples.

First, we verify the following lemma.

2.8. Lemma. Assume that λ_{κ(i)} ≰ λ and λ_{κ(j)} ≰ λ. Then α_{λ}^{(i)} = α_{λ}^{(j)}.

Proof. By (2*) we can assume that λ ∈ Z^{d}. Assume that α_{λ}^{(i)} ≠ α_{λ}^{(j)}. Then by the definition of λ^{(i,j)} one has λ^{(i,j)} ≤ λ. This together with λ_{κ(i)} ≰ λ implies (∗) λ_{κ(i)} ≰ λ^{(i,j)}. Therefore, by the definition of λ_{κ(i)} one has (∗∗) λ^{(i,j)} ∉ Z^{d}. Then (∗), (∗∗) and (2*) give α_{λ^{(i,j)}}^{(i)} = 0. Similarly α_{λ^{(i,j)}}^{(j)} = 0. These two vanishing contradict the definition of λ^{(i,j)}.

2.9. Proposition/Definition. (cf. [González01]) There is a change of variables as in 2.7 such that the new parametrizations have the following form:

ζ^{(i)} = α_{1}^{(i)}x^{λ_{κ(i)}} + ∑_{λ_{κ(i)}<λ} α_{λ}^{(i)}x^{λ} (α_{1}^{(i)} ≠ 0) (i ∈ I′).

If i ∈ I \ I′ then f^{(i)}(x, x′_{d+1}) = x′_{d+1} (i.e. ζ^{(i)} = 0).

A coordinate system which admits parametrizations as above is called “good”.

Proof. Choose an index i_{0} ∈ I so that λ_{κ(i_{0})} = max{λ_{κ(i)}}_{i∈I}. Then define φ(x) by ∑ α_{λ}^{(i_{0})}x^{λ}, where the sum is over λ ∈ Z_{≥0}^{d} with λ_{κ(i_{0})} ≰ λ. (In particular, if i ∈ I \ I′ then φ = ζ^{(i_{0})}.) Then the result follows from lemma 2.8 .

2.10. Corollary. Λ has a unique minimal element min Λ which equals min{λ_{κ(i)}}_{i}.

3. The partial toric resolution.

In this section we will assume that f is a quasi-ordinary singularity, and we will use the notations of the previous section. We will consider a toric modification π(Σ) : Z(Σ) → C^{d+1} whose restriction above a small ball B_{r} provides a partial resolution of (F, 0) := (f^{−1}(0), 0) ⊂ (C^{d+1}, 0). For simplicity and uniformity of the notation we will use C^{d+1} instead of B_{r}, but this will not affect the proofs. For the definition of π(Σ) associated with an arbitrary fan Σ, see [KKMS73] or [Oda88]. In the case of some special hypersurface singularities, π(Σ) is a natural (partial) embedded resolution, see e.g. [Varchenko76] or [GT00]. For irreducible quasiordinary singularities the modification π(Σ) was constructed in [González00a], and [González01] contains the reduced case. The presentation here combines [González00a], [González01] and [McN01].

3.1. The germs f_{1}^{(i)}. Assume that f is represented in a “good” coordinate system, which admits parametrizations as in 2.9 .

Define ζ_{1}^{(i)} := α_{1}^{(i)}x^{λ_{κ(i)}} for i ∈ I′ and ζ_{1}^{(i)} := ζ^{(i)} = 0 otherwise. Obviously, ζ_{1}^{(i)} determines an irreducible quasi-ordinary singularity f_{1}^{(i)}. If L denotes the fraction field of C{x}, then n^{(i)} = [L(ζ^{(i)}) : L] is the degree of f^{(i)}, and n_{1}^{(i)} = [L(x^{λ_{κ(i)}}) : L] is the degree of f_{1}^{(i)}. Since the field extension [L(ζ^{(i)}) : L] is generated over L by the distinguished tuples of f^{(i)} ([Lipman88], Lemma 5.7), 2.2 (2) implies that n_{1}^{(i)} divides n^{(i)}. Set e^{(i)} := n^{(i)}/n_{1}^{(i)} ∈ Z_{>0}. (If i ∉ I′ then n^{(i)} = n_{1}^{(i)} = e^{(i)} = 1.) Obviously

f_{1}^{(i)} = ∏_{j=1}^{n_{1}^{(i)}}(x_{d+1} − ω_{j}α_{1}^{(i)}x^{λ_{κ(i)}}) = x_{d+1}^{n_{1}^{(i)}} − γ^{(i)}x^{n_{1}^{(i)}λ_{κ(i)}}, with γ^{(i)} := (α_{1}^{(i)})^{n_{1}^{(i)}},

where {ω_{j}}_{j} are the n_{1}^{(i)}-roots of unity. Notice also that if κ(i) = κ(j) and γ^{(i)} = γ^{(j)} then f_{1}^{(i)} = f_{1}^{(j)}. The importance of the germ f_{1}^{(i)} is given by the fact that its Newton polyhedron determines the Newton polyhedron of f^{(i)} (see below).

3.2. The Newton polyhedron of f. Given f_{red} = ∑_{v} c_{v}x̄^{v} ∈ C{x̄}, its Newton polyhedron N (f_{red}) is the convex hull of the set ∑_{c_{v}≠0} v + R_{≥0}^{d+1}. Any vector u ∈ (R^{d+1})_{≥0}^{∗} defines the face F_{u} of N (f_{red}) by

F_{u} := {v ∈ N (f_{red}) : ⟨u, v⟩ = inf_{v′∈N (f_{red})}⟨u, v′⟩}.

N (f^{(i)}) has only two vertices, namely (0, …, 0, n^{(i)}) and (n^{(i)}λ_{k(i),1}, …, n^{(i)}λ_{k(i),d}, 0) provided that i ∈ I′. If i ∉ I′ then N (f^{(i)}) has only one vertex, namely (0, …, 0, 1). Finally, the Newton polyhedron N (f_{red}) of f_{red} is the Minkowski sum of the polyhedrons N (f^{(i)}) (i ∈ I). Recall that the set {λ_{κ(i)}}_{i∈I} is totally ordered, hence the compact faces of N (f_{red}) constitute a “monotone polygonal path” (using the terminology of [González00b] and [McDonald95]). Let F_{co} be the union of all compact faces of N (f_{red}). Then

f_{red}|_{F_{co}} = ∏_{i∈I}(f_{1}^{(i)})^{e^{(i)}}.

Above f_{red}|_{F} denotes the (symbolic) restriction of f_{red} to F, namely f_{red}|_{F} = ∑_{v∈F} c_{v}x̄^{v}.

3.3. The fan Σ(N (f_{red})) and its subdivision Σ in (R^{d+1})_{≥0}^{∗}. We say that two vectors in (R^{d+1})_{≥0}^{∗} are related if they define the same face in N (f_{red}). The fan Σ(N (f_{red})) is defined in such a way that the classes of the above relation are the relative interiors of the cones of Σ(N (f_{red})). In fact, the fan Σ(N (f_{red})) is obtained by subdividing the cone (R^{d+1})_{≥0}^{∗} with the linear subspaces l^{(i)} (i ∈ I′) given by the equations

l^{(i)}(v) := ∑_{t=1}^{d} λ_{k(i),t}v_{t} − v_{d+1} = 0 (i ∈ I′).

Since {λ_{κ(i)}}_{i∈I′} is totally ordered, for any i, j ∈ I′, κ(i) ≠ κ(j), one has

(∗) l^{(i)} ∩ l^{(j)} ∩ (R^{d+1})_{>0}^{∗} = ∅.

In general, Σ(N (f_{red})) is not nonsingular (for the terminology, see [Oda88] 1.4; in some articles “regular” is used for “nonsingular”). We will use the notation Σ for a nonsingular (simplicial) subdivision of Σ(N (f_{red})) supported by (R^{d+1})_{≥0}^{∗}. For the existence of such a fan Σ, see Theorem 11 (page 32) of [KKMS73].

For an arbitrary Σ it is not true that π(Σ) : Z(Σ) → C^{d+1} is an isomorphism above C^{d+1} \ Sing F (a property sometimes required for an embedded resolution). But, with a good choice of Σ, one has a slightly weaker property (which is still sufficient for us, cf. 4.2 ). More precisely, there exists a nonsingular fan Σ so that π(Σ) is an isomorphism above C^{d+1} \ F. For the proof see (3.15) of [McN01], a similar property was also used in [Varchenko76] and [González01].

In fact, in [González01], the first author considered those nonsingular subdivisions Σ which have the following additional property: any nonsingular cone of Σ(N (f_{red})) is a cone of Σ. The existence of such a fan is proved in [Cox00], Theorem 5.1. If Σ satisfies this property then π(Σ) is an isomorphism above C^{d+1} \ F (cf. [McN01]), and the restriction of π(Σ) to the strict transform F̃ of F is an isomorphism above F \ Sing F (fact shown in [González00a, González01]).

In the sequel, we will assume that Σ satisfies the above property.

3.4. The fan Σ supported by (R^{d+1})_{≥0}^{∗} defines a modification π(Σ) : Z(Σ) → C^{d+1}. The variety Z(Σ) is smooth, and it is covered by affine spaces {Z(σ)}_{dim σ=d+1} (i.e. each Z(σ) ≈ C^{d+1}). The inclusion σ ⊂ (R^{d+1})_{≥0}^{∗} induces a morphism π(σ) : Z(σ) → C^{d+1}. If σ = ⟨a^{1}, …, a^{d+1}⟩ and a^{k} has coordinates (a_{1}^{k}, …, a_{d+1}^{k}), then π(σ) has the form:

x_{1} = u_{1}^{a_{1}^{1}} ⋯ u_{d+1}^{a_{1}^{d+1}}, …, x_{d+1} = u_{1}^{a_{d+1}^{1}} ⋯ u_{d+1}^{a_{d+1}^{d+1}}.

The total transform of f = ∑_{v} c_{v}x̄^{v} by π(σ) is

f ∘ π(σ) = ∑_{v} c_{v}u_{1}^{⟨a^{1},v⟩} ⋯ u_{d+1}^{⟨a^{d+1},v⟩} = u_{1}^{m(a^{1})} ⋯ u_{d+1}^{m(a^{d+1})} · f̃_{σ},

where m(a) := inf_{v∈N (f_{red})}⟨a, v⟩ and f̃_{σ} defines the equation of the strict transform of f in the chart Z(σ) ⊂ Z(Σ).

3.5. The divisors D(a) in Z(Σ). To any primitive vector a of Σ^{(1)}(:= 1skeleton of Σ), we associate a divisor D(a) in Z(Σ). In any chart Z(σ), associated with a cone σ = ⟨a^{1}, …, a^{d+1}⟩, D(a^{k}) is given by {u_{k} = 0}. If b ∉ {a^{1}, …, a^{d+1}}, then D(b) ∩ Z(σ) = ∅. The divisor D := ∪_{a∈Σ^{(1)}}D(a) in the smooth variety Z(Σ) is a normal crossing divisor with smooth irreducible components.

By 3.4 one gets that π(Σ)(D(a^{k})) is the coordinate subspace defined by x_{t} = 0 for all t with a_{t}^{k} ≠ 0. If e^{1}, …, e^{d+1} denote the canonical basis on Z^{d+1}, then π(Σ)(D(e^{k})) = {x_{k} = 0}, but for any a ∈ Σ^{(1)} \ {e^{1}, …, e^{d+1}} one has dim π(Σ)(D(a)) ≤ d − 1. The critical locus (exceptional divisor) of π(Σ) is exactly the union of these divisors:

∪_{a∈Σ^{(1)}\{e^{k}}_{k}} D(a).

By the above discussion, D(a) is compact (i.e. D(a) ⊂ π(Σ)^{−1}(0)) if and only if a is an interior point of (R^{d+1})_{≥0}^{∗}. More generally, let A = {a^{1}, …, a^{s}} be a non-empty subset of Σ^{(1)}, and write D_{A} := ∩_{a∈A} D(a). The D_{A} ≠ ∅ if and only if {a^{1}, …, a^{s}} forms a cone σ_{A} in Σ. In this case D_{A} is compact if and only if σ_{A} ∩ (R^{d+1})_{>0}^{∗} ≠ ∅. In fact

π(Σ)^{−1}(0) = ⋃_{σ_{A}∩(R^{d+1})_{>0}^{∗}≠∅} D_{A}.

3.6. The strict transform F̃ of F. Set F^{(i)} = {f^{(i)} = 0} and for any space Y write Ỹ for its strict transform via π(Σ).

We start with the following remark. If we fix an index i ∈ I′, then all the properties of the germs f^{(i)}, f_{1}^{(i)} and their strict transforms follow from section 3 of [McN01], where the irreducible case is treated. Indeed, our fan Σ considered here satisfies all the restrictions required in [loc. cit.] (even some more, since it is compatible with all the linear spaces l^{(j)}). In the next paragraph we list those properties of F̃^{(i)} which will be needed later. For proofs see [McN01], [González00a] or [González01] (but, for the convenience of the reader, we provide some hints as well).

For a fixed i ∈ I′ the following facts hold:

(1) For any A = {a^{1}, …, a^{s}} ⊂ Σ^{(1)}, if D_{A} ∩ F̃^{(i)} ≠ ∅, then {a^{1}, …, a^{s}} should be a cone σ_{A} in Σ ∩ l^{(i)}. In this case, D_{A} ∩ F̃^{(i)} is compact if and only if σ_{A} ∩ l^{(i)} ∩ (R^{d+1})_{>0}^{∗} = σ_{A} ∩ (R^{d+1})_{>0}^{∗} ≠ ∅. In particular, D_{A} ∩ F̃^{(i)} is compact if and only if D_{A} is compact (cf. 3.5 ).

(2) We say that the “natural stratification” of D ∩ F̃^{(i)} is given by strata of type:

(∩_{a∈A}D(a) \ ∪_{a∉A}D(a)) ∩ F̃^{(i)}, (A ⊂ Σ^{(1)}).

Then for any stratum Ξ of D∩F̃^{(i)} with dim Ξ > 0, one has χ(Ξ) = 0. (This follows from the fact that Ξ is contined in an orbit of the toric variety Z(Σ) represented in some chart Z(σ) by an equation u_{d+1} =a non-zero constant.)

Next, we fix i, j ∈ I′ with i ≠ j. Then the assumption about Σ (cf. 3.3 ) and the above properties give:

(3) If κ(i) ≠ κ(j), then π(Σ)^{−1}(0) ∩ F̃^{(i)} ∩ F̃^{(j)} = ∅. (If π(Σ)^{−1}(0) ∩ F̃^{(i)} ∩ F̃^{(j)} is non empty, there is D_{A} ⊂ π(Σ)^{−1}(0) such that D_{A} ∩ F̃^{(i)} ∩ F̃^{(j)} is non empty, and σ_{A} ∩ (R^{d+1})_{>0}^{∗} ≠ ∅, cf. with the last formula of 3.5 . Then by (1) we have that σ_{A} ⊂ Σ ∩ l^{(i)} ∩ l^{(j)}. This implies that σ_{A} ∩ (R^{d+1})_{>0}^{∗} ⊂ l^{(i)} ∩ l^{(j)} ∩ (R^{d+1})_{>0}^{∗} is empty by 3.3 (∗), a contradiction.)

(4) If κ(i) = κ(j), then π(Σ)^{−1}(0) ∩ F̃^{(i)} = π(Σ)^{−1}(0) ∩ F̃^{(j)} provided that γ^{(i)} = γ^{(j)}; otherwise π(Σ)^{−1}(0) ∩ F̃^{(i)} ∩ F̃^{(j)} = ∅. (This follows again by the irreducible case, where the intersection π(Σ)^{−1}(0) ∩ F̃^{(i)} is described in terms of f_{1}^{(i)}, cf. [McN01], 3.20 and 3.21. See also the discussion in 3.7 , or Proposition 24 of [González01].)

(5) D(e^{d+1}) ∩ F̃^{(j)} = ∅ for any j ∈ I′. Since F̃^{(i)} = D(e^{d+1}) for i ∈ I \ I′, one obtains that for such an i, F̃^{(i)} ∩ F̃^{(j)} = ∅ for any j ∈ I′. (This follows from e^{d+1} ∉ ∪_{i∈I′}l^{(i)} and part (1).)

3.7. The total transform of x^{β}f. The divisor D ∩ F̃ has a “natural stratification” given by strata of type

(∩_{a∈A}D(a) \ ∪_{a∉A}D(a)) ∩ F̃, (A ⊂ Σ^{(1)}).

We are interested only in those strata which are situated in π(Σ)^{−1}(0). They are contained in compact intersections of type D_{A}.

Next, we fix β ∈ Z_{>0}^{d}, and consider the germ x^{β}f. Notice that π(Σ)^{−1}({x_{t} = 0}) = D(e^{t}) (for t = 1, …, d), hence (D ∪ {the total transform of {x^{β}f = 0}}) = (D∪F̃), and both of them have the same “natural stratification” (i.e. by introducing the factor x^{β}, we do not create new strata).

Consider such a strata Ξ with dim Ξ > 0. We claim that the total transform of x^{β}f along Ξ is an equisingular family of singularities. The argument is similar as in [González00a] and [González01] where it is proved that the total transform of f is equisingular along Ξ.

(By this we mean the following. The strict transform along the stratum Ξ defines a family of hypersurface quasi-ordinary singularities in such a way that each member of the family has the same distinguished tuples. The situation locally is described in [Lipman88] Appendix 7, where “natural stratification” of an irreducible quasi-ordinary hypersurface is studied. The important point is that this family is equiresolvable, a resolution is obtained from the embedded resolution of the total transform of f, see [González01]).

Now we concentrate our discussion on the zero-dimensional strata {Ξ}_{dim Ξ=0} of D ∩ F̃.

We fix an arbitrary d-cone σ′ = ⟨a^{1}, …, a^{d}⟩ of Σ ∩ l^{(i)} (i ∈ I′), and we consider the unique (d + 1)-cone σ = ⟨a^{1}, …, a^{d}, a^{d+1}⟩ of Σ with a primitive vector a^{d+1} satisfying l^{(i)}(a^{d+1}) > 0. Then, in the chart Z(σ) one can find a zero-dimensional stratum O_{σ′} := F̃ ∩ D(a^{1}) ∩ ⋯ ∩ D(a^{d}) of D ∩ F̃. In fact, if I = I′, then there is a one-to-one correspondence between the zero-dimensional strata of D ∩ F̃ and the points O_{σ′} where σ′ runs over the d-cones of Σ ∩ l^{(i)} for all i ∈ I′. If I ≠ I′ then D ∩ F̃ has some additional zero-dimensional strata on D(e^{d+1}) as well.

Using the above notations, in the chart Z(σ), the point O_{σ′} is given by the equations u_{1} = ⋯ = u_{d} = 0 and w := 1 − γ^{(i)}u_{d+1} = 0. The first author in [González01] has shown that the strict transform f̃_{σ} of f in the local coordinates (u_{1}, …, u_{d}, w) is a (not necessarily reduced) quasi-ordinary singularity. Moreover, the germ ((f̃_{σ})_{red}, O_{σ′}) has smaller complexity than f_{red} (e.g. #Λ is smaller, or its Eggers-Wall diagram is simpler).

In the proof of Theorem B, we will use an inductive step, which replaces the germ x^{β}f by its total transform at the points {O_{σ′}}_{σ′}. Obviously, the total transform of x^{β}f at the point O_{σ′} has similar form, namely u^{η}f̃_{σ} for some η (by 3.4 ). The point is that if β ∈ Z_{>0}^{d} then η ∈ Z_{>0}^{d} as well. Indeed, let us write ⟨a^{k}, β⟩ for ∑_{t=1}^{d} a_{t}^{k}β_{t}. Then by 3.4 one gets that η_{k} = m(a^{k}) + ⟨a^{k}, β⟩ for any k = 1, …, d. Notice that a^{k} ≠ e^{d+1} since e^{d+1} ∉ l^{(i)}. Hence ⟨a^{k}, β⟩ > 0 for each k.

4. The proof of Theorems A and B

4.1. The proof of Theorem B.. The proof is based on the facts listed in sections 2 and 3 and on the following result proved in [GLM97]. We mention, that in this paper, stratification means a (semi-analytic) pre-stratification, without any regularity assumption.

4.2. Theorem. Let g : (C^{d+1}, 0) → (C, 0) be the germ of an analytic function, and B_{r} a sufficiently small (Milnor) ball of it. Let ϕ : X → B_{r} be a birational modification such that ϕ is an isomorphism above the complement of {g = 0}. Let S be a stratification of ϕ^{−1}(0) such that along each stratum Ξ of S the zeta-function of the germ (g ∘ ϕ, x) at x ∈ Ξ does not depend on x ∈ Ξ. Denote this rational function by ζ_{Ξ}(t). Then:

ζ(g)(t) = ∏_{Ξ∈S} ζ_{Ξ}(t)^{χ(Ξ)}.

In order to prove Theorem B we will apply 4.2 for ϕ = π(Σ), where the fan Σ satisfies all the properties 3.3 . Since the image of the critical locus of π(Σ) is included in F, 4.2 can be applied for the germ g = x^{β}f. The divisor D has a natural stratification with the following strata:

{(∩_{a∈A}D(a) \ ∪_{a∉A}D(a)) \ F̃}_{A⊂Σ^{(1)}} , {(∩_{a∈A}D(a) \ ∪_{a∉A}D(a)) ∩ F̃}_{A⊂Σ^{(1)}}.

This stratification of D induces a stratification on π(Σ)^{−1}(0) which will be denoted by S. Each stratum Ξ of S is included in some compact intersection of type ∩_{a∈A}D(a) (cf. 3.6 (1)). By the very definition, the stratification is compatible with F̃. Denote by S ∩ F̃ (resp. by S \ F̃) the collection of those strata which are in F̃ (resp. are in π(Σ)^{−1}(0) \ F̃). The stratification S satisfies the assumptions of 4.2 by the results of 3.6 and 3.7 .

We recall that the divisor D = ∪_{a∈Σ^{(1)}}D(a) consists of the exceptional divisors {D(a)}_{a} corresponding to a ∈ Σ^{(1)} \ {e^{k}}_{1≤k≤d+1}, and the strict transforms of the coordinate hyperplanes {D(e^{k})}_{1≤k≤d+1}. On the other hand, the divisors {D(e^{k})}_{1≤k≤d} are components of the total transform of g. The divisor D(e^{d+1}) will require a special care. Therefore, we will distinguish two cases: (Case A) I ≠ I′ (i.e. x_{d+1} is a component of f); and (Case B) I = I′ (i.e. x_{d+1} is not a component of f).

We will analyze the contribution ζ_{Ξ}(t)^{χ(Ξ)} for each stratum Ξ.

(1) First assume that Ξ ∈ S \ F̃ is of dimension dim Ξ < d.

Recall that the total transform of x^{β}f outside of F̃ is a normal crossing divisor. Moreover, the zeta-function of a germ (u_{1}, …, u_{d+1}) ↦ u_{1}^{ρ_{1}} ⋯ d_{d+1}^{ρ_{d+1}} is 1, provided that at least two ρ_{k}’s are strictly positive.

In (Case A), along each Ξ there are at least two coordinate hyperplanes in the zero-set of the total transform of g. Hence the zeta-function ζ_{Ξ}(t) = 1. (Cf. also with the classical case [A’Campo75] when ϕ is an embedded resolution.)

In (Case B) D(e^{d+1}) ∩ F̃ = ∅ (cf. 3.6 (5)). Therefore, any stratum Ξ for which the argument presented in (Case A) is not valid, is an orbit of the toric variety Z(Σ) of dimension d − 1 and type D(e^{d+1}) ∩ D(a) \ ∪_{b}D(b). So χ(Ξ) = 0.

(2) Assume that Ξ ∈ S \ F̃ and dim Ξ = d. Then there is a unique stratum Ξ′ of D of type D(a) \ (∪_{b≠a}D(b)) such that Ξ ⊂ Ξ′. Moreover, Ξ is obtained from Ξ′ by eliminating its intersection with F̃, which is a union of (d − 1)-dimensional strata of type Ξ′′ ∈ S ∩ F̃. Now, Ξ′ is an orbit of the toric variety Z(Σ), hence χ(Ξ′) = 0. But for each Ξ′′, χ(Ξ′′) = 0 as well, because of 3.6 (2). Hence χ(Ξ) = 0.

(3) Now, assume that Ξ ∈ S ∩ F̃ of dimension dim Ξ > 0. Then χ(Ξ) = 0 by 3.6 (2).

(4) Finally, consider the zero-dimensional strata of Ξ ∈ S ∩ F̃. By 3.7 they corresponds exactly to the set of points O_{σ′} described in 3.7 , provided that I = I′. Otherwise there are some additional points on D(e^{d+1}), but in these points the total transform of g is a normal crossing divisor with zeta-function = 1.

Therefore, steps (1)-(4) give the inductive formula:

ζ(x^{β}f)(t) = ∏ ζ(total transform of x^{β}f at O_{σ′})(t),

where the product is over the d-cones σ′ ⊂ Σ ∩ (∪_{i∈I′}l^{(i)}).

In 3.7 we verified that total transform at O_{σ′} has the form u^{η}f̃_{σ} satisfying all the inductive properties: f̃_{σ} is quasi-ordinary in local coordinates (u, w) with smaller complexity than f, and η ∈ Z_{>0}^{d}. Therefore Theorem B follows by induction. (After a finite number of toric modifications, all the total transforms u^{η}f̃_{σ} will have the form u^{η}w^{η_{0}} with zeta-function 1.)

4.3. Now we will revise Theorem A. First we recall that Λ = ∅ if and only if f is irreducible and smooth. For such a germ f, Theorem A is obviously true (with ζ(f)(t) = 1 − t). Therefore, in the sequel, we will assume that Λ ≠ ∅. We reorder the variables (x_{1}, …, x_{d}) in such a way that the first entry of min Λ is non-zero.

For any index set K ⊂ {1, …, d} we define f|_{x_{K}=0} by f|_{x_{k}=0 for all k∈K}.

4.4. Theorem A′. Fix the above notations. Then for any index set K ⊂ {1, …, d} with 1 ∉ K 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}).

In fact, if min Λ has at least two non-zero entries, then ζ(f)(t) = 1 − t^{deg f}.

Proof of Theorem A′. Obviously we can assume that f is represented in a good coordinate system (cf. 2.9 ). The proof is based on the following fact.

The “weak splitting property” of the zeta-function. [McN01] Assume that for two germs f, g : (C^{d+1}, 0) → (C, 0) and any k ≥ 1 one has ζ(g^{k}f) = 1. Then ζ(f) = ζ(f|_{g=0}).

Using this and Theorem B we obtain that ζ(f) = ζ(f|_{x_{1}···x_{d}=0}). By a Mayer– Vietoris argument

ζ(f|_{x_{1}···x_{d}=0}) = ∏_{K} ζ(f|_{x_{K}=0})^{(−1)^{#K−1}},

where the product is over the non-empty subsets K of {1, …, d}. Hence

(∗_{1}) ζ(f) = ∏_{K} ζ(f|_{x_{K}=0})^{(−1)^{#K−1}}.

Notice that for any non-empty K, the restriction f|_{x_{K}=0} is a quasi-ordinary germ, so if #K ≤ d − 2 one can apply (∗_{1}) repeatedly for f|_{x_{K}=0}. Therefore, by repeated induction, one obtains that

(∗_{2}) ζ(f) = \frac{∏_{K:#K=d−1} ζ(f|_{x_{K}=0})}{(ζ(f|_{x_{1}=···=x_{d}=0}))^{d−1}}.

Now, if 1 ∈ K , then f|_{x_{K}=0} = x_{d+1}^{deg f} (cf. 2.9 ), hence (∗_{2}) becomes (∗_{3}) ζ(f) = ζ(f|_{x_{2}=···=x_{d}=0}). The first statement of the theorem follows from (∗_{3}) applied for both f and f|_{x_{K}=0}. Finally, again by 2.9 , if min Λ has at least two non-zero entries, then ζ(f|_{x_{2}=···=x_{d}=0}) = x_{d+1}^{deg f}, hence Theorem A′ follows.

4.5. Final remarks. (1) Assume that min Λ has only one non-zero entry. Then the set of elements of Λ with only one non-zero entry determine completely the topological type of g. This follows from the fact that the Eggers-Wall diagram (i.e. Λ) of a plane curve singularity determines its topological type. For details, see e.g. [GarcíaB00]. For a formula for ζ(g) in terms of the topology of g, see [EN85].

(2) ζ(f) forgets almost all the information about the distinguished tuples Λ of f_{red}. This behaviour is very different compared with the case of irreducible plane curve singularities where ζ(f) is a complete topological invariant.

(3) Theorem B of the present paper (applied for an irreducible f) is slightly weaker than Theorem B of the article [McN01]. Nevertheless, we decided to present this very version since its proof is simpler, more conceptual, and its statement still implies Theorem A.

(The exact analog of Theorem B [McN01] is the following: Assume that β ∈ Z_{≥0}^{d} is non-zero, and min Λ+β has at least two non-zero entries. Then ζ(x^{β}f)(t) = 1. Our conjecture is that this statement is true, but we did not verify all the details of its proof.)

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Université Paris 7, Institut de Mathématiques, Equipe “Geometrie et Dynamique”, Case 7012, 2, Place Jussieu, 75251 Paris Cedex 05, France

    E-mail address: gonzalez@math.jussieu.fr

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

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

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

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