The zeta function of singularities

András Némethi^{∗}

October 11, 1997

(J. Algebraic Geometry 2 (1993) 1-23)

1 Introduction

Let (X, x) ⊂ (C^{N}, x) be a germ of an analytic space and f : (X, x) ⟶ (C, 0) be an analytic germ defined in a neighbourhood of x. One important topological invariant of f is its Milnor fibration ([14, 17])

f_{ϵ,η} : B_{ϵ} ∩ X ∩ f^{−1}(S_{η}^{1}) ⟶ S_{η}^{1}

with Milnor fiber F = f_{ϵ,η}^{−1}(η) and geometric monodromy m_{f} : F ⟶ F. At the homological level m_{f} is encoded by the zeta function:

ζ_{f}(λ) = ∏_{q} det(1 − λm_{f,q})^{(−1)^{q+1}}

where m_{f,∗} : H_{∗}(F, C) ⟶ H_{∗}(F, C) is induced by m_{f}.

In the case (X, x) = (C^{2}, 0) the zeta function ζ_{f} was calculated by A’Campo [2] and Guzein-Zade [8, 9]. In the general dimension ((X, x) = (C^{N}, 0)) Milnor and Orlik [18] calculated it for quasihomogeneous isolated singularities, Varchenko [28] and Ehlers [6] calculated it in terms of the Newton diagram (with some restrictions), and A’Campo [3] determined it in terms of the invariants of the resolution. As a general reference for (hypersurface) singularities we refer to [4] and [16].

In this paper we develop a new approach for the computation of the zeta function, which can be regarded as a synthesis of the polar curve method of Lê Dũng Tráng [12, 13, 14] and the Jaco–Shalen–Johannson decomposition technique of the fibered graph multilinks [7]. This new method gives both a very explicit algorithm of the zeta function in more general setting and a natural explanation of the behaviour of the zeta function in the context of series of singularities, of approximations of singularities and of Sebastiani– Thom type problems.

∗Supported by the Netherlands Organization for Scientific Research (N.W.O).

The main idea is to write f : (X, x) ⟶ (C, 0) as a composed singularity f = P ∘ ϕ : (X, x) ⟶ (C^{2}, 0) ⟶ (C, 0) such that ϕ has a manageable discriminant space and P is a curve singularity. Then the zeta function ζ_{f} of f can be identified with the zeta function ζ_{(P,F)} of P with coefficients in the constructible complex F = Rϕ_{∗}C_{X}.

More generally, if G ∈ D_{c}^{b}(X) is a constructible complex then the zeta function ζ_{(f,G)} can be computed by ζ_{(P,F)} where F = Rϕ_{∗}G. On the other hand, for a curve singularity P and constructible complex F on (C^{2}, 0), we give a very precise formula in terms of the representations of the corresponding local systems on some well-determined link components parametrized by an Eisenbud–Neumann diagram.

In section 2 we present this reduction formula. Section 3 contains its proof. In section 4 we compute the zeta function ζ_{(f,F)} in terms of some zeta functions ζ_{(f_{v},F_{v})} where f_{v} is the composition: f ∘ i_{v} : (X_{v}, x) ⟶ (X, x) ⟶ (C, 0) where X_{v} are one-codimensional subspaces and F_{v} = F|X_{v}. This can be regarded as the inductive step in the explicit computation of ζ_{(f,F)}.

It is remarkable that in our context it is very easy to reprove (and improve) that ζ_{f} is a (rational) product of cyclotomic polynomials and if (X, x) = (C^{N}, 0) then the Lefschetz number L_{f} is zero.

In section 5 we illustrate our method in the semi-global context. In this case the basic EN-diagram is not an algebraic diagram. This gives a motivation to present the reduction formula (§ 2) in the general setting of graph links.

In section 6 we take the particular case f = P ∘ ϕ, where ϕ = (g, h) is a germ with separable variables and P is an arbitrary curve singularity. This is the most general Sebastiani–Thom type construction. For these singularities our zeta function is computed in terms of the monodromy representations of g and h as well as the several variable Alexander polynomial of P. This result was an intermediate step in our study of zeta functions [21].

The main objects of section 7 are composed singularities f_{k} = P_{k} ∘ ϕ, where ϕ is an isolated complete intersection singularity ( ICIS ) and P_{k} is a topological series belonging to P_{∞} (in the sense of Schrauwen). Since we are able to compute both ζ_{f_{k}} and ζ_{f_{∞}} ( f_{∞} = P_{∞} ∘ ϕ ), we can give the formula of the correction term ζ_{f_{k}}/ζ_{f_{∞}} and we are able to explain in principle the known particular results such as the formula of Siersma in the case of Iomdin’s series [26], the results of Schrauwen [23] about the zeta function of series of plane curves, as well as our earlier result [19, Theorem D].

In section 8, using the results of the previous section, we give a characterisation of the approximations of type f + ϵφ such that the correction term is of Siersma-type. Moreover, we illustrate that the apparently strong hypothesis in section 7 about ϕ (i.e. to be ICIS ) is not really an obstruction in its application in general situations.

This paper has been written during a six-week visit to the University of Utrecht and could not have been written without the warm stimulation of Professor D. Siersma and Professor J. H. M. Steenbrink. We thank them for their help. We also thank R. Schrauwen and M. G. M. van Doorn for the fruitful collaboration in this period. This material was presented in the Singularity Workshop organized by the Catholic University Nijmegen, 30 Nov.–1 Dec.1990.

(In [22], we try to clarify where the place of our zeta function in the multitude of the (topological) zeta functions is. More precisely, we show that our main theorems fit in the problems about the torsion invariants and Lefschetz type formulas.)

2 The reduction formula

2.1 Let Σ be an oriented homology 3–sphere and let

(Σ, m_{1} S_{1} ∪ … ∪ m_{r} S_{r} ) = L(m)

be a fibered graph multilink associated to the splice diagram Γ(m) (not necessary minimal)[7].

Let Γ^{·}(m) be the graph diagram obtained from Γ(m) by replacing the arrowhead vertices with zero multiplicities into boundary vertices. We denote by N the set of nodes of Γ^{·}(m) (or Γ(m) ) and by B the set of boundary vertices of Γ^{·}(m) .

If n ∈ N is a node, we define K_{n} as a nonsingular oriented fiber of the Seifert component corresponding to n ; if b ∈ B is a boundary vertex then we define K_{b} as the (possibly singular) oriented fiber corresponding to b in the corresponding Seifert component.

A fiber of Γ^{·}(m) meets the link components S_{i} (m_{i} = 0) transversely and the complement of its intersection with ⋃_{m_{i}=0} S_{i} can be identified with a fiber of Γ(m). If v ∈ V = N ∪ B is a non-arrowhead vertex then K_{v} meets the fiber of Γ^{·}(m) in

m_{Γ}(K_{v}) = ∑_{j=1}^{r} m_{j} · l(S_{j}, K_{v})

points ( l(,) denotes the linking number of two link components ). 2.2 In this paper we will suppose that m_{Γ}(K_{v}) ≠ 0 for each v ∈ V. (Condition which implies that the multilink is fibered [7, p.88].) We will introduce an orientation of the virtual link components K_{v} (v ∈ V) by the inequality m_{Γ}(K_{v}) > 0.

The set {K_{b}}_{b∈B} defines a stratification S = {S_{b}}_{b∈{0}∪B} of Σ − K (K = ⋃_{m_{j}≠0} S_{j}), with strata:

S_{0} = Σ − (K ∪ ⋃_{b∈B} K_{b}) and S_{b} = K_{b} if b ∈ B.

Let F = F^{·} ∈ D_{c}^{b}(Σ − K) be a bounded constructible complex whose cohomology sheaves are locally constant of finite rank on the strata of S. In particular, for each v ∈ V, we have a monodromy action:

M_{K_{v}}^{·} : H^{·}(F | K_{v}) ⟶ H^{·}(F | K_{v})

with zeta function

ζ_{(K_{v},F)}(λ) = ∏_{q} det(I − λM_{K_{v}}^{q} )^{(−1)^{q+1}} (v ∈ V)

Obviously ζ_{(K_{n},F)} does not depend on the choice of the nonsingular fiber K_{n} if n ∈ N.

On the other hand, we can define a global object, too. If p : Σ−K ⟶ S^{1} denotes the projection map of the fibered multilink L^{·}(m), associated to Γ^{·}(m) (which extends the fibration map of L(m) ), then R^{i}p_{∗}F is a local system on S^{1} for all i. Its monodromy will be denoted by M_{Γ}^{i}. The corresponding zeta function is defined by:

ζ_{(L,F)}(λ) = ∏_{q} det(I − λM_{Γ}^{q})^{(−1)^{q+1}}

Our reduction formula is: 2.3 Theorem

ζ_{(L,F)}(λ) = ∏_{v∈V} [ζ_{(K_{v},F)}(λ^{m_{Γ}(K_{v})})]^{2−δ(K_{v})}

where δ(K_{v}) is the degree of the corresponding vertices of Γ^{·}(m) .

3 The proof of Theorem 2.3

Suppose that L(m) is a result of splicing L′(m′) and L′′(m′′) :

L(m) = L′(m′) \underline{S′ S′′} L′′(m′′)

Let T^{2} be the torus along which the splicing is done. Then Σ has a decomposition:

Σ′ − T ub(S′) ∪_{T^{2}} Σ′′ − T ub(S′′)

where T ub(S′) resp. T ub(S′′) are small tubular neighbourhoods of S′ resp. S′′ in Σ′ resp. Σ′′. The restriction of F to Σ′ −K′ ∪T ub(S′) can be uniquely extended to Σ′ − K′ (K′ = (K ∩ Σ′) ∪ S′) as a local system on T ub(S′) − S′. Similarly we extend the restriction of F on Σ′′ − K′′ ∪ T ub(S′′) to Σ′′ − K′′.

Since the above decomposition of Σ − K is compatible with the fibering p, by Mayer-Vietoris theorem: ζ_{(L,F)} = ζ_{(L′,F)} · ζ_{(L′′,F)}/ζ_{(T^{2},F|T^{2})}. Here ζ_{(T^{2},F|T^{2})} is the zeta function associated with the locally constant sheaf R(p|T^{2})_{∗}(F|T^{2}) on S^{1} . Because the fibers of p|T^{2} are \underline{oriented} circles and the geometric monodromy of p|T^{2} , as permutation of these circles, preserves their orientation, we have ζ_{(T^{2},F|T^{2})} = 1 (see for example [19, (3.3.9)]).

On the other hand, if we denote by ζ_{(Γ,F)}(λ) the term which appears in the right hand side of the formula (2.3) then: ζ_{(Γ,F)} = ζ_{(Γ′,F)} · ζ_{(Γ′′,F)}. (Here Γ′, Γ′′ are the corresponding graphs of L′ and L′′.)

This is a consequence of the splicing condition [7, p.87].

Therefore, it is enough to prove the theorem for multilinks of one of the following types:

[figure: Type I; Γ^{·}(m); K_{1}; (m_{1}) S_{1}]

where m_{1} > 0 and S = {Σ − S_{1} ∪ K_{1}, K_{1}} or S = {Σ − S_{1}}.

[figure: Type II; Γ^{·}(m); K; a_{1}; a_{r}; ⋮; (m_{1}) S_{1}; (m_{r}) S_{r}]

where S = {S_{0} = Σ − ⋃_{i=1}^{r} S_{i}}. 3.1 The formula for graphs of type I

The fiber p^{−1}(1) can be identified with m_{1} disjoint copies of the twodimensional disc D_{i} with center O_{i} , such that

(p^{−1}(1), p^{−1}(1) ∩ K) = (∪_{i}D_{i}, ∪_{i}O_{i}).

Therefore H^{q}(p^{−1}(1), F) is equal to ⊕_{i}H^{q}(F_{O_{i}}). The zeta function of monodromy on this is the same as the zeta function of monodromy on ⊕_{i} ⊕_{q} F_{O_{i}}^{q}. The monodromy M_{Γ}^{q} can be identified to the matrix of the (m_{1}×m_{1}–blocks) :

( \begin{matrix}0 & 0 & … & 0 & M_{K_{1}}^{q} \\ I & 0 & … & 0 & 0 \\ 0 & I & … & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & … & I & 0 \end{matrix} ) .

Hence:

ζ_{(L,F)}(λ) = ∏_{q} det(I−λM_{Γ}^{q})^{(−1)^{q+1}} = ∏_{q} det(I−λ^{m_{1}} M_{K_{1}}^{q})^{(−1)^{q+1}} = ζ_{(K,F)}(λ^{m_{1}}).

3.2 Graphs of type II

Let ∗ be a base point on p^{−1}(1), and H = π_{1}(p^{−1}(1), ∗) and G = π_{1}(Σ − K, ∗). If G is a group then Z[G] denotes its integer group ring and Der(G, H) denotes the crossed homomorphisms of G into the G–module H.

In this second case H^{·} = H^{·}(F) is local a system, hence corresponds to a monodromy representation ρ^{·} : π_{1}(Σ − K, ∗) → Aut(H_{∗}^{·}).

Let us consider the following exact sequence of groups induced by the fibration p:

0 → π_{1}(p^{−1}(1), ∗) →^{i_{∗}} π_{1}(Σ − K, ∗) →^{p_{∗}} Z → 0

By i_{∗} the C–vector space H_{∗}^{·} becomes an H–module and H^{q}(p^{−1}(1), H_{∗}^{·}) = H^{q}(H, H_{∗}^{·}) (p^{−1}(1) is a K(H, 1)–space !). Since H is free H^{q}(H, H_{∗}^{·}) = 0 if q ≥ 2 [27].

Choose g ∈ G such that p_{∗}(g) = 1. The maps ρ^{·}(g) ∈ Aut(H_{∗}^{·}) and c_{g} : H → H, h → g^{−1}hg induce an automorphism of the following exact sequence:

00  00 H^{0}(H, H_{∗}^{·}) H^{0}(H, H_{∗}^{·})  H_{∗}^{·} H_{∗}^{·}  Der(H,H_{∗}^{·}) Der(H,H_{∗}^{·}) H^{1}(H, H_{∗}^{·}) H^{1}(H, H_{∗}^{·}) ⟶  ⟶  ⟶  ⟶  ⟶ ⟶  ⟶  ⟶  ⟶  ⟶ ↑  ↑  ↑  ↑ g_{0}^{·}  ρ^{·}(g)  g_{Der}^{·}  g_{1}^{·}

By [25], g_{i}^{·} (i = 0, 1) does not depend on the choice of g. These automorphisms correspond to the monodromy actions, i.e.:

(3.3) ζ_{(L,F)}(λ) = ∏_{q}[det(I−λg_{Der}^{q})/det(I−λ·ρ^{q}(g))]^{(−1)^{q}}

Since H is a free group (generated by the elements, say b_{1}, …, b_{N}) by the correspondence δ → (δ(b_{i})_{i=1}^{N}) of Der(H, H_{∗}^{·}) with (H_{∗}^{·})^{N}, g_{Der}^{q} can be identified with the block-matrix:

(3.4) [g_{Der}^{q}] : (H_{∗}^{·})^{N} → (H_{∗}^{·})^{N} ; [g_{Der}^{q}] = [s∘(ρ^{q})˜ (g·i_{∗}˜ (\frac{∂w_{i}}{∂b_{i}}))]_{ij}

where: w_{i} = c_{g}(b_{i}) ∈ H; i_{∗}˜ : Z[H] → Z[H] is induced by i_{∗}; (ρ^{q})˜ : Z[G] → Z[AutH_{∗}^{·}] is induced by ρ^{q}; s : Z[AutH_{∗}^{·}] → End_{C}H_{∗}^{·} ; s(∑_{i} c_{i}[a_{jk}^{i}]_{jk}) = [∑_{i} c_{i}a_{jk}^{i}]_{jk}; \frac{∂}{∂b_{i}} : Z[H] → Z[H] are derivations determined by the relations \frac{∂b_{i}}{∂b_{j}} = δ_{ij}.

We denote m_{Γ}(K) = ∑_{j=1}^{r} m_{j}a_{1} ⋯ â_{j} ⋯ a_{r} by m. Recall, by our initial condition (2.2): m > 0.

The Seifert link L(m) has a regular S^{1}–action. This action determines an m–fold cyclic cover of the fiber p^{−1}(e^{2πiθ}) over the r–punctured sphere (Σ − K)/S^{1} = B such that the geometric monodromy p^{−1}(1) → p^{−1}(1) of the fibration p is a generator of the group Z of the covering transformations [7, p.91].

Hence we have the following commutative diagram with exact rows and columns:

00  H H  0Z G L_{r−1} 0  0Z Z Z_{m} 0  00 ⟶  ⟶  ⟶  ⟶ j_{∗}  q_{∗} ⟶  ⟶  ⟶  ⟶ i_{∗}  p_{∗} ⟶ ↑  ↑ ↑ ↑ ↑  ↑ ↑ ↑ ↑ k_{∗} s_{∗} (3.5)

(s_{∗} : G → L_{r−1} = π_{1}(B) is induced by the action of S^{1} on Σ − K , k_{∗}(1) can be identified with the class of an orbit in Σ − K ).

Since L_{r−1} is a free group with (r−1)–generators, H has N = m(r−2)+1 generators [27].

It is not hard to prove that we can choose a base b_{1}, …, b_{r−1} in L_{r−1} such that q_{∗}(b_{1}) is a generator of Z_{m}. Then T = {1, b_{1}, …, b_{1}^{m−1}} is a Schneider transversal of H in L_{r−1}. If q_{∗}(b_{1}) = û_{1} and û · û_{1} = ˆ1 in Z_{m} , then the monodromy of H corresponds to the conjugation action h → b_{1}^{−u}hb_{1}^{u} in L_{r−1}. In order to simplify the notations we suppose that û = ˆ1. (The general case is similar). Then g ∈ G with p_{∗}(g) = 1 can be chosen such that s_{∗}(g) = b_{1}. Let b_{00} ∈ H be an element such that j_{∗}(b_{00}) = b_{1}^{m} . Then, using the Schneider transversal, in H can be found a base [27, p.182] :

{b_{00} ; b_{11}, b_{12}, …, b_{1m}; b_{21}, …, b_{2m}; …; b_{r−1,1}, …, b_{r−2,m}}

such that the conjugation by b_{1} ( the monodromy at π_{1}–level ) is the following (we omit the symbol j_{∗} ):

b_{1}^{−1}b_{ij}b_{1} = b_{i,j+1} i = 1, …, r − 2; j = 1, …, m − 1 b_{1}^{−1}b_{im}b_{1} = b_{00}^{−1}b_{i1}b_{00} i = 1, …, r − 2 b_{1}^{−1}b_{00}b_{1} = b_{00}

By an easy computation we obtain:

[\frac{∂w_{ij}}{∂b_{kl}} ] = ( \begin{matrix}1 & 0 & … & 0 \\ ∗ & M & … & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ ∗ & 0 & … & M \end{matrix} )

M = ( \begin{matrix}0 & & & & & & b_{00}^{−1} \\ 1 \\ & 1 \\ & & . & . & . \\ & & & & & 1 & 0 \end{matrix} )

(M ∈ Mat_{m×m})

Using (3.4) we get:

det(I − λ[g_{Der}^{q}]) = det(I − λ · ρ^{q}(g)) · [det(I − λ^{m} · ρ^{q}(g^{m}i_{∗}(b_{00}^{−1})))]^{r−2}

By the diagram (3.5) g^{m} · i_{∗}(b_{00}^{−1}) = k_{∗}(1), which corresponds to the Seifert fiber of the Seifert node. Hence, using (3.3):

ζ_{(L,F)}(λ) = ∏_{q} det(I − λ^{m} · ρ^{q}(K))^{(−1)^{q}(r−2)} = [ζ_{(K,F)}(λ^{m})]^{2−r}.

4 The zeta function of germs of analytic functions

Let (X, x) ∈ (C^{N}, 0) be the germ of an analytic subspace X of C^{N} and let f : (X, x) → (C, 0) be an analytic function (not identically zero on X ) with f(x) = 0. Then we have the following: 4.1 Theorem ( Lê Dũng Tráng [12, 13, 14]): If ϵ > 0 is small enough and 0 < η << ϵ, the mapping induced by f:

ψ_{ϵ,η} : B_{ϵ} ∩ X ∩ f^{−1}(D_{η} − {0}) → D_{η} − {0}

is a topological fibration.

(B_{ϵ} denotes the closed real ball in C^{N} of center x and radius ϵ > 0, D_{η} is a disc of C centered at 0 with radius η ).

The monodromy of f is the monodromy of this topological fibration. Let l : C^{N} → C be a sufficiently general affine form such that l(x) = 0 and set ϕ = (f, l) : (X, x) → (C^{2}, 0). Let Δ be the Cerf diagram of f relative to l = 0 [11] and Δ_{0} = Δ ∪ {c = 0} ( c and d are the coordinates in (C^{2}, 0).) 4.2 Theorem (Lê Dũng Tráng [12]) There exists a fundamental system of privileged polydiscs P_{α} = D_{α} × P′_{α} (α ∈ A) relative to the stratification of X and f = 0, centered at x, and Q_{α} = D_{α} × D′_{α} (α ∈ A) centered at 0 in C^{2} such that ϕ induces:

ϕ_{α} : P_{α} ∩ X ∩ ϕ^{−1}(Q_{α} − Δ_{0}) → Q_{α} − Δ_{0}

which is a topological fibration.

Moreover, f induces:

φ_{α} : P_{α} ∩ X ∩ ϕ^{−1}(Q_{α} − {0} × D′_{α}) → D_{α} − {0}

which is fiber homeomorphic to the fibration of Theorem (4.1) and

φ′_{α} : P_{α} ∩ X ∩ ϕ^{−1}({0} × (D′_{α} − {0})) → D′_{α} − {0}

which is fiber homeomorphic to the fibration (4.1) applied to the restriction of f to X ∩ {l = 0}.

We note that with a suitable change of the discs we can suppose that Δ ∩ ∂(D_{α} × D′_{α}) ⊂ ∂D_{α} × D′_{α} . Using this fact, we can replace the fibration from (4.2) by its restriction over a ball D_{η}^{2} = {(c, d) ∈ C^{2} :∥ (c, d) ∥< η}, D_{η}^{2} ⊂ D_{α}×D′_{α} , such that for δ sufficiently small over S_{δ}^{1} = {w ∈ C :| w |= δ}, f induces:

φ̃_{α} : P_{α} ∩ X ∩ ϕ^{−1}(P^{−1}(S_{δ}^{1}) ∩ D_{η}^{2}) ⟶^{P ∘ϕ_{α}} S_{δ}^{1}

which is a topological fibration fiber homeomorphic to the fibration (4.1).(Here P (c, d) = c.)

Moreover, using the fact that the restriction map

ϕ_{α} : P_{α} ∩ X ∩ ϕ^{−1}(Δ − {0}) → Δ − {0}

is a topological fibration, and using the Thom isotopy lemma, we get that

(4.3)  f̃_{α} : P_{α}∩X∩ϕ^{−1}(S_{η}^{3}−P^{−1}(0)) → S^{1}
f̃_{α} = f/ | f |= P ∘ ϕ_{α}/ | P ∘ ϕ_{α} |

is a topological fibration fiber homeomorphic to the Milnor fibration (4.1). In other words, ϕ_{α} can be interpreted as a bundle map between the fibration (4.3) and the fibration of P: P / | P |: S_{η}^{3} − P^{−1}(0) → S^{1}.

Let {Δ_{i}}_{i=1}^{r} be the irreducible components of Δ and set S_{i} = Δ_{i}∩S_{r}^{3} , i = 1, ..., r. We have a natural multilink

L(m) = ( S_{η}^{3} ; m_{1}S_{1} ∪ … ∪ m_{r}S_{r} ∪ m_{0}S_{0})

where m_{1} = … = m_{r} = 0 , m_{0} = 1 , S_{0} = {c = 0} ∩ S_{η}^{3}.

The restriction map

ϕ_{α}^{e} : P_{α} ∩ X ∩ ϕ^{−1}(S_{η}^{3} − {c = 0}) → S_{η}^{3} − {c = 0}

determines a constructible complex F = R(ϕ_{α}^{e})_{∗}C_{X} over S_{η}^{3} − S_{0} whose cohomology sheaves are locally constant over S_{η}^{3}−Δ_{0} and over S_{i} ; i = 1, ..., r. Obviously, the zeta function associated with the monodromy of the Milnor fibration (4.1) is ζ_{f}(λ) = ζ_{(L,F)}(λ).

Each non-arrowhead vertex v ∈ V corresponds to a germ g_{v} : (C^{2}, 0) → (C, 0) such that the corresponding virtual link component K_{v} is g_{v}^{−1}(0) ∩ S_{η}^{3}. Let n_{v} : (D, 0) → (g_{v}^{−1}(0), 0) ⊂ (C^{2}, 0) (v ∈ V) be the normalisations of these (irreducible) germs which determines the following pull-backs:

n_{v}^{∗}(ϕ) : (n_{v}^{∗}(X), n_{v}^{∗}(x)) → (C, 0)

Note, that all these maps are deformations of the analytic space-germ (X ∩ f^{−1}(0) ∩ l^{−1}(0), x).

By our reduction formula:

4.4 Theorem

ζ_{f}(λ) = ∏_{v∈V} [ζ_{n_{v}^{∗}(ϕ)}(λ^{m_{Γ}(K_{v})})]^{2−δ(K_{v})}

where Γ(m) is the corresponding (algebraic) graph diagram of L(m). (The numerical invariants m_{Γ}(K_{v}) can also be computed by (g_{i}, c)_{0}.)

Set (X_{v}, x) = (n_{v}^{∗}(X), n_{v}^{∗}(x)) and P ∘ n_{v} ∘ n_{v}^{∗}(ϕ) = f_{v}. Then the pair ((X_{v}, x), f_{v}) can be interpreted as a subobject of ((X, x), f), i.e. (X_{v}, x) is a 1–codimensional subspace and f_{v} = f|X_{v}. The normalisation map n_{v} can be chosen such that n_{v}(t) = (t^{(g_{v},c)_{0}} , d(t) ), i.e. P ∘ n_{v}(t) = t^{m_{Γ}(K_{v})}. Hence:

ζ_{n_{v}^{∗}(ϕ)}(λ^{m_{Γ}(K_{v})}) = ζ_{f_{v}}(λ)

By Theorem 4.4 we get:

4.5 Theorem

ζ_{f}(λ) = ∏_{v∈V} ζ_{f_{v}}(λ)^{2−δ(K_{v})}

4.6 Theorem 4.4 and 4.5 can be generalised in the following way: Let S_{X} be a stratification on X :

∅ = S_{X,−1} ⊂ S_{X,0} ⊂ ⋯ ⊂ S_{X,n} = X.

Let F_{X} ∈ D_{c}^{b}(X) be an S_{X}–constructible complex and f : (X, x) → (C, 0) be an analytic function as above. Then the construction of Lê Dũng Tráng can be realised in a way compatible with the stratification S_{X} . We obtain that for ϵ > 0 sufficiently small and 0 < η << ϵ the maps

ψ_{ϵ,η,i} : B_{ϵ} ∩ S_{X,i} ∩ f^{−1}(D_{η} − {0}) → D_{η} − {0}

are topological fibrations. In particular Rf_{∗}F_{X} is local system on D_{η} − {0} hence we can consider its zeta function ζ_{f,F} . For the relative construction we take l generic with respect to S_{X}. If we choose common polydiscs P_{α} resp. Q_{α} for {ϕ_{α}|S_{X,i}}_{i} then R(ϕ_{α}^{e})_{∗}F_{X} is Δ(S)–constructible, where Δ(S) = ⋃_{i} Δ(f|S_{X_{i}}) is the union of the Cerf diagrams of the restrictions f|S_{X_{i}}. Similarly as above we construct the germs n_{v}^{∗}(ϕ) for v ∈ V (Δ(S)) =the non– arrowhead vertices of the graph Γ(S) associated with Δ(S).

4.7 Theorem

ζ_{f,F}(λ) = ∏_{v∈V (Δ(S))}[ζ_{n_{v}^{∗}(ϕ),n_{v}^{∗}F}(λ^{m_{Γ(S)}(K_{v})})]^{2−δ(K_{v})}

or using the notations of (4.5):

ζ_{f,F}(λ) = ∏_{v∈V (Δ(S))}[ζ_{f_{v},F|X_{v}}(λ)]^{2−δ(K_{v})}

4.8 Example Let f : (C^{n+1}, 0) → (C, 0) be a homogeneous germ of degree q. Then Γ(m) has the following form:

[figure: Γ(m) :; {c = 0} (1); (0) S_{r}; (0) S_{1}; K_{d} = {d = 0}; K; 1; 1; 1; ⋮; q]

Therefore: ζ_{f}(λ) = ζ_{n_{K_{d}}^{∗}(ϕ)}(λ) · ζ_{n_{K}^{∗}(ϕ)}(λ^{q}) · ∏_{v=1}^{r} ζ_{n_{v}^{∗}(ϕ)}(λ^{q}).

Since the geometric monodromies of the singularities n_{v}^{∗}(ϕ) (v = 1, …, r) and n_{K}^{∗}(ϕ) are trivial (consider the natural C^{∗}–actions), we get:

ζ_{f}(λ) = ζ_{f|l=0}(λ) · (1 − λ^{q})^{χ}

By induction:

ζ_{f}(λ) = (1 − λ^{q})^{−χ(f)/q}

(χ(f) = the Euler–characteristic of the Milnor fiber of f).

In particular, the zeta function of the determinant map det : C^{n^{2}} = Mat_{n}(C) → C (n > 1) is trivial because χ(SL(n, C)) = 0. 4.9 Since dimn_{v}^{∗}(X) = dimX − 1, Theorem 4.4, 4.5 resp. 4.7 give an inductive algorithm to compute ζ_{f} ( resp. ζ_{f,F} ). If dimX = 1, then the geometric monodromy of f can be identified with a permutation of a finite number of points, hence ζ_{f}^{−1}(λ) is a product of cyclotomic polynomials.

By our inductive step we obtain: 4.10 Corollary ζ_{f} is a rational function of cyclotomic polynomials.

Moreover, if (X, S_{X}, F_{X}) is as in (4.6) then the zeta function has the following finite product form:

ζ_{f,F}(λ) = ∏_{k} det(I − λ^{m_{k}} M_{k})^{n_{k}}

where M_{k} are the algebraic monodromies of the local systems induced by F_{X} on some one–dimensional subspaces of X. 4.11 Suppose that the subspace (f^{−1}(0), x) resp. (X, x) ⊂ (C^{N}, x) is an n–resp. an (n + 1)–dimensional ICIS (isolated complete intersection singularity). Then f : (X, x) → (C, 0) is a deformation of (f^{−1}(0), x) (possibly not smoothing). Hence the fibers f^{−1}(t) are homotopically bouquet of n–spheres. By our construction this property is preserved: each pull-back n_{v}^{∗}(ϕ) : (n_{v}^{∗}(X), n_{v}^{∗}(x)) → (C, 0) is an n–dimensional ICIS (as a deformation of the (n − 1)–dimensional ICIS (n_{v}^{∗}(f^{−1}(0)), n_{v}^{∗}(x)). Our formula relates the corresponding characteristic polynomials. 4.12 If (X, x) = (C^{N}, 0) and df(0) = 0, then by [11] the components Δ_{i} of the Cerf diagram have the line {c = 0} as their tangent cone. Therefore m_{Γ}(K_{v}) > 1 for all v ∈ V except for {d = 0} ∩ S_{η}^{3} ( m_{Γ}({d = 0}) = 1). Therefore, the Lefschetz number L_{f} = −dζ/dλ(0) satisfies L_{f} = L_{f|l^{−1}(0)}. Since a singularity f : (C, 0) → (C, 0) with df(0) = 0 has Lefschetz number zero, we obtain: 4.13 Corollary (A’Campo [1], Lê Dũng Tráng [12]) The Lefschetz number of f : (C^{N}, 0) → (C, 0) is equal to zero provided that df(0) = 0.

Moreover, by induction using Theorem 4.5 we get: 4.14 Corollary (Attributed in [3] to P. Deligne) The zeta function of a germ f : (C^{N}, 0) → (C, 0) has the following form:

ζ_{f}(λ) = 1 + a_{f} · λ^{m(f)} + higher order terms

i.e. ord_{λ}(ζ_{f} −1) ≥ m(f) where m(f) is the multiplicity of f at the origin.

5 The zeta function of global maps.

The aim of this section is to show that our method can also be used in the semi–global case ( i.e. for applications with target a disc and with quasi– projective fibers). The (local) generic linear form l is replaced by a generic Lefschetz pencil. In order to simplify the construction we will present the generic affine case; the interested reader can transfer (using the technique of the projective Lefschetz pencils and dual varieties) to the projective case. 5.1 Let (V, V_{0}) be a pair of affine algebraic varieties in C^{N}, D a disc in C centered in 0 ∈ D and f : (V, V_{0}) → (D, 0) an analytic function such that:

a.) f^{−1}(0) = V_{0}

b.) f over D − {0} is a topological local trivial fibration. 5.2 Any linear form l in C^{N} defines an affine Lefschetz pencil {l = d}_{d∈C} . Suppose that the pair ϕ = (f, l) : f^{−1}(D_{α}) → D_{α} × C ( D_{α} is a sufficiently small disc centered at 0 ) has the following properties:

There exists an analytic set Δ ⊂ D_{α} × C such that:

a.) dimΔ = 1, Δ ∩ {0} × C = {A_{1}, …, A_{t}} is a finite set and the closure Δ̄ of Δ in D̄_{α} × C is compact.

b.) ϕ_{α} : ϕ^{−1}(D_{α} × C − Δ ∪ {0} × C) → D_{α} × C − Δ ∪ {0} × C is a topological fibration. 5.3 Remark The assumption 5.2 in general is not satisfied. Consider for example f : f^{−1}(D) ⊂ C^{2} → D , f(x, y) = x(xy − 1). For any linear form l the pair (f, l) has a component of the discriminant locus which admits {0} × C as asymptote. If f has generic behaviour at infinity, then 5.2 is always satisfied.

Let ϕ = (f, l) be as in (5.2). Then :

c.) P ∘ϕ : ϕ^{−1}(∂D_{α}×C) → ∂D_{α} is exactly the fibration of f. Its geometric monodromy is a lifting of the geometric monodromy of the following fibration of pair of spaces: P : (∂D_{α} × C, ∂D_{α} × C ∩ Δ) → ∂D_{α}.

d.) F = Rϕ_{∗}C is a constructible complex. 5.4 If D_{α} is sufficiently small, then we can choose small neighbourhoods D_{α} × D′_{i} of A_{i} (i = 1, …, t) such that

a.) in D_{α} × D′_{i} the space Δ ∩ D_{α} × D′_{i} has cone-structure

b.)Δ ∩ ∂(D_{α} × D′_{i}) ⊂ ∂D_{α} × D′_{i}

In D_{α} × D′_{i} the algebraic curve Δ ∩ (D_{α} × D′_{i}) ∪ {0} × C determines the graph link L_{i} with diagram Γ_{i} :

[figure: Γ_{i} :; {0} × C (1); D_{α} × {A_{i}}; …; (0); (0); Δ_{i1}; Δ_{is_{i}}]

Consider the Seifert link (Σ(0, 1, …, 1), S_{0} ∪ S_{1} ∪ … ∪ S_{t}) with multiplicities (1, 0, …, 0). Consider the graph Γ(m) , resp. the multilink L(m) obtained by splicing Σ(0, 1, …, 1) with the diagrams Γ_{i}:

[figure: Γ(m):; S_{0} (1); 0; 1; 1; ⋮; D_{α} × {A_{1}}; D_{α} × {A_{t}}; (0); (0); (0); (0); Δ_{11}; Δ_{t1}; Δ_{1s_{1}}; Δ_{ts_{t}}; …; …]

By construction : ζ_{f}(λ) = ζ_{(L,F)}(λ).

Using similar notations as in section 4, we get: 5.5 Theorem

ζ_{f}(λ) = ∏_{v∈V} [ζ_{n_{v}^{∗}(ϕ)}(λ^{m_{Γ}(K_{v})})]^{2−δ(K_{v})}

5.6 We remark that the link L(m) is not an algebraic multilink.

6 Sebastiani–Thom type results

Let g : (X, x) ⊂ (C^{N}, 0) → (C, 0) and h : (Y, y) ⊂ (C^{M}, 0) → (C, 0) be germs of analytic functions and let P : (C^{2}, 0) → (C, 0) be an analytic germ with two variables. Our aim is to determine the zeta function of the composed germ f = P (g, h) : (X × Y, (x, y)) → (C, 0).

Let us consider the Milnor fibrations (4.1) for g and h:

ψ_{ϵ,α}^{g} : B_{ϵ}^{N} ∩ X ∩ g^{−1}(D_{α} − {0}) → D_{α} − {0}
ψ_{ϵ,α}^{h} : B_{ϵ}^{M} ∩ Y ∩ h^{−1}(D′_{α} − {0}) → D′_{α} − {0}

Let D_{η}^{2} be a 2–dimensional disc centered at (0, 0) such that : a.) D_{η}^{2} ⊂ D_{α} × D′_{α} b.) D_{η}^{2} is a Milnor–ball for P.

Denote ϕ = (g, h) : B_{ϵ}^{N} × B_{ϵ}^{M} ∩ X × Y → C^{2} 6.1 Lemma a.) For δ sufficiently small, P ∘ϕ : ϕ^{−1}(P^{−1}(S_{δ}^{1})∩D_{η}^{2}) → S_{η}^{1} is a topological fibration which is fiber homeomorphic to the fibration of f. b.) Moreover, this fibration is fiber homeomorphic to the fibration:

\frac{P ∘ ϕ}{| P ∘ ϕ |} : ϕ^{−1}(S_{η}^{3} − P^{−1}(0)) → S^{1}.

Proof: The proof of the first part is similar as in [21] (where (X, x) = (C^{N}, 0) and (Y, y) = (C^{M}, 0)). The second part follows from the Thom isotopy lemma.

Let Γ(m) be the graph of the multilink

L(m) = (S_{η}^{3} , m_{c}S_{c} ∪ m_{d}S_{d} ∪ m_{1}S_{1} ∪ … ∪ m_{r}S_{r})

where:

m_{c} (resp. m_{d}) = \{ \begin{matrix}0 & if c (resp. d) is not a factor of P. \\ m & if P = c^{m} · P′ (P = d^{m} · P′) and c |/P′ (d |/P′). \end{matrix}

S_{c} = {c = 0} ∩ S_{η}^{3} , S_{d} = {d = 0} ∩ S_{η}^{3} ; P = c^{m_{c}} · d^{m_{d}} · P_{1}^{m_{1}} · . . . · P_{r}^{m_{r}} ( m_{c} ≥ 0 , m_{d} ≥ 0 , m_{i} ≥ 1 for i = 1, …, r) is the (formal) factorization of P in C[[c, d]]; S_{i} = P_{i}^{−1}(0) ∩ S_{η}^{3} for i = 1, …, r.

Again, if we take F = Rϕ_{∗}C_{X×Y}, then ζ_{f} = ζ_{(L,F)}. We note that F is defined on all S_{η}^{3} and is locally constant on S_{η}^{3} −{cd = 0} and S_{η}^{3} ∩{cd = 0}. Obviously, ζ_{(S_{c},F)} = ζ_{h} and ζ_{(S_{d},F)} = ζ_{g}.

Because the fundamental group π_{1}(S_{η}^{3} − {cd = 0}) = Z × Z is abelian we have for (other) non–arrowheads (for more details see [20]):

ζ_{(K_{v},F)}(λ) = ∏_{q} det(I − λM_{K_{v}}^{q})^{(−1)^{q+1}} = ∏_{q} det(I − λE_{q,g}^{l(K_{v},S_{c})} · E_{q,h}^{l(K_{v},S_{d})})^{(−1)^{q+1}}

where E_{q,g} ; E_{q,h} ∈ Aut H^{q}(G × H, C) , G resp. H are the Milnor fibers of g resp. h; E_{q,g} = ⨁_{i+j=q}(M_{g})_{i} ⊗ (I_{h})_{j} ; E_{q,h} = ⨁_{i+j=q}(I_{g})_{i} ⊗ (M_{h})_{j} ; I_{h} resp. I_{g} are the identities; M_{g} resp. M_{h} are the algebraic monodromies of g resp. h.

Using the formula [7, p.96] for the several variable Alexander polynomial Δ of the link L = S_{c} ∪ S_{d} ∪ ⋃_{i=1}^{r} S_{i} (the variables λ_{c}, λ_{d}, λ_{1}, …, λ_{r} of Δ(λ_{c}, λ_{d}, λ_{1}, …, λ_{r}) correspond to the link components S_{c}, S_{d}, S_{1}, ⋯, S_{r} ), we obtain:

∏_{v∈V (Γ(m))} [ζ_{(K_{v},F)}(λ^{m_{Γ}(K_{v})})]^{2−δ(K_{v})} =

= ∏_{q} detΔ(λ^{m_{c}}E_{q,g} , λ^{m_{d}}E_{q,h} , λ^{m_{1}}I , …, λ^{m_{r}}I)^{(−1)^{q}}.

The set V (Γ^{·}(m)) contains V (Γ(m)) and possibly S_{c} if m_{c} = 0 , resp. S_{d} if m_{d} = 0.(In this case m_{Γ}(S_{c}) = (P, c)_{0} resp. m_{Γ}(S_{d}) = (P, d)_{0}. )

If m_{c} ≠ 0 ( resp. m_{d} ≠ 0 ) then m_{Γ}(S_{c}) ( resp. m_{Γ}(S_{d}) ) is not defined; in this case we introduce the notation ζ_{h}(λ^{m_{Γ}(S_{c})}) = 1 (resp. ζ_{g}(λ^{m_{Γ}(S_{d})}) = 1).

By these notations: 6.2 Theorem

ζ_{f}(λ) = ζ_{g}(λ^{m_{Γ}(S_{d})}) × ζ_{h}(λ^{m_{Γ}(S_{c})})×

× ∏_{q} detΔ(λ^{m_{c}}E_{q,g} , λ^{m_{d}}E_{q,h} , λ^{m_{1}}I , …, λ^{m_{r}}I)^{(−1)^{q}}

6.3 Example If f = g + h, then Δ(λ_{c} , λ_{d} , λ_{c+d}) = 1 − λ_{c}λ_{d}λ_{c+d} .

Therefore : ζ_{f}(λ) = ζ_{g}(λ)·ζ_{h}(λ)·∏_{q}( det(1−λ ⨁_{i+j=q}(M_{g})_{i}⊗(M_{h})_{j} ))^{(−1)^{q}} = = ζ_{g}(λ) · ζ_{h}(λ) · ζ_{g×h}^{−1}(λ) = ζ_{g∗h}(λ). (The last term is the zeta function of the monodromy defined on the join space of the fibers).

7 The zeta function of topological series of composed singularities

7.1 Let (g, h) : (C^{n+1}, 0) → (C^{2}, 0) be a germ of an analytic function which defines an ICIS. If ϕ = (g, h) : (X , 0) → D_{η}^{2} is an “excellent representative” of the ICIS [16], then F = Rϕ_{∗}C_{X} on D_{η}^{2} is a constructible sheaf: F is locally constant on D_{η}^{2} − Δ and on Δ − {0} where Δ is the (reduced) discriminant locus of ϕ. Denote the irreducible components of Δ by Δ_{1}, …, Δ_{s} and the corresponding link components in S_{η}^{3} by K_{i}^{Δ} = S_{η}^{3} ∩ Δ_{i} , i = 1, …, s. Recall that each fiber of ϕ has the homotopy type of a bouquet of (n − 1)–spheres, in particular H^{q}(F) = 0 if q ∉ {0, n − 1}. If n ≥ 2 then H^{0}(F) = C_{D_{η}^{2}}. 7.2 Let P : (C^{2}, 0) → (C, 0) be an analytic germ in two variables with factor decomposition P = P_{1}^{m_{1}} ⋯ P_{r}^{m_{r}}. If the discriminant locus Δ of ϕ is given by Δ = (P^{Δ})^{−1}(0), where P^{Δ} = P_{1}^{Δ} · . . . · P_{s}^{Δ} , then after a rearrangement we can suppose that :

(7.3) P = P_{1}^{m_{1}} ⋯ P_{t}^{m_{t}}·P_{t+1}^{m_{t+1}} ⋯ P_{r}^{m_{r}} ; P^{Δ} = P_{1}^{Δ} ⋯ P_{t}^{Δ}·P_{t+1}^{Δ} ⋯ P_{s}^{Δ}.

and the common components are P_{1} = P_{1}^{Δ}, …, P_{t} = P_{t}^{Δ} .

Denote K_{i} = P_{i}^{−1}(0) ∩ S_{η}^{3} for i = 1, …, r. Obviously K_{i} = K_{i}^{Δ} if i = 1, …, t. 7.4 If (g, h) and P are as above we can define f = P (h, g) : (C^{n+1}, 0) → (C, 0). In general f admits a very big singular space Singf^{−1}(0) nevertheless the Milnor fiber F of f has a very nice description [19]. Moreover: 7.5 Theorem [19] a.) If η and 0 < δ << η are sufficiently small, then the maps φ_{P} = P / | P |: (S_{η}^{3} − P^{−1}(0), S_{η}^{3} ∩ Δ − P^{−1}(0)) → S^{1} P : (P^{−1}(S_{δ}^{1}) ∩ D_{η}^{2}, P^{−1}(S_{δ}^{1}) ∩ D_{η}^{2} ∩ Δ) → S_{δ}^{1} are fiber homeomorphic locally trivial fibrations of pairs of spaces. b.) The composition : φ_{P} ∘ ϕ : X ∩ ϕ^{−1}(S_{η}^{3} − P^{−1}(0)) → S^{1} is a topological fibration fiber homeomorphic to the Milnor fibration of f. Moreover, the geometric monodromy of ϕ∘φ_{P} can be considered as a lifting of the geometric monodromy of φ_{p} .

The fiber of φ_{P} can be identified with a (s−t)–punctured minimal Seifert surface of the multilink :

(7.6) L(m, 0) = ( S_{η}^{3} , m_{1}K_{1} ∪ … ∪ m_{r}K_{r} ∪ m̄_{t+1}K_{t+1}^{Δ} ∪ … ∪ m̄_{s}K_{s}^{Δ}) with multiplicities (m_{1}, …, m_{r}, m̄_{t+1}, …, m̄_{s}) , m̄_{t+1} = … = m̄_{s} = 0.

This multilink has the following schematic EN-diagram [7]:

[figure: Γ(P, Δ)(m,) :; …; …; …; (0); (0); K_{t+1}^{Δ}; K_{s}^{Δ}; (m_{1}); (m_{t}); (m_{t+1}); (m_{r}); K_{1}; K_{t}; K_{t+1}; K_{r}]

Obviously we can apply our general theorem (2.8) to compute ζ_{f}. In fact: 7.7 Corollary ζ_{f}(λ) = ζ_{(L,F)}(λ).

Our interest in this section is to compare the zeta function of a series to the zeta function of the “limit” of the series. More precisely: to relate the correction term of the zeta functions with some correction diagrams of the series. 7.8 Definition The topological series of composed singularities belonging to f = P (h, g) consist of all topological types of composed singularities f′ = P′(h, g) such that the multilinks with splice–diagram Γ(P′, Δ)(m′, 0) associated with ((P′)^{−1}(0) ∪ Δ) ∩ S_{η}^{3} (similarly as in (7.6) ) arise as splice of the splice diagram Γ(P, Δ)(m, 0) with some other connected diagrams Γ_{i} ; i = 1, …, r; i.e.:

[figure: Γ(P′, Δ)(m′, 0) =; Γ(P, Δ)(m, 0); Γ_{1}; Γ_{r}; K̃_{1}; K̃_{r}; K_{1}; K_{r}; …]

where Γ_{i}(m_{ij}) has the following form:

[figure: K_{i}˜ (m_{i}˜ ); (m_{i1}) K_{i1}; (m_{is_{i}}) K_{is_{i}}; ⋮]

7.9 Some comments are necessary:

a.) If i = 1, …, t; then K_{i1} = K_{i}^{Δ} . The multiplicity m_{i1} is zero if P_{i}^{Δ} does not appear in the decomposition of P′, otherwise is the multiplicity of P_{i}^{Δ} in the factor decomposition of P′.

b.) We do not require the minimality of our diagrams, but we always require the \underline{semi-minimal algebraicity condition} (S.A.C) : in a portion of a diagram as follows:

[figure: a_{0}; a′_{0}; a_{r}; a_{1}; ⋮; a′_{1}; a′_{s}; ⋮]

we must have: a_{0}a′_{0} ≥ a_{1} ⋯ a_{r}a′_{1} ⋯ a′_{s}.

c.) We permit that some of the splice-diagrams Γ_{i} may be void (i.e. we do not operate in those places).

d.) Recall the splice-conditions [7]:

(7.10) \begin{matrix}m_{i} & = ∑_{1≤j≤s_{i}} m_{ij} · l(K̃_{i}, K_{ij}) & i = 1, …, r. \\ Nm̃_{i} & = ∑_{i≠j≤r} m_{j} · l(K_{i}, K_{j}) & i = 1, …, r. \end{matrix}

7.11 Let Γ_{i}^{·}(m_{ij}) be the diagram obtained from Γ_{i}(m_{ij}) by changing the arrowhead vertices with zero multiplicities in boundary vertices (if i = t + 1, …, r then Γ_{i}^{·} = Γ_{i}) and V^{i} denotes the set of non-arrowhead vertices of Γ_{i}^{·}. Then theorem 2.8. applied to f and f′ implies: 7.12 Theorem

ζ_{f′}(λ) = ζ_{f}(λ) · ∏_{i=1}^{r} ∏_{v∈V^{i}}[ζ_{(K_{v},F)}(λ^{m_{Γ_{i}}(K_{v})})]^{2−δ(K_{v})}

7.13 Example Suppose that for i = 1, …, t the diagram Γ_{i} is the following:

[figure: K̃_{i} (m̃_{i}); (m_{i2}) K_{i2}; (0); K_{i}^{Δ} = K_{i1}; q; 1; 1; ^{¯K̄^{i}}; Γ_{i} :; (q satisfies S.A.C)]

Denote the irreducible components of the critical locus of ϕ over Δ_{i} by C_{i1}, …, C_{it_{i}} and by d_{ij} the topological degree of the branched covering ϕ | C_{ik} : C_{ik} → Δ_{i} ; k = 1, …, t_{i} . Let F_{ik} be the Milnor fiber of the ICIS ϕ : (C^{n+1}, z) → (C^{2}, ϕ(z)) , z ∈ C_{ik}−{0}, and M_{ik} resp. L_{ik} ∈ AutH^{n+1}(F_{ik}, C) the horizontal resp. the vertical monodromies (for more detail see [19, 26] ). Then by the Mayer-Vietoris theorem the correction term given by Γ_{i} is:

ζ_{(K_{i}^{Δ},F)}(λ^{m̃_{i}+qm_{i2}})/ζ_{( ¯K̄_{i},F)}(λ^{m̃_{i}+qm_{i2}}) = ∏_{k=1}^{t_{i}} det(I−λ^{(m̃_{i}+qm_{i2})d_{ik}}L_{ik}M_{ik}^{qd_{ik}})^{(−1)^{n+1}}.

8 Approximation of f by f + ϵφ

8.1 Let f : (C^{n+1}, 0) → (C, 0) be a germ of an analytic function with onedimensional singular locus (Σ, 0). Let (Σ_{1}, 0), …, (Σ_{r}, 0) be the branches of (Σ, 0). It is a well-known fact that the transversal μ–type of f along Σ_{i} is constant at all points of a punctured neighbourhood of 0 in Σ_{i} [10, 15]. This isolated singularity μ–class has Milnor number μ_{i}^{∘} and monodromy M_{i} ∈ Aut(Z^{μ_{i}^{∘}}), which is in fact the horizontal monodromy of f along Σ_{i}. The vertical monodromy L_{i} ∈ Aut(Z^{μ_{i}^{∘}}) is the characteristic mapping of the local system of the sheaf of vanishing cycles [5]. Note that L_{i}M_{i} = M_{i}L_{i} . 8.2 Let φ : (C^{n+1}, 0) → (C, 0) be a germ of an analytic function such that: (IS) f + ϵφ is an isolated singularity for any ϵ in a punctured neighbourhood B_{α}^{·} of the origin.

If the zero set V (φ) of φ contains some components of the singular locus of f, we add a term φ′ to φ with sufficiently high order such that f +ϵ(φ+φ′) is an isolated singularity topologically equivalent to f + ϵφ for ϵ ∈ B_{α}^{·}, and V (φ + φ′) ∩ Σ = {0}.

Hence, (from the topological point of view) we can suppose that φ has the property (IS) and (Φ) : V (φ)∩Σ = {0}. Denote by d_{i}(φ) the topological degree of the restriction of φ on Σ_{i}.

Note that if B_{α}^{·} is sufficiently small, then we can choose a common order of determinacy and all the germs {f + ϵφ}_{ϵ∈B_{α}^{·}} have the same topological type.

Moreover, it is not hard to prove the following: 8.3 Lemma There exists a germ φ′ such that:

(1) the germs f + ϵφ and f + ϵ(φ + φ′) are topologically equivalent;

(2) d_{i}(φ) = d_{i}(φ + φ′) i = 1, …, r ;

(3) the pair (f, φ + φ′) defines an ICIS. 8.4 Our aim is to compare the topological invariants of f + ϵφ with the corresponding invariants of f. By the general theory (Section 4) using the polar curve method separately for f and f + ϵφ we can compute the corresponding zeta functions, but in this case it is hard to compare their Cerf diagrams. But, using Lê ’s attaching formula [11] (or [19]) and our results (section 7 and Lemma 8.3), we get the following: 8.5 Theorem

a.) Let φ be a germ with properties (IS) and (Φ). Then

μ(f + ϵφ) ≤ (−1)^{n}[χ(f) − 1] + ∑_{i=1}^{r} μ_{i}^{∘} · d_{i}(φ)

b.) If in the above formula we have equality, then we also have the following formula for the zeta functions:

ζ_{f+ϵφ}(λ) = ζ_{f}(λ) · ∏_{i=1}^{r} det(I − λ^{d_{i}}L_{i}M_{i}^{d_{i}})^{(−1)^{n+1}}

8.6 Remark Obviously, the “zeta function formula” (b.) implies the “Milnor number formula” ( a.) with equality). Moreover, these are also equivalent to the following equivalent facts:

c.) there exists a germ φ′ such that (8.3.1)-(8.3.3) is valid and (f, φ + φ′) has a “nice discriminant” i.e. {c = 0} does not appear as a tangent cone of the discriminant components, excepting the component {c = 0}.

d.) if φ′ is a germ such that (8.3.1)-(8.3.3) is valid then (f, φ + φ′) must have a “nice discriminant”.

8.7 Remark The equality in (8.5.a) in general does not hold. For example: f(x, y) = y^{3}, φ(x, y) = x^{2k+1}y + x^{N}, N ≥ 4k + 1.

On the other hand, in a lot of cases the equality holds. D. Siersma has conjectured that if the tranversal type of each Σ_{i} is A_{1} and the multiplicity of φ is enough big then the equality holds.

References

[1] A’Campo, N., Le nombre de Lefschetz d’une monodromie, Indag. Math. 35(1973), 113-118.

[2] A’Campo, N., Le groupe de monodromie du deploiement des singularités isolées de courbes planes I, Math. Ann. 213 (1975), 1-32.

[3] A’Campo, N., La fonction zeta d’une monodromie, Commentarii Mathematici Helvetici 50 (1975), 233-248.

[4] Arnold, V. I., Guzein-Zade, S. M. and Varchenko, A. N., Singularities of differentiable maps, Vols I and II, Birkhauser, 1988.

[5] Deligne, P., Le formalisme des cycles évanescents, SGA VII^{2}, Exp. XIII, Lectures Notes in Math. 340(1973), 82-115.

[6] Ehlers, F., Newtonpolyeder und die Monodromie von Hyperflachensingularitaten, Bonner Mathematische Schriften No. 111, Bonn (1978).

[7] Eisenbud, D. and Neumann, W., Three-dimensional link theory and invariants of plane curve singularities, Ann. of Math. Studies 110, Princeton Univ. Press, (1985).

[8] Gusein-Zade, S. M., Intersection matrices for certain singularities of functions of two variables, Funk. Anal. Pril. 8(1)(1974), 11-15.

[9] Gusein-Zade, S. M., Dynkin diagrams of singularities of functions of two variables, Funk. Anal. Pril. 8(4)(1974), 23-30.

[10] Iomdin, I. N., Complex surfaces with a one-dimensional set of singularities, Sibirsk. Mat. Z. 15(5)(1974), 1061-1082.

[11] Lê Dũng Tráng, Calcul du nombre de cycles évanouissants d’une hypersurface complexe, Ann. Inst. Fourier (Grenoble) 23(1973), 261-270.

[12] Lê Dũng Tráng, Le monodromie n’a pas de points fixes, J. Fac. Sci. Univ. Tokyo Sec IA. Math. 22(1975), 409-427.

[13] Lê Dũng Tráng, The geometry of the monodromy, Studies in Mathematics, No 8.

[14] Lê Dũng Tráng, Some remarks on relative monodromy, Nordic Summer School / NAVF, 1976.

[15] Lê Dũng Tráng, Ensembles analytiques complexes avec lieu singulier de dimension un (d’apres I. N. Iomdin), Séminaire sur les singularités, Publ. Math. Univ. Paris VII, 87-95(1980)

[16] Looijenga, E. J. N., Isolated singular points on complete intersections, London Math. Soc. Lect. Note Series 77, Cambridge Univ. Press, (1984).

[17] Milnor, J., Singular points of complex hypersurfaces, Annals of Math. Studies 61, Princeton Univ. Press, (1968).

[18] Milnor, J. and Orlik, P., Isolated singularities defined by weighted homogeneous polynomials, Topology, Vol 9(1970), 385-393.

[19] Némethi, A., The Milnor fiber and the zeta function of the singularities of type f = P (h, g), Compositio Mathematica, 79(1991), 63-97.

[20] Némethi, A., Global Sebastiani–Thom theorem for polynomial maps, J. Math. Soc. of Japan, Vol 43, No 2 (1991).

[21] Némethi, A., Generalised local and global Sebastiani–Thom type theorems, Compositio Mathematica, 80(1991), 1-14.

[22] Némethi, A., Algebraic torsion, zeta function and Dirichlet series for graph links in homology 3-spheres, submitted.

[23] Schrauwen, R., Topological series of isolated plane curve singularities, Eiseignement Mathématique 36(1990), 115-141.

[24] Schrauwen, R., Series of singularities and their topology, Thesis Rijksuniversiteit Utrecht, (1991).

[25] Serre, J.-P., Local fields, Graduate texts in math. 67(1979)

[26] Siersma, D., The monodromy of a series of hypersurface singularities, Comment. Math. Helvetici 65(1990), 181-197.

[27] Suzuki, M., Group theory I, Grundlehren der Mathematischen Wissenschaften 247.

[28] Varchenko, A. N., Zeta functions of monodromy and Newton’s diagram, Invent. Math. 37(1976), 253-262.

(Institute of Math. of the Romanian Academy, Bucharest, Romania) Current address: The Ohio State University, Department of Mathematics,

231 West 18th Avenue, Columbus, OH 43210-1174, U.S.A. e-mail: nemethi@function.mps.ohio-state.edu