Sébastien Marinesque

ITERATIVE RECONSTRUCTION METHODS FOR WAVE EQUATIONS

Some iterative techniques are defined to solve reversible inverse problems and a common formulation is explained. Numerical improvements are suggested and tests validate the methods.

Résumé. Nous définissons des techniques itératives pour la résolution de problèmes inverses réversibles et en fournissons une formulation commune. Après avoir suggéré des améliorations pour leur implémentation, des expérimentations sont présentées qui valident ces méthodes.

Introduction

The classical ThermoAcoustic Tomography (TAT) problem formulates as follows. Consider an object f 0 contained in an open set Ω of R n which emits an acoustic pressure wave at t = 0, considered as a Dirac pulse. This wave is modeled as a solution of:    Lp = 0 in (0, T ) × Ω, p(0) = f 0 , ∂ t p(0) = 0, [START_REF] Agranovsky | On reconstruction formulas and algorithms for the thermoacoustic and photoacoustic tomography[END_REF] where L is an operator modeling an acoustic wave phenomenon. Then this pressure wave is observed (e.g. thanks to piezoelectric sensors) and a set of observations is obtained from the solution p. That can be expressed thanks to an observation operator C mapping a solution p to observations Cp. The inverse TAT problem consists in developing and studying methods to reconstruct f 0 from Cp and to define situations in which this reconstruction is possible.

In the three past decades many techniques have been developed, offering effectual results (see works from authors of [START_REF] Agranovsky | On reconstruction formulas and algorithms for the thermoacoustic and photoacoustic tomography[END_REF], [START_REF] Finch | Determining a function from its mean values over a family of spheres[END_REF] [START_REF] Paltauf | Photoacoustic tomography using a Mach-Zehnder interferometer as acoustic line detector[END_REF], [START_REF] Qian | A new numerical algorithm for thermoacoustic and photoacoustic tomography with variable sound speed[END_REF], [START_REF] Xu | Photoacoustic imaging in biomedecine[END_REF] among others). The new techniques we propose in section 1 rely on the following idea, influenced by [START_REF] Auroux | Back and Forth Nudging algorithm for data assimilation problems[END_REF]: if the system we consider is reversible in time, then the initial state to reconstruct can be seen, backward in time, as a state to reach, so that usual control and filtering techniques can be used to solve this inverse problem. For this purpose, we first used the Back and Forth Nudging algorithm (see [START_REF] Auroux | Back and Forth Nudging algorithm for data assimilation problems[END_REF]) in [START_REF] Bonnefond | Application of a nudging technique for thermoacoustic tomography[END_REF]. With filtering techniques, as the Kalman filter defined in [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF] and one of its reduced rank formulation, the SEEK filter (cf. [START_REF] Pham | A singular evolutive extended Kalman filter for data assimilation in oceanography[END_REF]), we deal here with possible improvements of this method.

Of course, many assumptions are necessary to obtain a favorable observation situation: the way the wave propagates, depending on the media and on the kind of the wave, the final time and the number, size and position of the sensors recording the wave command the information contained in the data (see e.g. [START_REF] Kuchment | Mathematics of thermoacoustic tomography[END_REF], [START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF] and the references therein). Moreover, even if the continuous problem is well set, numerical issues still put up some resistance, as considering noisy data, algorithmic complexity or apparition of spurious high-frequency oscillations during numerical implementations (the paper [START_REF] Zuazua | Propagation, observation, and control of waves approximated by finite difference methods[END_REF] surveys this point).

Observation and stabilization of the wave equation

We assume that L = ∂ tt -∆ is the D'Alembert operator. If necessary, we can suppose that f 0 = 0 and consider the stabilization problem for the initial value problem related to L instead of the inverse TAT problem. Similar considerations are also valid for linear variable speed (reversible) wave equations. Define p = p p ′ , for any p ∈ C 0 ((0, T ); H 1 0 ) ∩ C 1 ((0, T ); L 2 ), and

A = 0 I ∆ 0 on H = H 1 0 × L 2 , then D(A) = (H 1 0 ∩ H 2 ) × H 1 0 and equation (1) writes:    p ′ = Ap, p(0) = p 0 = f 0 0 . (2)
The observations are defined in a Hilbert space U . It is convenient, when working with wave equations, to consider the time derivative of the observations, so that we assume that C ∈ L(L 2 , U ) and use equally Cp ′ or Cp. In the practice of TAT, we only get observations from p and use its time derivative when needed.

Concerning stabilizability and controllability of wave equations, we have the fundamental criterion:

Definition 1.1. The observation inequality is satisfied if there exists T, M > 0 such that:

T 0 Cp ′ 2 U dt ≥ M p 0 2 H , (3)
for all p 0 ∈ H, where p is the solution of (1) with initial data p 0 . Scalar M is called observability constant.

Indeed, in [START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF], one finds the following result:

Proposition 1.2. The three following propositions are equivalent: (i) The observation inequality (3) is satisfied.

(ii) For every positive-definite self-adjoint operator T ∈ L(U ), the operator A -C ⋆ T C generates an exponentially stable C 0 -semigroup on H. (iii) The system (A, C ⋆ ) is exactly controllable.

Many geometrical interpretations to the observation inequality have been presented, mostly known as Geometric Optics Condition (GOC from [START_REF] Bardos | Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary[END_REF]). In particular, these results explain this heuristic situation: when Cp ′ = 1 ω p ′ , where ω is an open subset of Ω, then enough energy from p 0 has to pass through ω to get enough information to reconstruct p 0 . It depends on many parameters such as the position of the sensors, the speed map of the wave equation, the final time T , etc.

In this context, we introduce a first reconstruction method, the Kalman-Bucy filter.

The Kalman-Bucy filter

We recall the main results concerning the (continuous) Kalman-Bucy filter. It yields a way to approximate the real state that minimizes the error variance in the following situation (see [START_REF] Kalman | New result in linear filtering and production theory[END_REF]):

Assume that the true state solves the linear differential equation x t ′ = Mx t + ν and the theoretical model is governed by x f ′ = Mx f . Given data y o , we denote the observation error by ε, so as ε = y o -Cx t . Errors ε and ν are null mean white Gaussian noise processes and their respective covariance matrices are Q and R. Definition 1.3. Given x f (0) and P f (0), the Kalman-Bucy filter consists in the two following differential equations, one to estimate the state x and one for the covariance matrix P, of a differential Riccati type:

x ′ = Mx + PC T R -1 (y o -Cx) , P ′ = MP + PM T + Q -PC T R -1 CP T ,
and the Kalman gain K is given by

K = PC T R -1 .
Let us explain some links between subsections 1.1 and 1.2. In the filters formulations, the feedback is realized thanks to an operator written P C ⋆ T C as feedbacks write C ⋆ T C and P = Id H in subsection 1.1. Thus one can consider here that P weights the feedback in comparison to the model. See [START_REF] Zabczyk | Remarks on the algebraic Riccati equation in Hilbert space[END_REF] for some results similar to Proposition 1.2 about feedbacks P C ⋆ T C.

We are studying different ways to define the stabilizing operator P , first with the nudging operator P C ⋆ T C = kC ⋆ C, where k > 0, that is in the framework of Proposition 1.2, then with filters. It leads to the following back and forth reconstruction algorithms.

Iterative initial data reconstruction methods for inverse problems

We get benefits from the reversibility of the wave equations, go back in time and use data again during a backward evolution, from t = T to t = 0, to deduce an approximation of f 0 . Such an idea has lead D. Auroux and J. Blum to define the Back and Forth Nudging algorithm in [START_REF] Auroux | Back and Forth Nudging algorithm for data assimilation problems[END_REF], then D. Auroux and E. Cosme improved it with the Back and Forth SEEK (from private communication, see below for a description of the BF-SEEK). These techniques can be formulated for any sequences of positive operators (P f k ), (T f k ), (P b k ), (T b k ) as follows: Definition 1.4. Given a set of observations Cp o and a rough estimate p 0 , define p b 0 (0) = p 0 . An iterative reconstruction method consists in iterating a back and forth process. The forward solution is given by:

∂ tt p f k = ∆p f k -P f k C ⋆ T f k C(p f k -p o ) in (0, T ) × Ω, (4)
with initial data p f k (0) = p f k-1 (0) and p f k ′ (0) = 0 and the backward solution solves:

∂ tt p b k = ∆p b k + P b k C ⋆ T b k C(p b k -p o ) in (0, T ) × Ω, (5)
with

final data p b k (T ) = p f k (T ) and p b k ′ (T ) = p f k ′ (T ). The process is iterated for k ≥ 1. Concerning P C ⋆ T C, it is left implicit that time or space derivatives of (p f k -p o ) and (p b k -p o ) can be considered. The sign preceding P b k C ⋆ T b k C(p b k -p o
) changes for convenience of notation as one can notice that, when T contains a first-order time derivative, then the backward equation ( 5) writes forward:

∂ tt p b k = ∆p b k -P b k C ⋆ T b k C(p b k -p o ),
thanks to the variable substitution t → T -t. Finally, the correcting term still helps to stabilize the system. Note that unlike many usual methods, the model is considered here as a weak constraint, which can be useful since it may not be well known (e.g. refer to [START_REF] Jin | Thermoacoustic tomography with correction for acoustic speed variations[END_REF] about simulation of inhomogeneous acoustic speed model and relative issues in TAT).

Finally, this kind of algorithms yields successive estimates (p b k (0)) k≥0 of the initial object to reconstruct. As explained by Proposition 1.2 one knows that, under favorable observation conditions, both forward and backward equations are related to exponentially stable semigroups, that leads to the convergence of the algorithm.

Numerical experiments

Discretization and numerical considerations

Consider the 1-D domain Ω = (-1/2, 1/2) and (0, T ) = (0, 1) both uniformly gridded with steps δx = 1/100 and δt = 1/200. We deal with the classical finite difference time domain theta-scheme (θ-FDTD) to simulate the wave equation. Sensors are located periodically every δ data grid points from the first one.

Upper discussion about observability, stabilizability and controllability have their matching in this situation: a discrete observation condition occur, similar to (3), which is equivalent too to the discrete stabilizability of the system (we omit details, see [START_REF] Zuazua | Propagation, observation, and control of waves approximated by finite difference methods[END_REF] and the reference therein when θ = 0). In order to compensate for possibly damaged observation conditions, due either to sensors configuration or to noise, we carry out the solution suggested in [START_REF] Ramdani | Uniformly exponentially stable approximations for a class of second order evolution equations, Application to LQR problems[END_REF], adding an artificial viscous heating attenuation term in the θ-FDTD, that leads to:

p n+1 -2p n + p n-1 δt 2 = ∆ θ δx (p n-1 , p n , p n+1 ) + ε∆ δx p n -p n-1 δt ± P C ⋆ T C p n-1 -p o n-1 p n -p o n , where ∆ θ δx (p n-1 , p n , p n+1 ) = θ∆ δx p n-1 + (1 -2θ)∆ δx p n + θ∆ δx p n+1 , θ = 0.25, ε = 0 or ε = (δx) α , α ∈ (1, 2
), and ± stands to express both forward and backward implementations.

The term

p n-1 -p o n-1 p n -p o n
T allows us to consider derivatives of the correcting term in the feedbacks.

From Kalman to SEEK filter

Only main results about the Kalman filter are given and we describe then how to derive the SEEK filter from it (as in [START_REF] Rozier | A reduced-order Kalman filter for data assimilation in physical oceanography[END_REF]). These algorithms divide into two steps, a forecast one and an analysis one, in which are taken account the observations to correct the forecast, and two different kinds of parameters are considered, first the states (x f and x a ), then the relative error covariance matrices (P f and P a ) or their square root (S f and S a ). Definitions of the Kalman and SEEK filters are explained respectively on left and right columns below. , one iterates:

Kalman Filter (KF) SEEK Filter Analysis step Analysis step

K n = P f -1 n + C T R -1 C -1 C T R -1 , K n = S f n I r + (CS f n ) T R -1 (CS f n ) -1 (CS f n ) T R -1 , x a n = x f -K n Cx f n -y o n , x a n = x f n -K n Cx f n -y o n , P a n = [I n -K n C] P f n , S a n = S f n I r + (CS f n ) T R -1 (CS f n ) -1/2 , Forecast step Forecast step x f n+1 = Mx a n , x f n+1 = Mx a i , P f n+1 = MP a n M T + Q. S f n+1 = MS a n .
where I k is the k × k identity matrix (in the discrete state space if k = n and in the reduced rank space if k = r) and K n is the Kalman gain, minimizing the trace of the error covariance on x a i , or the reduced rank Kalman gain in SEEK.

Theoretically, one has

P f n = S f n S f T n +Q in SEEK, but since Q is not well known, we define P f n = (1+γ)S f n S f T n
, where γ ∈ (0, 1), which avoids an additional decomposition in SEEK. It is set similarly in KF.

In KF, if the state space has a range of n, the forecast step necessitates 2n model evolution steps to get the forecast error covariance matrix from the analysis error covariance matrix, such that a less optimal gain may be considered to reduce calculation cost. This is the purpose of the SEEK to yield such a gain by considering a reduced rank gain more simple to obtain. Since the error covariance matrices are symmetric and positive-definite, Pham et al. [START_REF] Pham | A singular evolutive extended Kalman filter for data assimilation in oceanography[END_REF] suggested to consider the following decomposition: if P is a symmetric positive-definite n × n matrix whose r ≤ n larger eigenvalues are λ 1 ≥ . . . ≥ λ r > 0 and their corresponding eigenvectors V 1 , . . . , V r , the reduced decomposition of P is defined as the n × r matrix S = [

√ λ 1 V 1 . . . √ λ r V r ].
Since SEEK consists in using the reduced decomposition S f of P f , r model runs define the forecast error covariance matrix.

Results

Four methods are compared: Time Reversal (TR), Back and Forth Nudging (BFN), Back and Forth SEEK (BF-SEEK) and Kalman Filter (KF). Let us set R = 0.3I, γ = 0.01 for both BF-SEEK and KF, and the reduced rank r = 120, which is reduced when Rank I + (CS f i ) T R -1 (CS f i ) < r. An additive white Gaussian noise of level ν is added to the data when said. The nudging gain is set to 0.9δt for implementation limitations.

Table 1 and Fig. 1 and2 show RMS errors and some relative reconstructions obtained in various situations. The object to reconstruct is shown on the upper left part of figure 1 1. RMS errors and number of iterations to converge (in brackets) for 10 (δ data = 10), 2 (δ data = 99) and 1 (δ data > 99) sensor(s), noisy (ν = 30%) or noiseless data, with or without the numerical attenuation δx α . As KF reacts quite well to noise addition, it shows much more sensitivity to the number of sensors and easily fails. BF-SEEK offers obvious improvements for both calculation cost and reconstruction error (Fig. 1). When 2 sensors are left, one can observe possible interesting effects of the attenuation term against noise with TR and BF-SEEK. Nevertheless, it may damage the reconstruction (with BFN) or have insignificant consequences (with TR). When only 1 sensor is left, only BFN and BF-SEEK are robust enough to yield a good approximation of the object, but BF-SEEK needs to be corrected with the attenuation to keep stable (Fig. 2).

Conclusions

A common formulation for iterative stabilization of reversible evolution systems is given. It is used to define methods which solve some inverse problems for wave equations. Experiments show that the techniques we introduced may offer an alternative to usual inverse methods for TAT problem. In applications, knowledge about the quality of the sensors, the background and the model approximations can be used by filters, but in this case they need to be precisely tuned.

The use of an artificial attenuation is motivated by good results, and allows one to consider lossy medium in back and forth implementations when the physical loss does not exceed the numerical attenuation.

When space dimension increases, we first face an excessive calculation cost (one BF-SEEK iteration with rank 60 equals almost one thousand BFN in computation time). So one would get interested in hybrid methods, e.g. getting first estimates with TR and BFN and then reconstructing the solution with filters. A solution is offered by Perfectly Matched Layers to reduce the space domain of calculation and for which a first order discrete scheme formulation is necessary for the filters.

Definition 2 . 1 .

 21 Given x f 0 , P f 0 and S f 0 = P f 1/2 0

Figure 1 .

 1 Figure 1. Object to reconstruct and reconstructions by TR, BFN, BF-SEEK and KF for δ data = 10 and ν = 30%.

Figure 2 .

 2 Figure 2. BF-SEEK reconstructions. Case δ data = 99 and ν = 30%: α = 0 (a) and α = 2 (b). Case δ data > 99: α = 0 (c) and α = 1.8 (d).

Table

  then, to the right, one sees TR and BFN reconstructions, followed by BF-SEEK and KF reconstructions.

	Settings	TR	BFN	BF-SEEK	KF
	δ data = 10	5.4 0.5 (100)	0.6 (3)	6
	δ data = 10, ν = 30%	38.8 16.3 (5)	10.3 (2)	15.7
	δ data = 99	8.3 2.6 (100)	6.3 (1)	44.2
	δ data = 99, ν = 30%	14.1 10.8 (14)	19.9 (1)	46.4
	δ data = 99, ν = 30%, α = 2 9.8 20.7 (8)	12.2 (1)	42
	δ data > 99	51.1 20.7(7)	83.3 (3) 31.8 if α = 1.8 (3)	95

Université Paul Sabatier, Institut de Mathématiques de Toulouse, 118 route de Narbonne, F-31062 Toulouse Cedex c EDP Sciences, SMAI


Teissier
email: teissier@math.jussieu.fr

In order to obtain invariants of the geometry of germs of complex hypersurfaces "up to equisingularity" we extend the map µ ("jacobian multiplicity") which to a hypersurface associates its Milnor number to a map ν j ("jacobian Newton polygon") which takes values in a monoid which is, in a way, the simplest monoid after the integers: the monoid of Newton polygons.

We show below some geometric reasons to do this, and summarize some results which show that ν j ([X 0 ]) unites together many invariants of hypersurfaces.

In an appendix, we sketch some of the structure of the monoid of Newton polyhedra, with emphasis on the polygon (d = 2) case, and show some of the connections with the theory of multiplicities, mixed multiplicities, and integral closure of ideals.

The results and notations of the appendix are used in the text.

INTRODUCTION

We are interested 1 in finding invariants of the geometry "up to equisingularity" (see below) of a complex-analytic hypersurface f (z 1 , ..., z n ) = 0 in C n , in the neighborhood of a singular point which we assume to be 0 ∈ C n .

To obtain invariants, we combine two ideas:

(1) The first is to compare the foliation defined by f (in a neighborhood of 0 ∈ C n ) -that is, the foliation having as leaves the level hypersurfaces f = t, with the foliation defined by some "known" function g : (C n , 0) → (C, 0) for example a general linear function on C n : We can extend this idea to the use of "known" foliations of codimension ≥ 1, say defined by a regular sequence (g 1 , ..., g k ) on C{z 1 , ..., z n } that is, having leaves of dimension n -k defined by g 1 = u 1 , ..., g k = u k .

The most natural way of comparing two foliations is to study the space of points where the leaves are not in general position, that is, in our case, the zero-space of the differential form df ∧dg 1 ∧• • • dg k . This space is defined by the ideal generated by the (k + 1)-minors of the jacobian matrix of (f, g 1 , ..., g k ) and is nothing but the critical subspace of the map:

p : C n → C k+1 (coordinates t, u 1 , ..., u k ) defined by t • p = f , u i • p = g i (1 ≤ i ≤ k).
If we assume that f = g 1 = • • • = g k = 0 has an isolated singularity, this critical subspace C is finite over C k+1 and therefore we can define its image space D ⊂ C k+1 (by the Fitting ideal F 0 (p * O C )) which is a hypersurface in C k+1 (see [START_REF] Teissier | The hunting of invariants in the geometry of discriminants[END_REF], §1). It is simpler to study D than to study C, and to do it we introduce the second idea:

(2) The second idea is to study a hypersurface by means of its Newton polyhedron. Recall that the Newton polyhedron of a power series, say

H(v 1 , ..., v N ) = A∈N N c A v A
is defined to be the (boundary of the) convex hull in N N of the set cA =0 (A + N N ).

The Newton polyhedron is not at all invariant by change of the coordinates v 1 , ..., v N and one must always specify which coordinates one chooses to compute it.

The procedure we propose is to take as invariants of f the collection of Newton polyhedra of the equations defining the discriminants D of maps p as in [START_REF] Teissier | Cycles évanescents, sections planes et conditions de Whitney[END_REF], with respect to the specified coordinates (t, u 1 , ..., u k ) and for well chosen sequences (g 1 , ..., g k ).

Here are three basic examples: a) Case k = n, g i = z i (1 ≤ i ≤ n): then our map p : C n → C n+1 and its discriminant is nothing but its image, which is the hypersurface in C n+1 defined by: t -f (u 1 , ..., u n ) = 0. In this case, our D is the graph of f , and its Newton polyhedron is obtained directly from the Newton polyhedron of f in the coordinates z 1 , ..., z n in a way which the reader can see immediately . So this case is only the direct application of (2) to f without going through [START_REF] Teissier | Cycles évanescents, sections planes et conditions de Whitney[END_REF]. b) Case k = 1, g 1 = z 1 , a general linear function on C n taken as coordinate; here "general" means (cf. [START_REF] Teissier | Cycles évanescents, sections planes et conditions de Whitney[END_REF], Chap II) that the direction of the hyperplane z 1 = 0 is not a limit direction of tangent hyperplanes to f = 0 at non-singular points. For simplicity, we assume that f (z 1 , ..., z n ) = 0 has an isolated singularity. In any case what we have to study now is the plane curve D ⊂ C 2 , discriminant of the map p : C n → C 2 defined by t = f , u 1 = z 1 . From now on, I use freely the notations and results of the Appendix on Newton polygons at the end of this note.

The Newton polygon of our plane curve D in the coordinates (t, u 1 ) is given by

N (D) = l q=1 e q + m q m q ,
where l is the number of irreducible components of the curve in C n defined by ( ∂f ∂z2 , ..., ∂f ∂zn ), m q is the multiplicity at 0 of the q-th component of that curve, and e q + m q is the intersection multiplicity at 0 of the same q-th component with the hypersurface f = 0. Note that by elementary properties of the intersection multiplicity, this last intersection multiplicity is ≥ m q , hence e q ≥ 0. In fact it can be shown that e q ≥ m q , (see [CEW] chap II) because e q is the intersection multiplicity of the q-th component of the curve in C n defined by the ideal ( ∂f ∂z2 , ..., ∂f ∂zn ) with the hypersurface ∂f ∂z1 = 0; the intersection multiplicity (Γ, H) 0 of a curve Γ and a hypersurface H in a non-singular space always satisfies (Γ, H) 0 ≥ m 0 (Γ) • m 0 (H), hence we have e q ≥ m q . c) Case k = 0: we consider f : (C n , 0) → (C, 0), and assume that its fibre (X 0 , 0) = (f -1 (0), 0) has an isolated singularity. In this case, the critical subspace is defined by the ideal ( ∂f ∂z1 , ..., ∂f ∂zn ) = j(f ), and the discriminant is defined in (C, 0) by t µ C{t} where µ = dim C C{z 1 , ..., z n }/j(f ) is the Milnor number of f . Here, the "Newton polygon" is given as the "convex hull" of the set µ + N: it is just the point with abscissa µ. Looking at a), b) and c), we see that even in the special case where we take the g j to be linear functions, the set of Newton polyhedra we obtain contains in particular:

• The polyhedra in R n+1 = R × R n which are obtained as convex join of the point (1, 0, ..., 0) with a Newton polyhedron of f , in all possible coordinate systems. • Polygons in R 2 , about which we shall see more below.

• The Milnor number of f . Our idea is to consider the monoid of integers, where the invariant "Milnor number of f " takes its values, as a special case of others monoids, namely the monoids of Newton polyhedra. One of the advantages of this viewpoint is that it gives a natural frame to extend the results for isolated singularities to the general case, but we will not discuss this here, and will restrict ourselves to the isolated singularity case. §1. In this lecture, I only want to list some of the invariants which are obtained by considering the case b) above, i.e., k = 1 and a general linear function. Obviously this is the simplest case after k = 0, i.e., the invariant we obtain (if indeed it is an invariant...) is the simplest one after the Milnor number.

First of all we must make clear that the Newton polygon we study can be obtained, up to elementary transformations, in these different ways: first, of course there is the way given in b) above:

i) l q=1
e q + m q m q is the Newton polygon of the discriminant of the map

C n → C 2
given by t = f, u = z 1 (in the coordinates t, u).

ii)

l q=1

e q + m q m q is the Newton polygon of a general vertical plane section of the discriminant of a miniversal unfolding of f (or of a versal deformation of (X 0 , 0) = (f -1 (0), 0), in the coordinates given by the natural decomposition of the miniversal unfolding space C µ = C × C µ-1 . See [START_REF] Teissier | The hunting of invariants in the geometry of discriminants[END_REF].

iii)

l q=1
e q m q is the Newton polygon of the image of the curve Γ in C n defined by the ideal ( ∂f ∂z2 , ..., ∂f ∂zn ) by the map ϕ : C n → C 2 given by t = ∂f ∂z1 , v = z 1 , the Newton polygon of the plane curve ϕ * (Γ) is taken in the coordinates t, u. [Here, the image ϕ * (Γ) is defined by the Fitting ideal F 0 (ϕ * O Γ ), see [START_REF] Teissier | The hunting of invariants in the geometry of discriminants[END_REF]. It is important that the images should be compatible with base change].

We shall work with this last polygon e q m q , but clearly the datum of any of the three is equivalent to the datum of any other. We shall denote e q m q by ν j ([X 0 ]). The fact that it depends only upon X 0 follows from the results in the appendix which imply that (with the notations of the appendix):

ν j ([X 0 ]) = ν j(f ) (m),
where j(f ) = ( ∂f ∂z1 , ..., ∂f ∂zn ), m = (z 1 , ..., z n ) ⊂ C{z 1 , ..., z n }. The first basic fact is that ν j ([X 0 ]) is in fact an invariant of equisingularity, for some notion of equisingularity. Definition 1.1. Let (X 0 , 0) ⊂ (C n , 0) be a germ of a complex-analytic hypersurface. For each integer i, 0 ≤ i ≤ n, consider the Grassmanian G (i) of i-dimensional linear spaces through 0 in C n , and min H∈G (i) µ(X 0 ∩ H, 0). It can be proved that the set of i-planes for which this minimum is obtained is a Zariski open dense subset of G (i) . The corresponding value is denoted by µ (i) and called the Milnor number of a general i-plane section of X 0 .

Note that it can be shown that µ (i) is also the smallest possible Milnor number for the intersection of X 0 with a non-singular subspace of dimension i through 0. In all this we agree that if the singularity is not isolated, its Milnor number is +∞.

In the same vein, one can easily define the topological type of a general i-plane section of X 0 (as embedded in the i-plane). Definition 1.2. We call the total topological type of (X 0 , 0) the datum of the topological type of all the general i-plane sections of X 0 , (0 ≤ i ≤ n).

Theorem 1.1. (See [START_REF] Teissier | Variétés polaires I, invariants polaires des hypersurfaces[END_REF]) Let π : (X, 0) → (S, 0) be a family of (n -1)-dimensional complex analytic hypersurfaces with isolated singularities, where S is non-singular. Assume that each fibre X s = π -1 (s) contains a point x(s) such that (X s , x(s)) has the same total topological type as (X 0 , 0). Then, for a sufficiently small representative of π, we have that x(s) is the only singular point of X s , that there exist a complex analytic section σ : S → X of π with σ(s) = x(s) and that:

ν j ([X s ]) is independent of s ∈ S
In short, if we have a family of hypersurfaces where the total topological type is constant, ν j ([X s ]) is constant.

It should be remarked here that any "reasonable" notion of equisingularity should imply the condition above, so that our ν j ([X 0 ]) will be constant in an equisingular deformation, once we know what that term should mean. We remark also that the condition "total topological type constant" implies, and in fact is equivalent to the condition: the sequence µ * (X s ) of the Milnor number of general i-plane section of

X s at x(s) is independent of s, i.e., µ * (X s ) = (µ (n) (X s ), µ (n-1) (X s ), ..., µ (1) (X s ), µ (0) (X s ))
is constant. §2. Now here is a partial list of the geometric features of X 0 which one can read from ν j ([X 0 ]) and which are therefore constant in a deformation where the total topological type is constant:

1) The length ℓ(ν j [X 0 ]) is equal to µ (n) (X 0 ), the Milnor number of X 0 .

2) The height h(ν j [X 0 ]) = µ (n-1) (X 0 ), the Milnor number of a general hyperplane section of X 0 .

3

) Therefore ℓ(ν j [X 0 ]) + h(ν j [X 0 ]) = µ (n) (X 0 ) + µ (n-1) (X 0
) is equal to the diminution of class which the presence of singularity isomorphic with (X 0 , 0) imposes on a projective hypersurface.

This means that if X ⊂ P n is a projective hypersurface such that the closure in Pn of the set of points representing the tangent hyperplanes to X at non-singular points is again a hypersurface X ⊂ Pn (the dual hypersurface of X) we have, setting d = degree of X, ď = degree of X and assuming that X has only isolated singular points, the formula (cf.

[13], Appendix 2):

ď = d(d -1) n-1 - x∈SingX (µ (n) (X, x) + µ (n-1) (X, x))
Remark 1.1. For a plane curve, it gives, remarking that µ (1)

(X 0 ) = m 0 (X 0 ) -1: ď = d(d -1) - x∈SingX (µ(X, x) + m(X, x) -1)
where µ(X, x) is the Milnor number at x and m(X, x) is the multiplicity.

So far, we have used from our Newton polygon only very trivial features, namely its height and length, and the gist of theorem 1.1 is as follows: if we look at a family of hypersurfaces, and the corresponding family of image curves (as in iii) of §1), we know that the Newton polygons of the fibers of that family of curves all have the same length and height since these are µ (n) and µ (n-1) respectively. But this by no means implies in general that the Newton polygon is constant. For example consider the family of plane curves (depending on λ) t a -u b + λt p u q = 0 for given a, b, say ≥ 3, where p > 0, q > 0,

p a + q b < 1.
For all values of λ the Newton polygon has height a and length b, but it is not constant. We see that our family of curves is of a rather special type, since as soon as height and length are constant the whole polygon is constant. Probably this is linked with the fact that the normalization of the discriminant of a versal unfolding is non-singular.

Anyway, now come some results which do use slopes of edges of the jacobian Newton polygon, albeit mostly that of the last edge:

4) Given f (z 1 , ..., z n ) with isolated singularity at 0 and denoting by m the maximal ideal of C{z 1 , ..., z n }, consider the invariant e q m q associated with f as in §1, iii).

Then we have: For an integer N, the following properties are equivalent:

(a). N > sup q ( eq mq ) (b). Any function g ∈ C{z 1 , ..., z n } such that g -f ∈ m N +1 has the same topological type as f (as a germ of mapping (C n , 0) → (C, 0)). 5) The smallest possible exponents in the Łojasiewicz inequalities

|gradf (z)| ≥ C 1 |f (z)| θ1 , |gradf (z)| ≥ C 2 |z| θ2 (as z → 0)
are given by: θ 1 = sup q ( eq eq+mq ), θ 2 = sup q ( eq mq ). 6) In the note [START_REF] Teissier | Sur la version catastrophique de la règle des phases de Gibbs[END_REF], I introduced the following invariant of f : δ(f ), or δ(X 0 , 0), is the maximum number of singular points which can appear in the same fibre of an arbitrary small perturbation of f (resp. deformation of X 0 ). Mr. I.N. Iomdin showed that.

sup q ( e q m q ) < 2δ ≤ µ (n) + µ (n-1)
In fact, he showed that as long as k < sup q ( eq mq ), one can find arbitrarily close to X 0 , a singularity of type A k i.e., isomorphic to:

z k+1 1 + z 2 2 + • • • + z 2 n = 0. In modern notation: k < sup( e q m q ) =⇒ "A k ←-X 0 (arrow=generalization). (hence 2δ(X 0 ) ≥ 2δ(A k ) which is k if k is even, k -1 if k is odd.
However, this bound is not the best possible, i.e., one cannot reverse the implication above: For example the singularity E 6 (z 2 1 + z 3 2 + z 4 3 = 0) has all eq mq equal to 3 by the results in [START_REF] Teissier | Variétés polaires I, invariants polaires des hypersurfaces[END_REF] but we have

E 6 → A 5 .
Since the eq mq of A 5 are all equal to 5, this also shows that there is no upper or lower semi-continuity of sup q ( eq mq ) in a family of hypersurfaces, in general. (Note that ν j (X t ) is upper-semi-continuous in the sense of polygons "being above".) However , I do not know a counter-example to the possibility that:

"In a family where µ (n) is constant, sup q ( eq mq ) is also constant". (All the unimodular families are, up to suspension, families of curves with µ (2) constant, hence µ * constant, hence constant jacobian Newton polygon.) Remark 1.2. The invariant δ(X 0 , 0) is interesting for the following two reasons: first, in the case where (X 0 , 0) is a plane curve, with ring

O 0 = C{z 1 , z 2 }/(f (z 1 , z 2 )) we have that: δ(X 0 , 0) = dim C O 0 /O 0 where O 0 is the integral closure of O 0 in
its total ring of fractions. Therefore, δ(X 0 , 0) is nothing but the "diminution of genus" which the presence of a singularity isomorphic to (X 0 , 0) would impose on a projective curve. Secondly, in any dimension, δ(X 0 , 0) gives a lower bound for the codimension of the Thom stratum of the origin in the base space of a versal unfolding. This result is one half of the "Gibbs phase rule" (see [START_REF] Teissier | Sur la version catastrophique de la règle des phases de Gibbs[END_REF], [START_REF] Teissier | The hunting of invariants in the geometry of discriminants[END_REF]).

Question: Does the constancy of µ (n) (X t , 0) imply the constancy of δ(X t , 0) for a family of hypersurfaces (X t , 0)?

For the case n = 2, i.e., plane curves, the answer is yes, as was proved in [START_REF] Teissier | Résolution simultanée; II: Résolution simultanée et cycles évanescents, Séminaire Demazure-Pinkham-Teissier sur les singularités des surfaces[END_REF] (Probably the result is still true for curves which are complete intersections, but here we deal only with hypersurfaces).

Remark 1.3. From its definition and because the vertices of a Newton polygon have integral coordinates it is clear that

sup q e q m q ≤ µ (n) (X 0 ).
so that if we like to use only the Milnor number for bounds, we may say that the (µ (n) (X 0 )+ 1)-th Taylor polynomial of f determines the topological type. However the result given above is much sharper; in fact, the equality sup eq mq = µ (n) characterizes singularities of type A k (i.e., with µ (n-1) = 1), as a glance at ν j ([X 0 ]) will show.

7)

Coming back to the study of the foliation of C n by the level hypersurfaces f = t, we now describe some work of Rémi Langevin. Given a C ∞ -manifold of even dimension V 2n ⊂ R M and a point x ∈ V , one will consider a normal vector n ∈ N V,x and the orthogonal projection π n : R M → T ⊕ R.n, where T V,x ⊕ R.n is the affine subspace of R M through x spanned by T V,x and n near x. The image π n (V ) is a hypersurface in this 2n + 1-dimensional affine space, having Gaussian total curvature K n (x). We can take n of unit length, and the average of K n (x) on the unit sphere of N V,x has a meaning. It is called the Lipschitz-Killing curvature of V 2n and depends only upon the induced metric on V 2n . It is denoted by K(x). (See [START_REF] Langevin | Courbure et singularités complexes[END_REF] and [START_REF] Langevin | Courbure, feuilletages et singularités algébriques, Seminar on Singularities[END_REF]).

If we are given a Riemannian manifold M foliated by even-dimensional submanifolds, we can define a map K : M → R by: K(m) = Lipschitz-Killing curvature at m of the leaf of the foliation going through m.

In the case of the foliations by f = t, the level manifolds are of even real-dimension, and all this is applicable, as follows:

Let

B ǫ = {z ∈ C n /|z| ≤ ǫ}, let for ζ ∈ R, ζ > 0, T ζ = {z ∈ C n /|f (z)| < ζ}, and X t = {z ∈ C n /f (z) = t}.
Proposition 1.1. (Langevin, see [START_REF] Langevin | Courbure et singularités complexes[END_REF])

lim ǫ→0 lim t→0 Xt∩Bǫ |K|dv = B n • (µ (n) (X 0 ) + µ (n-1) (X 0 ))
where B n is a constant depending only on n. (dv= volume element) (The double limit must be understood to mean that |t| → 0 "much" faster than |ǫ|). Proposition 1.2. (Langevin, see [START_REF] Langevin | Courbure, feuilletages et singularités algébriques, Seminar on Singularities[END_REF]). For a fixed ǫ, small enough, the integral T ζ ∩Bǫ |K|dw has a Puiseux expansion as a function of ζ, and the first term is given by:

T ζ ∩Bǫ |K|dw = γ ǫ • ζ 2 mq 0 eq 0 +mq 0 + • • • ,
where eq 0 mq 0 = sup q eq mq , and γ ǫ ∈ R is a positive real number (dw: volume element on T ζ ).

The proofs of both results use the fact that the intersection multiplicity of the polar curve with respect to a general hyperplane direction, which is our curve Γ defined by 1) , and that the e q , m q appear in the parametrization of the branches of Γ. (See [START_REF] Teissier | Cycles évanescents, sections planes et conditions de Whitney[END_REF], ch. II, [START_REF] Teissier | Introduction to equisingularity problems[END_REF], ch. III) 8) Next, the filtration on the relative homology H n-1 (X t , X t ∩ H, Z) [where X t : f = t, H : z 1 = 0, where H is general, and we look at everything in a small ball B ǫ ] which is described in [START_REF] Teissier | Introduction to equisingularity problems[END_REF], ch. III suggests that there could be some connection between the exponent of oscilation associated to the critical point of f at the origin, and the sequence of jacobian Newton polygons of f and its restrictions to general i-planes through the origin (1 ≤ i ≤ n). Indeed, since it was proved that all the general hyperplane sections of X 0 are (c)-cosecant (see [START_REF] Teissier | Variétés polaires I, invariants polaires des hypersurfaces[END_REF]) the jacobian Newton polygon of a general hyperplane sections is well defined, and then we can go on to lower dimensions.

( ∂f ∂z2 , ..., ∂f ∂zn ), if z 1 = 0 is a general hyperplane, is equal to µ (n) + µ (n-
Let γ(t) be a horizontal family of homology classes of dimension n -1 in X t (t ∈ D η , η sufficiently small), and let ω ∈ Γ(C n , Ω n-1

C n /Dη ) [where C n stands for f -1 (D η ) ∩ B ǫ ]. Consider the integral

I(t) = γ(t) ω (cf. the paper [6] of Malgrange)
Then for small t there is an expansion

γ(t) ω = α∈Q,0≤q≤n-1 c α,q (γ, ω)t α (log t) q
and the lower bound of the set of α's such that there exist γ, ω, q with c α,q = 0 is called the exponent of the Gauss-Manin connection of f , à la Arnol'd, and denoted by σ(f ). Now let us denote by θ (i) the rational number sup q ( eq mq ) (i) , where

ν j X 0 ∩ H (i) = e q m q (i)
is the jacobian Newton polygon of the intersection of X 0 with a general iplane H (i) through 0 in C n . Question: Does one have the inequality

σ(f ) ≥ n i=1 1 θ (i) + 1 ?
The underlying idea is that there should be a rather simple connection between σ(f ) and σ(f |H) where H is a general hyperplane through 0. A more optimistic version of the same question is the following:

Question: Is there an expression

σ(f ) -σ(f |H) = φ(ν j ([X 0 ]))
where φ is a function from the set of Newton polygons to the rationals? In view of the fact that the behavior of σ(f ) under the Thom-Sebastiani operation is known, i.e., if

f ⊕ f ′ is f (z 1 , ..., z n ) + f ′ (z ′ 1 , ..., z ′ n ′ ) then σ(f ⊕ f ′ ) = σ(f ) + σ(f ′ )
, we are led to the following Question: Can one compute the sequence ν

* j ([X 0 ] ⊥ [X ′ 0 ]) of the jacobian Newton poly- gons ν j (([X 0 ] ⊥ [X ′ 0 ]) ∩ H (i)
) of the sections by general i-dimensional planes of the hypersurface defined by

f ⊕ f ′ = 0 (1 ≤ i ≤ n + n ′ ) from the corresponding sequences ν * j ([X 0 ]) and ν * j ([X ′ 0 ])? 9)
Finally we describe the only result so far which uses the totality of the jacobian Newton polygon, and which shows that at least in special case, the jacobian Newton polygon contains a lot of information.

Let (X 0 , 0) ⊂ (C 2 , 0) be a germ of an irreducible plane curve, given parametrically by z 1 = z 1 (t), z 2 = z 2 (t). Then O X0,0 = C{z 1 (t), z 2 (t)} ֒→ C{t} and it is known (cf. [START_REF] Zariski | Modules des branches planes[END_REF]) that the topological type (or equisingularity type) is completely determined by, and determines, the semi-group Γ of the orders in t of the elements of O X0,0 .

Let Γ = β0 , β1 , ..., βg be a minimal system of generators for this semi-group. Set l i = ( β0 , ..., βi ) (greatest common denominator and define n i by l

i-1 = n i • l i . Since l g = 1 we have l 0 = β0 = n 1 • • • n g . Theorem 1.2. (M. Merle [7]) ν j ([X 0 ]) = g q=1 (n q -1) βq -n 1 • • • n q-1 (n q -1) n 1 • • • n q-1 (n q -1) ,
and therefore ν j ([X 0 ]) is completely determined by Γ and determines it: in this case, ν j ([X 0 ]) is a complete invariant of the topological type of (X 0 , 0).

Jacobian Newton Polygons and quasi-homogeneous singularities:

One should be aware of the fact that the jacobian Newton polygon is not determined by the weights in a quasi-homogeneous hypersurface with isolated singularity: the following example was found by Brianc ¸on and Speder:

Let (X t ) (t ∈ C) be the family of surfaces defined by the equation:

z 3 2 +tz α 1 z 2 +z β 1 z 3 + z 3α
3 where α, β are given integers such that α ≥ 3 and 3α = 2β + 1. Then

ν j ([X 0 ]) = 2β 2 + 2β(2β -2) 2β -2 , ν j ([X t ]) = 2β(2β -1) 2β -1 for t = 0.
Remark 1.4. In this example the usual Newton Polyhedron in the coordinates z 1 , z 2 , z 3 is not the same fot t = 0 and t = 0, however, the plane supporting the only compact face remains the same of course.

An optimistic question is the following: Question: Given a hypersurface consider the totality of its Newton polyhedra with respect to all coordinate systems. From each remember only the support hyperplanes of the compact faces. Is the set of possible configurations of these support hyperplanes determined by the topology (or the "equisingularity class") of the hypersurface?

Check it for the case n = 2.

APPENDIX ON NEWTON POLYHEDRA

Let K be a field, and

f = A∈N k f A z A ∈ K [[x 1 , . . . , x k ]]. Define Supp(f ) = {A ∈ N k |f A = 0}, consider the set ∆(f ) = A∈Supp(f ) (A + N k )
and N (f ) its convex hull in R k , which we call the Newton Polyhedron (in fact the boundary of this convex region is also called the Newton Polyhedron) of the series f. Let N c (f ) denote the union of the compact faces of N (f ).

Remark 2.1. Assume that f ∈ K [x 1 , . . . , x k ] ; the definition above is well adapted to the study of f near the origin. Historically people also considered the convex hull of Supp(f ), which when f is a polynomial is a compact convex body in R k , and used it to obtain information on the singularities of the hypersurface {f = 0} not only at the origin, but at the other singular points, including those at infinity.

Let us define the sum of two Newton polyhedra N 1 and N 2 as follows: N 1 + N 2 is the (boundary of the) convex hull in R k of the set of points of the form p 1 + p 2 where p i lies in the convex region of R k bounded by N i (i = 1, 2). Then the sum is clearly commutative and associative. Exercise 1:

Show that if f 1 , f 2 ∈ K [[x 1 , . . . , x k ]] then N (f 1 f 2 ) = N (f 2 ) + N (f 2 ).
Exercise 2: Define a Newton Polyhedron to be elementary if it has at most one compact face of dimension k -1, and its non compact faces of dimension k -1 lie in coordinate hyperplanes, i.e., it is the standard simplex up to affinity. Say that a Newton polyhedron has finite volume if the volume of the complement in the positive quadrant of the convex region it bounds is finite. Give an example of a polyhedron of finite volume which is not a sum of elementary polyhedra. Our viewpoint is to consider that Newton polyhedra, with the addition described above, constitute a very natural generalization of the monoid of integers.

2.1. Study of the case k = 2 i.e., Newton Polygons. Historically Newton first used the Newton Polygon to describe a successive approximation procedure for the computation of the roots y = y(x) (fractional power series) of an algebraic equation f (x, y) = 0 near 0, assumed to be an isolated singular point of f ∈ K [[x, y]] , K = R or C. Before describing Newton's result, we need some notation: Any Newton polygon enclosing a finite area determines a length ℓ(N ) which is the length of its projection on the horizontal axis (by convention) and a height h(N ), which is the length of its projection on the vertical axis:

h(N ) ↑ Exponent of z 2 Exponent of z 1 J J J J a a a a a a a a a a → ℓ(N )
We need a notation for an elementary polygon as follows: an elementary polygon is entirely described by its length and height, and we write it ℓ(N ) h(N ) :

ℓ(N ) h(N ) ←→ h(N ) Z Z Z Z Z Z Z Z Z Z ℓ(N )
Let N be a Newton polygon enclosing a finite area: it can be written (non-uniquely) as a sum of elementary polygons:

N = l i=1 ℓ(N i ) h(N i )
.

However, if we require that i = j =⇒ ℓ(

h(Nj) the decomposition becomes unique, and we call it the canonical decomposition of N .

First use of the Newton Polygon: The

Newton-Puiseux Theorem. Consider f ∈ C{z 1 , z 2 } (or C [[z 1 , z 2 ]]
) such that f (0, z 2 ) is not the function zero. By the Weierstrass preparation theorem, up to multiplication by a unit, which changes neither the germ at 0 of the zero set {f = 0} no the Newton polygon, we may assume that

f ∈ C{z 1 } [z 2 ].
Since C{z 1 } is an henselian ring, if we denote by K its field of fractions and by v the (z 1 )-adic valuation, v has a unique extension to every finite extension L of K, [and this extension takes its values in 1 d Z where d = [L : K] , by a theorem of Puiseux, but we do not need this, only the fact that it takes values in a submonoid of Q isomorphic with Z].

Let m ρ be the number of roots of f = 0 in a splitting extension L for f, which have a given valuation ρ.

Theorem 2.1 (Newton-Puiseux). I. N (f ) = ρ∈Q m ρ ρ m ρ . II. If f is irreducible in C{z 1 , z 2 } or C{z 1 } [z 2 ] , N (f ) is elementary,

and the canonical decomposition of N (f ) into elementary polygons can be realized by a decomposition

f = f 1 . . . f k . Corollary 2.1. Let U ⊂ C{z 1 } [z 2 ]
be the submonoid (for the multiplication) consisting of unitary polynomial in z 2 , not divisible by z 2 and let N U the set of Newton polygons which meet both axis, a submonoid of the additive monoid of all the Newton polygons (see the second exercise after definition 2.3 below) and let ∼ be the equivalence relation on U defined by: p 1 ∼ p 2 if p 1 and p 2 have the same number m ρ of roots with a given valuation ρ in any finite Galois extension L of K. Then f -→ N (f ) induces an isomorphism of monoids U/ ∼ -→ N U (remark that ∼ is compatible with the multiplication on U ).

2.3. Second use of the Newton Polygon: The Newton Polygon of two ideals.

2.3.1.

Let O be a Cohen-Macaulay reduced analytic algebra, and let n 1 and n 2 be two ideals of O which are primary for the maximal ideal. Consider a representative W of the germ of complex analytic space corresponding to O, and let π : W ′ -→ W be the normalized blowing up of the ideal n 1 . Since O is Cohen-Macaulay this normalized blowing up is also the normalized blowing-up of an ideal n 1 is an ideal generated by d "sufficiently general" elements of n 1 . Therefore n

[d]
1 is generated by a regular sequence, say (f 1 , . . . , f d ) and its blowing up π 0 : W ′ -→ W can be identified with the restriction of the first projection to the subspace W ′ ⊂ W × P d-1 defined by the ideal

(f i T j -f j T i , 1 ≤ i < j ≤ d). The exceptional divisor is set- theoretically P d-1 = {0} × P d-1 ⊂ W ′ (since n ⊂ m , the maximal ideal of O = O W,0 )
and therefore, each irreducible component of the exceptional divisor D of π (defined by n 1 .O W ′ ) maps to a divisor on W ′ which is set-theoretically P d-1 : the normalization map n : W ′ -→ W ′ induces a surjective map D i,red -→ P d-1 . Let degD i,red denote the degree of this map. Of course it can also be computed directly from n 1 .O W ′ , as follows: setting I = n 1 .O W ′ , a π-ample invertible sheaf, we have

degD i,red = deg(I/I 2 ⊗ OD O Di,red ),
where the degree of an ample invertible sheaf L on X is given by:

χ(L ⊗ν ) = degL (dimX)! ν dimX + O(ν dimX-1 ).
Now since W ′ is normal, and D is a divisor, at a "general" point of each D i,red we can define the order of vanishing of an element of the form f • π, f ∈ O along D i : at a general point z of D i both W ′ and D i,red are non-singular, and D i is defined by v ν .O W ′ ,z where v is part of a local coordinate system. Define v Di (f • π, z) as the highest power of v

dividing f • π in O W ′ ,z .
This number is obviously locally constant, hence depends only upon D i , since z lie in a Zariski open subset of D i , which is connected since D i is irreducible. Therefore we have associated with each component

D i of D an application v Di : O -→ N f -→ v Di (f • π, z)
where z stays in some Zariski open subset of D i . We define, for an ideal n of O, v Di (n) = min h∈n {v Di (h)}.

Definition 2.1. Let n 1 and n 2 be two primary ideals of O. The Newton polygon of n 1 and n 2 is defined to be:

ν n1 (n 2 ) = r i=1 degD i,red v Di (n 1 ) v Di (n 2 ) z 0 | Γj = δ j t mj j + . . .
This implies that we can express the value of z 0 at the intersection points of Γ and f d -v = 0 by a Puiseux expansion as a function of v on each Γ j :

(2) z 0 = ǫ j v m j e j + . . .

In all there are e j intersection points on Γ j , and j e j = e(n 1 ). If we take O = C{z 0 , . . . , z d }, the expansion (2) above tells us that our Newton polygon

ν n1 (m) = r j=1 e j m j
describes precisely the collection of "vanishing rates" of (a "generic" coordinate of) the intersection points of Γ with f d -v = 0, as functions of v, whereas e(n 1 ) is only the number of these points.

O ⊗ C O ′ of the ideal N = n 1 ⊗ 1 + 1 ⊗ n ′
1 is given by: e(N ) = e(n 1 ).e(n ′ 1 ). From our viewpoint it is natural to ask whether one can have a similar product formula for the Newton polygons of two ideals, with a suitable definition of the product of two Newton polygons. The answer is yes: Definition 2.2. Let P = ℓ(P ) h(P ) and Q = ℓ(Q) h(Q) be two elementary Newton polygons; define the product P * Q by

P * Q = ℓ(P )ℓ(Q) min(ℓ(P )h(Q), ℓ(Q)h(P ))
, and for any two polygons P, Q of finite volume, taking decompositions P = i P i , Q = j Q j into elementary polygons, define

P * Q = i,j P i * Q j .
(The result is independent of the choice of decompositions)

Exercise: Check that * is commutative, associative and distributive (!)

Proposition 2.2. Let O, O ′ be two Cohen-Macaulay analytic algebras, n 1 , n 2 in O primary ideals, n ′ 1 , n ′ 2 in O ′ also. Set N 1 = n 1 ⊗ 1 + 1 ⊗ n ′ 1 , N 2 = n 2 ⊗ 1 + 1 ⊗ n ′ 2 , both ideals on O ⊗ C O ′ (
note that we could just as well take the analytic tensor product O ⌢ ⊗ C O ′ ). Then we have:

ν N1 (N 2 ) = ν n1 (n 2 ) * ν n ′ 1 (n ′ 2
). The proof follows the construction given in [START_REF] Teissier | Variétés polaires I, invariants polaires des hypersurfaces[END_REF].

by taking the convex hull of the Minkowski sum of the convex sets bounded by our Newton polyhedra. However, the construction is of course quite similar, and the proof of the above result is obtained by induction on d after reduction to the case whose all the N i are elementary polyhedra, thanks to the following fact: let N be a Newton polyhedron of finite volume, and let σ 1 , . . . , σ k be its (d -1)-dimensional faces. Let h i be the distance from the origin to the hyperplane containing σ i . then we have:

d • Vol(N ) = k i=1 h i Vol(σ i ),
where of course Vol(N ) is the d-dimensional volume of the part of R d bounded by N in the positive quadrant, and Vol(σ i ) is the (d -1)-dimensional volume of the convex set σ i .

An interesting special case is r = 2:

Vol(λ 1 N 1 + λ 2 N 2 ) = d i=0 d i Vol(N [i] 1 , N [d-i] 2 )λ i 1 λ d-i 2 ,
and in the case of polygons, d = 2:

Vol(λ 1 N 1 + λ 2 N 2 ) = Vol(N 1 )λ 2 1 + 2Vol(N [1] 1 , N [1]
2 )λ 1 λ 2 + Vol(N 2 )λ 2 2 . Proposition 2.4. Let N 1 and N 2 be two Newton polygons of finite volume. Then we have of course ℓ(N 1 * N 2 ) = ℓ(N 1 )ℓ(N 2 ), but also:

h(N 1 * N 2 ) = 2Vol(N [1] 1 , N [1]
2 ). Proof: Since clearly h(N 1 * N 2 ) is bilinear with respect to the sum of Newton polyhedra, it is sufficient to prove that h(N

1 * N 1 ) = 2Vol(N 1 ). Now if N 1 = s i=1 ℓ i h i
it is an exercise on the convexity of the Newton polygon to check that:

h(N 1 * N 1 ) = i,j min(ℓ i h j , ℓ j h i ) = 2Vol(N 1 ). Corollary 2.4. If N 1 = i ℓ i h i , N 2 = j ℓ ′ j h ′ j 2Vol(N [1] 1 , N [1] 2 ) = i,j min(ℓ i h ′ j , ℓ ′ j h i ). Corollary 2.5. Let f 1 , f 2 ∈ C{z 1 , z 2 } be such that f 1 (0, z 2 ), f 2 (0, z 2
) are = 0, and satisfy the condition in proposition 2.4 with respect to their Newton polygons N 1 and N 2 . Then we have:

dim C (C{z 1 , z 2 }/(f 1 , f 2 )) = e(f 1 , f 2 ) = 2Vol(N [1] 1 , N [1]
2 ).

(This is because by proposition 2.4, and the definition of the resultant polynomial Res C{z1} f1,f2 (z 2 ), the height of N 1 * N 2 is the valuation of the resultant, which is equal to dim C (C{z 1 , z 2 }/(f 1 , f 2 ))).

(See [START_REF] Teissier | The hunting of invariants in the geometry of discriminants[END_REF], §1).

Remark 2.4. It should be noted that there is no unit element in general for the product * . However there is an interesting subset of the set of Newton polygons, which is stable by * and in restriction to which * has a unit: Definition 2.3. A Newton Polygon P = i ℓ i h i is special if ℓ i ≥ h i for all i.

Exercise: Let 1 = 1 1 . Show that P is special if and only if we have P * 1 = P .

Exercise: Enlarge the set of Newton polygons on which * is defined to all Newton polygons, not only those of finite area, and then show that the Newton polygon one wants to write 1 ∞ is a unit.

The next illustration was added in 2012.

Z Z Z Z Z h ℓ = ℓ h ℓ ′ = ℓ ′ ∞ ℓ h + ℓ ′ ∞ = h Z Z Z Z Z ℓ + ℓ ′ ℓ ′ ℓ h * 1 ∞ = ℓ h Exercise: ℓ h * ∞ h ′ = ∞ ℓh ′ , n n * n ′ n ′ = nn ′ nn ′
We mention in closing that the use of Newton polyhedra, and the use of the theory of multiplicities and integral dependence in the study of singularities are very closely related. One aspect of this is the link between the Newton polygon of two ideals, multiplicity theory and integral dependence which we saw above, but I like to mention also that there is a very close analogy between the theory of multiplicities and the theory of mixed volumes, this time in the sense of Minkowski (see [2], which suggested this to me; I thank Tadao Oda for inducing me to read it).

Here is the analogy: ) then vi = V (K

[i] 1 , K [d-i] 2
) then e i e i-1 ≥ e i-1 e i-2 , (2 ≤ i ≤ d) [START_REF] Teissier | Sur une inégalité à la Minkowski pour les multiplicités[END_REF]

v i v i-1 ≤ v i-1 v i-2 , (2 ≤ i ≤ d) (Fenchel-Alexandrov)
Minkowski type inequality Brünn-Minkowski inequality e(n1.n2) This analogy is not too difficult to explain, in two steps: First take O = C{z 1 , . . . , z d }. Then if an ideal n is generated by monomials, e(n) = d!Vol(N ) where N is the Newton Polyhedron defined by these monomials and from that it is very easy to see that if n 1 , n 2 are both generated by monomials e(n

[i] 1 , n [d-i] 2 ) = d!Vol(N [i] 1 , N [d-i] 2
),

1 ⊂

 1 n 1 where d = dimO, and n [d]

2. 3 . 3 .

 33 The product formula. Let O and O ′ be two analytic algebras (in fact two noetherian local rings), and let n 1 ⊂ O, n ′ 1 ⊂ O ′ be ideals which are primary for the respective maximal ideals. Then the multiplicity in the tensor product

Some

  Cohen-Macaulay analytic algebra O of dimension d Affine space R d An ideal n ⊂ O primary for the maximal ideal A convex body K ⊂ R d of finite volume Product n1.n2 Minkowski (pointwise) sum K1 + K2 Multiplicity e(n) Volume V (K) αi = |α| = d by with αi = |α| = d by e(n ν 1 1 . . . n νr r ) = V (ν1K1 + . . . + νrKr)

Equality ⇐⇒ n a 1 = n b 2

 2 (a, b ∈ N) [12] Equality ⇐⇒ K1, K2 are homothetic [E].

I am grateful to Lilia Alanis López and Carlos Guzman, from CIMAT in Guanajuato, Mexico, who transcribed my old handwritten notes into LaTeX. 1

where the exceptional divisor D of the normalized blowing up π : W ′ -→ W of n 1 is D = r i=1 D i , with D i irreducible. generated by d -1 "general" elements of n. It defines a curve Γ ⊂ W which is reduced. Let Γ = l j=1 Γ j where each Γ j is an irreducible germ of curve in W . By normalization we get a composed map Γ j nj -→ Γ j ⊂ W and therefore, since O Γj,0 = C{t j }, a new order function

Now let us consider an ideal n

for each irreducible component Γ j of Γ. Again we can define v Γj (n) for an ideal n in O.

Using the construction explained in [START_REF] Teissier | On a Minkowski type inequality for multiplicities II[END_REF] it is not difficult to prove:

Proposition 2.1.

(1)

Corollary 2.2. We have the two equalities

where e(n

) are the "mixed multiplicities" introduced in ([1], Chap. I) and in particular e(n 1 ) = r i=1 degD i,red v Di (n 1 ), which yields a "projection formula" which I used, in the case d = 2, in [START_REF] Teissier | Sur une inégalité à la Minkowski pour les multiplicités[END_REF].

Proof:

where h is a "general" element of n 2 , and this is equal to the intersection multiplicity of {h = 0} with Γ, which is e(n

2 )

where n

where the bar denotes the integral closure of ideals.

by the valuative criterion of integral dependence.

The Newton polygon of two ideals as a dynamic version of intersection multiplicities.

We want to explain informally what type of information ν n1 (n 2 ) contains compared to e(n 1 ). Suppose that we have reduced an intersection multiplicity problem to the computation of the multiplicity of an ideal n 1 = (f 1 , . . . , f d ) generated by a regular sequence in a Cohen-Macaulay analytic algebra O. Then by flatness, one may compute e(n 1 ) as the number of intersections of the hypersurface f d -v = 0 with the curve Γ :

) are sufficiently general these will be transversal intersections of nonsingular varieties for v = 0 sufficiently small (i.e., f d -v = 0 meets Γ at non-singular points of Γ, and transversally). Now on a component Γ j of Γ, we have an expansion for

) where t j is the local coordinate on Γ j . Now let m be the maximal ideal of O, and m j = v Γj (m), so there is a z 0 ∈ m such that 2.3.4. Comments on the product of Newton polygons. One could say that the definition of the product of Newton polygons is just ad-hoc for the proposition above. However, it has some interesting features, of which I now show two: First, we try to find an operation on polynomials which induces, bye taking Newton polygons, the product * . Let V be a ring (here C{z 1 }) and consider the following operation on the ring V [T ] (here T = z 2 ).

Res V P1,P2 (T ) = Res U (P 1 (T + U ), P 2 (U )), where Res U means the resultant with respect to the new indeterminate U . We assume now that P 1 and P 2 are unitary. We have that Res P1,P2 (T ) ∈ V [T ] is a polynomial of degree degP 1 .degP 2 and Res V P1,P2 (0) = Res T (P 1 (T ), P 2 (T )). If we go to a splitting extension for P 1 and P 2 of the fraction field of V , and let V be the integral closure of V in it;

where α i , β j are the roots of P 1 , P 2 .

Proposition 2.3. Let N 1 , N 2 be two Newton polygons of finite volume. For "almost all" pairs of elements

For any finite edge γ of the Newton Polygon of P ∈ C{z 1 } [z 2 ] define P γ to be the sum of those terms in P having an exponent which is on γ. Then "almost all" means that for any pair γ 1 , γ 2 of finite edges on N 1 , N 2 , the (quasi-homogeneous) polynomials P 1,γ1 and P 2,γ2 have no common zero outside 0, (in a neighborhood of 0). Then is clearly a Zariski-open condition on the coefficients of

The proof is straightforward from the expression of Res V P,l (T ) and the fact that the Newton polygon does not change when we go to an algebraic extension.

Remark 2.2. It should be emphasized that the binary operation P 1 , P 2 -→ Res V P1,P2 (T ) does not have all the nice properties like associativity, and that it is by no means the only one which gives a new polynomial having (almost always) N (P 1 ) * N (P 2 ) as Newton polygon.

Another feature of N 1 * N 2 is that the height h(N 1 * N 2 ) is twice the mixed volume of N 1 and N 2 in the following sense:

Let N 1 , . . . , N r be Newton polyhedra of finite volume in (the first quadrant of) R d . Then, for λ i ≥ 0 in R there is a polynomial expression for the volume of the polyhedron λ 1 N 1 + . . . + λ r N r (i.e., of the complement in the positive quadrant of the region bounded by r i=1 λ i N i ):

where α! = α 1 ! . . . α d ! and Vol(N ) is (by definition) the mixed volume of index α = (α 1 , . . . , α r ) of N 1 , . . . , N r . We see that Vol(N Remark 2.3. These mixed volumes should not be confused with the mixed volumes of Minkowski which are used in the isoperimetric and other external problems in the theory of convex sets. The sum of Newton polyhedra is not the Minkowski sum, but rather obtained