R El Moubtahij
email: redouane.el.moubtahij@univ-poitiers.fr

B Augereau

C Fernandez-Maloigne

H Tairi
email: htairi@yahoo.fr

A polynomial texture extraction with application in dynamic texture classification

Keywords: Texture extraction, Dynamic Textures, Polynomial decomposition, Video classification

Geometry and texture image decomposition is an important paradigm in image processing. Following to Yves Meyer works based on Total Variation (VT), the decomposition model has known a renewed interest. In this paper, we propose an algorithm which decomposes color image into geometry and texture component by projecting the image in a bivariate polynomial basis and considering the geometry component as the partial reconstruction and the texture component as the remaining part. The experimental results show the adequacy of using our method as a texture extraction tool. Furthermore, we integrate it into a dynamic texture classification process.

INTRODUCTION

Various works, more or less recently, have tackled the problem of image decomposition that still is one of the major aims in image processing. In fact, a lot of domain are concerned, from denoising to pattern recognition. In this paper, we investigate the possibility of having a representation space of the image information, space adapted to video classification and more especially dynamic textures classification. After a brief reminding of the principle of image decomposition, we detail our method of texture extraction : firstly presenting the Polynomial Transform, then the decomposition step. After that, our decomposition method is used to construct a video descriptor for dynamic texture classification.

IMAGE DECOMPOSITION

Yves Meyer [START_REF] Meyer | Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures[END_REF] has proposed a model of image decomposition using the algorithm of Rudin-Osher-Fatemi. [START_REF] Rudin | Nonlinear total variation based noise removal algorithms[END_REF] According to this model, an image is split in two parts, one containing the structure u, the other one containing the texture u. The result is provided by the minimization of the functional

F(u, v) = f F + λ g G (1)
where f ∈ F , g ∈ G and λ is the parameter of the model. More precisely, F is the space of functions with bounded variations and G the space of oscillating functions with the property that more a function is oscillating, more its standard norm g G will be low. This model can be solved numerically due to the formulation proposed in J-F.Aujol, [START_REF] Jean-François | Image decomposition: application to textured images and SAR images[END_REF][START_REF] Jean-François | Structure-texture image decompositionmodeling, algorithms, and parameter selection[END_REF] by the introduction of an additional parameter µ corresponding to the maximum norm of textures in the space G. The use of non-linear projectors defined by A.Chambolle 5 provides the decomposition of the image by an iterative algorithm (see [START_REF] Jean-François | Image decomposition: application to textured images and SAR images[END_REF][START_REF] Jean-François | Structure-texture image decompositionmodeling, algorithms, and parameter selection[END_REF] for more details).

A. Buades 6 has created a method that, as we know, is the fastest and most efficient implementation of the theory given Yves Meyer. [START_REF] Meyer | Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures[END_REF] It is a fast approximate solution to the original variational problem obtained by applying non-linear filtering to the image. For each image pixel, a decision is made whether it belongs to the geometric part or to the texture part. This decision is made by computing a local total variation of the image around the point, and comparing it to the local total variation after a low pass filter has been applied. In fact, edge points in an image tend to have a slowly varying local total variation when the image is convoluted by a low pass filter while textural points instead show a strong decay of their local total variation. After the selection of the points belonging to the geometrical part, the texture part is considered as the difference between the original image and the geometrical part. (See Figure 1 for image decomposition with Buades 6 method). In fact, there is no unique decomposition and the algorithm relies on an important parameter, the scale parameter which is directly related to the granularity of textures distinguished.

IMAGE DECOMPOSITION WITH POLYNOMIAL TRANSFORM

A Bivariate Polynomial (BP) of degree d is a function of x = (x 1 , x 2 ) ∈ R 2 given by

P (x) = (d1,d2)∈[0;d] 2 d1+d2≤d a d1,d2 x d1 1 x d2 2 (2)
with any a d1,d2 ∈ R.

Bivariate polynomial basis

Considering a finite set of pairs D = {(d 1 , d 2 )} ⊂ N 2 , we represent by E D the space of all BP such as a d1,d2 ≡ 0 if (d 1 , d 2 ) / ∈ D and by K D the subset of monomials

K D = K d1,d2 (x) = x d1 1 x d2 2 (d1,d2)∈D (3)
Obviously K D satisfies the linear independence and spanning conditions and so, K D is a basis of E D , the canonical basis. In our context of color image decomposition, we look for bases with more suitable properties such as orthogonality or normality. So, to construct a discrete orthonormal BP finite basis we first have to consider the underlying discrete domain

Ω = x (u,v) = x 1,(u,v) , x 2,(u,v) (u,v)∈D (4)
where D will now represent the set of pairs associated to Ω. Starting from K D we intend to construct a new orthonormal basis applying the Gram-Schmidt process. That implies that we need some product and norm for functions defined on Ω. Given two bivariate functions, F and G, their discrete extended scalar product is defined by

F |G = (u,v)∈D ω(x (u,v) ) F (x (u,v) ) G(x (u,v) ) (5)
with ω a real positive function over Ω [Legendre, Chebichev, Hermite, ...]. Then, the actual construction process of an orthonormal basis

B D,ω = {B d1,d2 } (d1,d2)∈D (6) is a recurrence upon (d 1 , d 2 ) T d1,d2 (x) = K d1,d2 (x) - (l1,l2)≺2(d1,d2) K d1,d2 |B l1,l2 ω B l1,l2 (x) (7)
B d1,d2 (x) = T d1,d2 (x) |T d1,d2 | ω (8)
where ≺ 2 is the lexicographical order and | | ω the norm induced by | ω . The resulting set of B polynomials verifies

B d1,d2 |B l1,l2 ω = 0 if (d 1 , d 2 ) = (l 1 , l 2 ) 1 if (d 1 , d 2 ) = (l 1 , l 2 ) (9)
and so B D,ω is effectively an orthonormal basis with respect to a weighting function ω. A special case, later used in this paper, is the complete base where D exactly represents the set of pairs associated to Ω, that is

D = [0; N 1 ] × [0; N 2 ] ( 10
)
The space E D being dense in the space of functions over Ω, it allows to well approximate any bivariate function I by an appropriate combination of elements of a B D,ω orthonormal basis

P I (x) = {(d1,d2)}⊂D b d1,d2 B d1,d2 (x) (11)
where b d1,d2 is the scalar resulting of the projection b d1,d2 = I|B d1,d2 ω . In fact, with a complete orthonormal basis, the polynomial approximation of I is a first order osculatory polynomial interpolation : for all points of the domain we have P I (x) = I(x). An other nice property is that the projection on polynomial B d1,d2 can be considered as an approximation of the partial derivation ∂ d1 ∂ d2 . Finally and in practice, the discrete projection process supposes that both I and B d1,d2 can be evaluated on the common domain Ω. So, the set of collocation points can be obtain by uniform or non-uniform discretization of given intervals. For example, with [-1; 1] 2 and referring to equation (10), the collocation points obtained by uniform discretization are

x 1,(u,v) = -1 + 2u N 1 x 2,(u,v) = -1 + 2v N 2 (12)

Polynomial Transform

Now we describe the Polynomial Transform algorithm which is founded on piecewise discrete polynomial approximation and the principle of Wavelet Packet. At a given level of this multi-resolution transform, lets consider a function U defined on a domain Ω of size n 1 × n 2 , and a basis B M,ω defined on a support M of size h 1 × h 2 , the transform process is defined as follows :

1. definition of a covering set of the discrete domain Ω with sub-domains Ω M of size h 1 × h 2 This method provides some flexibility especially in the choice of the resolution factors, depending of the subdomains size and of their offset, the transform can be perform with juxtaposed or overlapped sub-domains. Moreover, the choice of the weighting function ω allows, at the same time, to perform a multi-scale and a multiresolution transformation. As an illustration, Figure 2 shows an example of a first level transformation using a 4 × 4 Hermite complete basis.

Image decomposition by partial reconstruction

A Polynomial Transform performed with a complete basis is perfectly reversible. However, it is possible to obtain many kind of approximations by selecting the coefficients during the reconstruction phase, i.e. partial reconstruction. This choice of coefficients may follow various strategies, among them we have : (a) brutal restriction to a given subset, for example the polynomials of degree less than a threshold; (b) restriction based on energies, for example by using the normality of the basis to assimilate the absolute value of its coefficients to a part of the energy of a sub-domain, then sort the coefficients and finally retain a fixed number of these coefficients or those satisfying a certain condition (cf. PCA).

In our case, to decompose the image into geometric and texture component, we assume that the geometrical part is given by a partial reconstruction Ĩ of the original image I in an overlapped Polynomial Transform context. As seen before, this transform is very flexible, so there are many conceivable solutions. In order to get a compromise between quality and computation time, we choose to use an Hermite basis, to set the subdomain size to 3 × 3, with an offset of 2, and to select the three dominant coefficients for each sub-domain. The partial reconstruction of a color image I = (I j ) j=1•••3 , i.e. the construction of the geometrical part, can then be summarize by

Ĩj (x) = 1 c(x) {Ω M x}   Ψ(Ω M ) ω(x M ) (d1,d2)∈P j,M b j,d1,d2 (I j,M ) B d1,d2 (x M )   ( 13
)
where x is a point referring to the global image domain Ω, x M is the same point referring to a given sub-domain Ω M , I j,M the restriction of I j to sub-domain Ω M , P j,M the set of selected polynomials for I j,M approximation, ω the weighting function of the scalar product and b j,d1,d2 the coefficient of the projection of I j,M on the basis polynomial B d1,d2 . A degree of anisotropy Ψ(Ω M ) is assigned to each sub-domain Ω M and c(x) is the sum of x contributions, c(x)

= {Ω M x} Ψ(Ω M ) ω(x M ).
The degree of anisotropy is evaluated according to

Ψ(Ω M ) = 1 1 + λ r ( 14
)
where λ is the largest eigenvalue of a color structure tensor composed with the approximations of partial derivatives, projections on the basis polynomials of degree one

S = j (b j,1,0 ) 2 j b j,1,0 b j,0,1 j b j,0,1 b j,1,0 j (b j,0 ,1 ) 2 (15)
The balance between isotropic and anisotropic reconstruction is adjust by the parameter r that controls the degree of anisotropy in a range of 0.25 for isotropic (gaussian) to 2 for highly isotropic. By doing that, we assure a real color process and avoid marginal treatment deficiencies. Finally, the texture component I T is simply deduce from the partial reconstruction by considering that it is the residual part of the image

I T = I -Ĩ (16)
The results of image decomposition into geometry and texture components by partial reconstruction after an Hermite Polynomial Transform as defined in equation( 13), with the parameter r of equation ( 14) set to 0.75, are shown in Figure 3.

EXPERIMENTAL RESULTS

To illustrate the efficiency of our method, we will use it in a dynamic texture classification scheme which is among challenging research themes in the image analysis field. The database used in our application is Dyntex++ database, 7 see Figure 4. Because it is composed of images that represent textures in motion, it therefore seems appropriate for the present study. Since the videos are not evenly distributed among the 36 classes in this database, the pretreatments [START_REF] Péteri | DynTex : a Comprehensive Database of Dynamic Textures[END_REF] were realized so that each class contains the same number of sequences. We finally have 3600 dynamic textures, grouped into 36 classes and each sequence, i.e. each instance of a dynamic texture, has a size of 50 × 50 × 50 pixels.

In our approach, we use the video sequences without converting them in grey level. Then, we both extract the temporal information provided by the color displacement fields 14 (ν 1 , ν 2 ) and the spatial information provided by the color texture components I T using our method of Polynomial Transform decomposition. Thereafter, we modelize ν 1 , ν 2 and I T by approximating them with a complete Hermite bivariate basis of degree 2. In our experiment the datas were randomly split into two equal size training and test sets. [START_REF] Scovanner | A 3-dimensional sift descriptor and its application to action recognition[END_REF][START_REF] Zhao | Dynamic texture recognition using local binary patterns with an application to facial expressions[END_REF][START_REF] Ehsan | Directional space-time oriented gradients for 3d visual pattern analysis[END_REF][START_REF] Xu | Dynamic texture classification using dynamic fractal analysis[END_REF][START_REF] Baktashmotlagh | Discriminative non-linear stationary subspace analysis for video classification[END_REF] The random split was repeated 10 times and the average classification accuracy is reported in Table 1. The classification is performed with a SVM [START_REF] Chih-Chung | Libsvm: A library for support vector machines[END_REF] as classifier and RBF functions as kernels. From Table 1, we can see that our descriptor performs quite well comparatively to the usual methods of dynamic textures classification. Moreover, the application of our approach provides a very significant gain in computation time because all main computations can be realized through convolutions.

CONCLUSION

In this paper, we have proposed a new approach for texture extraction from color image sequence, by using Polynomial Transformations. Partial reconstruction and global approximation are used to build descriptors used in a classification process of dynamic textures. In addition to the simplicity of implementation, we provide a computing time which is especially fast compared to most of methods based on the theory of Yves Meyer due to the cost of the minimization of the total variation. The experimental results show that the proposed approach achieves a very good recognition rate for the Dyntex++ database. This shows the relevance of our texture extraction method in the context of classification of dynamic textures. In some future, we will continue to improve our image decomposition method in order to extract the noise coefficients ignored in the partial reconstruction of the image. We will also investigate the abilities of a derived method which only relies on three dimensional transformations for our classification process of dynamic textures.

Figure 1 .

 1 Figure 1. A. Buades method with scale parameter set to 3 : (a and d) original images, (b and e) images of geometry, (c and f) images of texture.

2 .Figure 2 .

 22 Figure 2. First level of Polynomial Transform with a 4 × 4 Hermite complete basis.

Figure 3 .

 3 Figure 3. Our decomposition method with r = 0.75 : (a and d) original images, (b and e) images of geometry, (c and f) images of texture.

Figure 4 .

 4 Figure 4. Examples of Dyntex++ database. VLBP (Volume Local Binary Patterns) 8 61.1% SIFT-3D (3 Dimensional Scale Invariant Feature Transform) 9 63.7% LBP-TOP(Local Binary Patterns on Three Orthogonal Planes) 10, 11 71.2%

Table 1 .

 1 Classification results for Dyntex++ database.


Delphine Pol

On the values of logarithmic residues along curves

We consider the germ of a reduced curve, possibly reducible. F.Delgado de la Mata proved that such a curve is Gorenstein if and only if its semigroup of values is symmetrical. We extend here this symmetry property to any fractional ideal of a Gorenstein curve. We then focus on the set of values of the module of logarithmic residues along plane curves, which determines the values of the Jacobian ideal thanks to our symmetry Theorem. Moreover, we give the relation with Kähler differentials, which are used in the analytic classification of plane branches. We also study the behaviour of logarithmic residues in an equisingular deformation of a plane curve.

val(J

with 1 = (1, . . . , 1) and γ ∈ N p the conductor of the curve, that is to say the minimal γ ∈ N p such that γ + N p ⊆ val(O D ). The set of values of Kähler differentials is a major ingredient used in [HH11] and [HHH15] to study the problem of the analytic classification of plane curves with one or two branches. A.G.Aleksandrov and A.Tsikh develop in [AT01] the theory of multi-logarithmic differential forms and multi-residues along reduced complete intersections. Since our symmetry Theorem is true for any Gorenstein curve, it is in particular true for complete intersections. We mention in section 3.4 several properties of reduced complete intersection curves which generalize the properties of plane curves.

The last section is devoted to the study of the behaviour of logarithmic residues in an equisingular deformation of a plane curve. In particular, we define a stratification by the values of the logarithmic residues, which is the same thanks to section 3 as the stratification by Kähler differentials. We prove that this stratification is finite and constructible, but it does not satisfy the frontier condition.

Acknowledgments. The author is grateful to Michel Granger for many helpful discussions on the

Introduction

Let (D, 0) be the germ of a reduced hypersurface in (C n , 0) defined by f ∈ C{x} := C {x 1 , . . . , x n } and with ring O D = C{x}/(f ). In his fundamental paper [START_REF] Saito | Theory of logarithmic differential forms and logarithmic vector fields[END_REF], K.Saito introduces the notions of logarithmic vector fields, logarithmic differential forms and their residues. A logarithmic differential form is a meromorphic form with simple poles along (D, 0) such that its differential has also simple poles along (D, 0). A logarithmic q-form ω may be written as:

gω = df f ∧ ξ + η
where g ∈ C{x} does not induce a zero divisor in O D , ξ is a holomorphic (q -1)-form and η is a holomorphic q-form. Then, the logarithmic residue res q (ω) of ω is defined as the coefficient of df f , that is to say:

res q (ω) = ξ g ∈ Ω q-1 D ⊗ O D Q(O D )
with Ω q-1 D the module of Kähler differentials on D and Q(O D ) the total ring of fractions of O D . We denote by R D the module of logarithmic residues of logarithmic 1-forms.

In [START_REF] Granger | Normal crossing properties of complex hypersurfaces via logarithmic residues[END_REF], M.Granger and M.Schulze prove that the dual over O D of the Jacobian ideal is R D . If moreover (D, 0) is free, that is to say, if the module of logarithmic differential 1-forms is free, the converse also holds: the dual of R D is the Jacobian ideal. They use this duality to prove a characterization of normal crossing divisors in terms of logarithmic residues: if the module R D is equal to the module of weakly holomorphic functions on D then D is normal crossing in codimension 1, and the converse implication was already proved in [START_REF] Saito | Theory of logarithmic differential forms and logarithmic vector fields[END_REF].

The purpose of this paper is to investigate the module of logarithmic residues along plane curves. Plane curves are always free divisors, and they are the only singular free divisors with isolated singularities. Indeed, A.G.Aleksandrov proves in [START_REF] Aleksandrov | Nonisolated Saito singularities[END_REF]§2] that the singular locus of a singular free divisor is of codimension 2 in the ambient space. We will also study the case of complete intersection curves, for which a notion of multi-residues has been introduced by A.G.Aleksandrov and A.Tsikh in [START_REF] Aleksandr | Tsikh : Théorie des résidus de Leray et formes de Barlet sur une intersection complète singulière[END_REF].

Let (D, 0) be the germ of a reduced Gorenstein curve in (C m , 0). In particular, (D, 0) may be a plane curve or a complete intersection curve. A parametrization of (D, 0) is given by the normalization of the local ring O D , and induces a map val : Q(O D ) → (Z ∪ (∞)) p called the value map, which associates to a fraction g ∈ Q(O D ) the p-uple of the valuations of g along each irreducible components of (D, 0). For a fractional ideal I ⊂ Q(O D ), we denote by val(I) the set of values of the non-zero divisors of I.

We first show in section 2 that the values of the Jacobian ideal and the values of the module of logarithmic residues determine each other. The statement is easy for irreducible curves (see Remark 2.10):

v ∈ val(R D ) ⇐⇒ c -v -1 / ∈ val(J D )
where c ∈ N is the conductor of the curve, i.e. c = min {c ∈ N; c + N ⊆ val(O D )}. It is a generalization of the well-known symmetry of the semigroup for Gorenstein curves (see for example [START_REF] Kunz | The value-semigroup of a one-dimensional Gorenstein ring[END_REF]). Nevertheless, the semigroup of a reducible curve also satisfies a certain symmetry property which has been proved by F.Delgado de la Mata in [START_REF] Delgado De La Mata | Gorenstein curves and symmetry of the semigroup of values[END_REF]. The statement of this symmetry needs more notations and the proof is more difficult than in the irreducible case. We prove here that there is also a symmetry between the values of the module of logarithmic residues and the values of the Jacobian ideal, which is in fact satisfied by any fractional ideal and its dual. Whereas the symmetry is immediate for irreducible curves, the proof of this generalization of Delgado's Theorem is much more subtle. It leads to the main result of this section, namely Theorem 2.4, which generalizes to any Gorenstein curve and any fractional ideal the Theorem 2.4 of [START_REF] Pol | Logarithmic residues along plane curves[END_REF].

In section 3, we give some properties of the set of values of the module of logarithmic residues and of the Jacobian ideal for plane curves. We show how to determine the set of values of the logarithmic residues of a reunion of branches of D from the set val(R D ). We also prove that the modules of logarithmic vector fields of the branches determine the set of values of the zero divisors of J D . Moreover, we give the relation between the Jacobian ideal and the Kähler differentials:

The symmetry of values

This section is devoted to the main Theorem 2.4, which is a generalization of the symmetry Theorem 2.8 of [START_REF] Delgado De La Mata | Gorenstein curves and symmetry of the semigroup of values[END_REF]. The statement is true for any fractional ideal of Q(O D ), so that it is in particular true for the Jacobian ideal and the module of logarithmic residues, which are studied in part 3 and 4.

We first introduce some definitions and notations inspired by [START_REF] Delgado De La Mata | Gorenstein curves and symmetry of the semigroup of values[END_REF], and then give the statement of the theorem. We give here a detailed proof of this theorem, and then a consequence on the Poincaré series associated to a fractional ideal of a Gorenstein curve (see Proposition 2.25).

Preliminaries

We recall here some results and notations of [START_REF] Delgado De La Mata | Gorenstein curves and symmetry of the semigroup of values[END_REF]. Let (D, 0) ⊂ (C m , 0) be the germ of a reduced analytic curve, with p irreducible components D 1 , . . . , D p . The ring O D i of the branch D i is a one-dimensional integral domain, so that its normalization O D i is isomorphic to C {t i } (see for example [dJP00, Corollary 4.4.10]). By the splitting of normalization (see [dJP00, Theorem 1.5.20]), the ring

O D of the normalization of D is O D = p i=1 C {t i }.
Moreover, the total rings of fraction of O D and O D are equal. We denote it by Q(O D ). We then have :

Q(O D ) = Q(O D ) = p i=1 C {t i } 1 t i
The normalization of D i gives a parametrization of the branch D i , which is denoted by

ϕ i : C {t i } → D i t i → (x i,1 (t i ), . . . , x i,m (t i )) Definition 2.1. Let g ∈ Q(O D ).
We define the valuation of g along the branch D i as the order in t i of g • ϕ i (t i ). We denote it by val i (g) ∈ Z ∪ {∞}, with the convention val i (0) = ∞.

We then define the value of g by val(g) = (val 1 (g), . . . , val p (g))

∈ (Z ∪ {∞}) p . Definition 2.2. Let I ⊂ Q(O D ) be a O D -module. It is called a fractional ideal if it is of finite type over O D and if it contains a non-zero divisor of Q(O D ).
For a fractional ideal I, we define val(I) = {val(g); g ∈ I non-zero divisor} ⊂ Z p . We also set

val(I) = {val(g); g ∈ I} ⊂ (Z ∪ {∞}) p .
Remark 2.3. We will prove in part 3 that the set val(I) determines the set val(I) (see Proposition 3.7).

Let I ⊂ Q(O D ) be a fractional ideal. We denote by I ∨ the dual of I, which is by definition

I ∨ = Hom O D (I, O D ). Let g ∈ I be a non-zero divisor. The morphism given by ϕ ∈ Hom O D (I, O D ) → ϕ(g) g ∈ Q(O D ) induces an isomorphism between Hom O D (I, O D ) and {h ∈ Q(O D ); hI ⊆ O D } (see [dJP00,
Proof of Lemma 1.5.14]). We can therefore consider I ∨ as a subset of Q(O D ).

We define the conductor ideal of the curve by

C D := Ann O D (O D /O D ). It is a fractional ideal of Q(O D )
. Moreover, it is also an ideal in O D , so that there exists γ = (γ 1 , . . . , γ p ) ∈ N p , called the conductor, such that C D = t γ O D , where for α ∈ Z p , we set t α = t α 1 1 , . . . , t αp p . By definition,

C D = O ∨ D .
We also introduce the following notations, which are analogous to the notations of [START_REF] Delgado De La Mata | Gorenstein curves and symmetry of the semigroup of values[END_REF]. Let M ⊆ Z p and v ∈ Z p . For i ∈ {1, . . . , p}, we define:

∆ i (v, M ) = {α ∈ M ; α i = v i and ∀j = i, α j > v j } and ∆(v, M ) = p i=1 ∆ i (v, M ). For a fractional ideal I ⊂ Q(O D )
, we write ∆(v, I) instead of ∆(v, val(I)). We consider the product order on Z p , so that for α, β ∈ Z p , α β means that for all i ∈ {1, . . . , p}, α i β i and inf(α, β) = min(α 1 , β 1 ), . . . , min(α p , β p ) .

We set α -1 = (α 1 -1, . . . , α p -1).

We can now state the main Theorem of this section, which generalizes [Pol15, Theorem 2.4]:

Theorem 2.4. Let D be the germ of a reduced curve with ring O D . Then, O D is a Gorenstein ring if and only if for all fractional ideal I ⊂ Q(O D ) and for all v ∈ Z p the following property is satisfied:

v ∈ val(I ∨ ) ⇐⇒ ∆(γ -v -1, I) = ∅ (1)
We notice that if we suppose the condition (1) satisfied for all fractional ideals I ⊂ Q(O D ), it is in particular satisfied by the ring O D , whose dual is O ∨ D = O D , so that we recognize Theorem 2.8 of [START_REF] Delgado De La Mata | Gorenstein curves and symmetry of the semigroup of values[END_REF]. Therefore, the curve is indeed a Gorenstein curve, which proves one of the implications of Theorem 2.4.

Remark 2.5. It is not sufficient to check if (1) is satisfied for one fractional ideal I to prove that the curve is Gorenstein. Indeed, by definition, for every curve, the equivalence (1) is satisfied by

I = O D and I ∨ = C D .

Proof of Theorem 2.4

From now on, we assume that O D is a Gorenstein ring. For example, D can be a plane curve or a complete intersection curve. We recall here a property of Gorenstein curves: Proposition 2.6 ([Eis95, Theorem 21.21], [dJP00, lemma 5.2.8]). Let I ⊂ Q(O D ) be a fractional ideal of a Gorenstein curve. Then it is a maximal Cohen-Macaulay module over O D and:

• The dual I ∨ of I is also a fractional ideal, and I ∨∨ = I.

• If I ⊂ J are fractional ideals, J ∨ ⊆ I ∨ and moreover, dim C J/I = dim C I ∨ /J ∨ .
In particular, for a Gorenstein curve,

C ∨ D = O D , so that for all α ∈ Z p , t α O D ∨ = t -α C D = t γ-α O D .
The following proposition comes from the definition of a fractional ideal, and will be very useful:

Proposition 2.7. Let I be a fractional ideal. Then there exist ν and λ in Z p such that

t ν O D ⊆ I ⊆ t λ O D (2)
In particular, it implies that ν + N p ⊆ val(I) ⊆ λ + N p . Moreover, we can replace in (2.7) λ by any element of Z p lower than λ, and ν by any greater element of Z p .

For a Gorenstein curve, by Proposition 2.6, we have the following sequence of inclusions:

t γ-λ O D ⊆ I ∨ ⊆ t γ-ν O D (3) Let I ⊂ Q(O D
) be a fractional ideal. We first prove the implication ⇒ of (1).

Proposition 2.8. Let v ∈ Z p . Then:

v ∈ val(I ∨ ) ⇒ ∆(γ -v -1, I) = ∅ Proof. Let v = (v 1 , . . . , v p ) ∈ val(I ∨ ), and g ∈ I ∨ with v = val(g). We assume ∆(γ -v -1, I) = ∅.
For the sake of simplicity, we may assume that ∆ 1 (γ -v -1, I) = ∅, which means that there exists h ∈ I with val(h) = (γ 1 -v 1 -1, w 2 , . . . , w p ) ∈ I and for all j 2, w j > γ j -v j -1. Since gh ∈ O D , we have

(γ 1 -1, w 2 + v 2 , . . . , w p + v p ) ∈ val(O D ), with w j + v j γ j . Therefore, ∆ 1 (γ -1, O D ) = ∅.
Nevertheless, from Corollary 1.9 of [START_REF] Delgado De La Mata | Gorenstein curves and symmetry of the semigroup of values[END_REF], ∆(γ -1, O D ) = ∅, which leads to a contradiction. Therefore, ∆(γ -v -1, I) = ∅. Notation 2.9. We set

V = {v ∈ Z p ; ∆(γ -v -1, I) = ∅}.
The set V contains the values of I ∨ , but a priori it may be bigger. In particular, it is not obvious that V is the set of values of a O D -module. Our purpose here is to prove that V is indeed equal to val(I ∨ ).

Remark 2.10. For irreducible curves, the statement of Theorem 2.4 can be rephrased as follows: for all v ∈ Z, v ∈ val(I ∨ ) if and only if γ -v -1 / ∈ val(I), which is a generalization of the Theorem of E.Kunz (see [START_REF] Kunz | The value-semigroup of a one-dimensional Gorenstein ring[END_REF]). The proof is easy for irreducible curves. Indeed, we have:

dim C I ∨ /t γ-λ C {t} = Card val(I ∨ ) ∩ (γ -λ + N) c dim C t λ C {t} /I = Card (λ + N) ∩ (val(I)) c
Since by Proposition 2.6, dim C I ν /t γ-λ C {t} = dim C t λ C {t} /I, we have the result.

The proof for a reducible curve is also based on a dimension argument, but which is much more subtle than in the irreducible case. First of all, we need the following properties, which should be compared with 1.1.2 and 1.1.3 in [START_REF] Delgado De La Mata | Gorenstein curves and symmetry of the semigroup of values[END_REF].

Proposition 2.11. For a fractional ideal

I ⊂ Q(O D ), if v, v ∈ val(I), then inf(v, v ) ∈ val(I).
Similarly, if v, v ∈ val(I), then inf(v, v ) ∈ val(I).

Remark 2.12. If v ∈ val(I) and v ∈ val(I), then inf(v, v ) ∈ val(I).

Proposition 2.13. Let v = v ∈ val(I). If there exists i ∈ {1, . . . , p} such that v i = v i , then there exists v ∈ val(I) such that:

1. v i > v i 2. v j min(v j , v j ) 3. If moreover v j = v j , then v j = min(v j , v j )
The proposition 2.11 is a consequence of the fact that the valuation of a general linear combination of two elements is the lowest one. The proposition 2.13 comes from the fact that a convenient linear combination will increase the valuation on the component D i , but we cannot say what happens on the other components where the equality holds.

Dimension and values

Let v ∈ Z p . We set I v = {g ∈ I; val(g) v} and (v, I) = dim C I/I v . Then (v, I) < ∞. We denote by (e 1 , . . . , e p ) the canonical basis of Z p . For M ⊆ Z p and v ∈ Z p , let

Λ i (v, M) = {α ∈ M ; α i = v i and α v}
We then have (see [START_REF] Delgado De La Mata | Gorenstein curves and symmetry of the semigroup of values[END_REF]Proposition 1.11]):

Proposition 2.14. For all v ∈ Z p , (v + e i , I) -(v, I) ∈ {0, 1} and (v + e i , I) = (v, I) + 1 if and only if Λ i (v, val(I)) = ∅.

Thanks to this proposition we can compute some dimensions from the set of values:

Corollary 2.15. Let ν, λ ∈ Z p such that ν + N p ⊆ val(I) ⊆ λ + N p (see Proposition 2.7). Let (α (j) ) 0 j M +1 be a finite sequence of elements of Z p satisfying:

• α (0) = λ and α (M +1) = ν
• For all j ∈ {0, . . . , M }, there exists i(j) ∈ {1, . . . , p} such that α (j+1) = α (j) + e i(j)

Then: dim C I/t ν O D = (ν, I) = Card j ∈ {0, . . . , M }; Λ i(j) (α (j) , val(I)) = ∅ (4)

Preliminary steps

We recall that V = {v ∈ Z p ; ∆(γ -v -1, I) = ∅}, and this set contains val(I ∨ ). The purpose of this section is to show that if the inclusion val(I ∨ ) ⊆ V is strict, then it has some combinatorial and numerical consequences. We then prove that it leads to a contradiction, which finishes the proof of Theorem 2.4.

First step

We first show that if V = val(I ∨ ), then there is an element w ∈ V \val(I ∨ ) which satisfies some properties which will be used in the next steps.

Let us assume that V = val(I ∨ ), and let w (0) ∈ V \val(I ∨ ) be "an intruder". By Proposition 2.7, there exist λ, ν ∈ Z p satisfying ν +N p ⊆ val(I) ⊆ λ+N p and γ -ν w (0) γ -λ. For the remaining of the proof, we fix such λ, ν.

The following Proposition gives an essential property of w (0) : Proposition 2.16. There exists j ∈ {1, . . . , p} such that Λ j (w (0) , val(I ∨ )) = ∅. Moreover, the corresponding coordinate satisfies w

(0) j < γ j -λ j .
Proof. If for all i ∈ {1, . . . , p}, Λ i (w (0) , val(I ∨ )) = ∅, then for all i ∈ {1, . . . , p} there exists 0) , val(I ∨ )), which contradicts the emptiness.

α (i) ∈ val(I ∨ ) such that α (i) i = w (0) i and α (i) j w (0) j . As a consequence, by Proposition 2.11, inf(α (1) , . . . , α (p) ) = w (0) ∈ val(I ∨ ), which is a contradiction. It gives the existence of a j ∈ {1, . . . , p} such that Λ j (w (0) , val(I ∨ )) = ∅. It is immediate to see that w (0) j < γ j -λ j since if w (0) j = γ j -λ j , then γ -λ ∈ Λ j (w (

Second step

For the sake of simplicity, we assume that Λ p (w (0) , val(I ∨ )) = ∅. The Corollary 2.15 together with a convenient finite sequence (α (j) ) can be used to compute the dimension of the quotient I ∨ /t γ-λ O D . We define a number as the cardinality of the set obtained by replacing val(I ∨ ) by V in (4). This number may a priori depend on the chosen sequence (α (j) ).

In order to compute dim C I ∨ /t γ-λ O D , we consider a sequence (α (j) ) 0 j n 0 with n 0 = p i=1 ν i -λ i satisfying:

• α (0) = γ -ν and α (n 0 ) = γ -λ
• for all j ∈ {0, . . . n 0 -1}, there exists i(j) ∈ {1, . . . , p} such that α (j+1) = α (j) + e i(j)

• there exists j 0 ∈ {0, . . . , n 0 -1} such that α (j 0 ) = w (0) and α (j 0 +1) = w (0) + e p

The existence of such a sequence follows from Proposition 2.16. Moreover, this sequence satisfies the required properties of Corollary 2.15.

For example, when p = 2, we can choose a sequence α as follows:

val 1 val 2 w (0) α (0) γ -ν α (n 0 ) γ -λ Figure 1
Remark 2.17. In [START_REF] Pol | Logarithmic residues along plane curves[END_REF] we choose a sort of "canonical" sequence, but in order to be less technical, we impose less conditions on α.

From Corollary 2.15, we have:

dim C I ∨ /t γ-λ O D = Card j ∈ {0, . . . , n 0 -1} ; Λ i(j) (α (j) , val(I ∨ )) = ∅ (5)
We want to compare this dimension with the number previously announced:

= Card j ∈ {0, . . . , n 0 -1} ; Λ i(j) (α (j) , V ) = ∅ (6)
Lemma 2.18. For the sequence α defined above, we have:

1 + dim C I ∨ /t γ-λ O D (7) Proof. It is clear that Λ i(j) (α (j) , val(I ∨ )) = ∅ ⇒ Λ i(j) (α (j) , V ) = ∅.
Moreover, since there exists j 0 such that α (j 0 ) = w (0) and α (j 0 +1) = α (j 0 ) + e p , Λ p (α (j 0 ) , V ) = ∅, but the assumptions on w (0) implies Λ p (α (j 0 ) , val(I ∨ )) = ∅. Hence the inequality.

From now on, our sequence α is fixed.

Third step

The purpose of this third step is to compare this number to dim C I/t ν O D . For i ∈ {0, . . . , n 0 } we set β (i) = γ -α (n 0 -i) . The sequence β satisfies the properties of Corollary 2.15, so that it can be used to compute the dimension dim C I/t ν O D .

For the sequence α of Figure 1, we represent the sequence β on the following diagram, where

v (0) = γ -w (0) -1: val 1 val 2 v (0) β (0) λ ν β (n 0 ) Figure 2
The following proposition gives a relation between and dim C I/t ν O D : Proposition 2.19. With the above notations we have:

dim C I/t ν O D p i=1 (ν i -λ i ) -
To prove this proposition, we need the following lemma: Lemma 2.20. Let w ∈ Z p and i ∈ {1, . . . , p}. Then:

Λ i (w, V ) = ∅ ⇒ Λ i (γ -w -e i , val(I)) = ∅ Proof. Let w ∈ Λ i (w, V ). By the definition of V , we have ∆(γ -w -1, val(I)) = ∅. Moreover, (γ-w -e i ) i = γ i -w i -1 and for j = i, (γ-w -e i ) j = γ j -w j γ j -w j . Thus Λ i (γ-w -e i , val(I)) = ∆ i (γ -w -1, val(I)) = ∅.
Proof of Proposition 2.19. We first notice that the two sequences α (j) and β (j) has the same number of terms, namely n 0 + 1 = p i=1 (ν i -λ i ) + 1. By Corollary 2.15, we have:

dim C I/t ν O D = Card j ∈ {0, . . . , n 0 -1} ; Λ i(n 0 -j-1) (β (j) , val(I)) = ∅ (8)
We notice that for all j ∈ {0, . . . ,

n 0 -1}, γ -α (j) -e i(j) = γ -α (j+1) = β (n 0 -(j+1)) . Therefore, by the previous Lemma, if Λ i (α (j) , V ) = ∅ then Λ i β (n 0 -(j+1)) , val(I) = ∅.
We then obtain the result by comparing (8) and (6).

End of the proof of Theorem 2.4

We can now finish the proof of Theorem 2.4.

The inclusion val(I ∨ ) ⊆ V is given by Proposition 2.8. It remains to prove that this inclusion cannot be strict.

The Proposition 2.19 gives:

dim C I/t ν O D p i=1 (ν i -λ i ) - Moreover, since p i=1 (ν i -λ i ) = dim C t λ O D /t ν O D = dim C t λ O D /I + dim C I/t ν O D ,
we have by Proposition 2.6:

dim C I ∨ /t γ-λ O D = p i=1 (ν i -λ i ) -dim C I/t ν O D Hence the inequality: dim C I ∨ /t γ-λ O D (9)
However, by Lemma 2.18, if V = val(I ∨ ), then 1 + dim C I ∨ /t γ-λ O D , which contradicts (9). Therefore, we have V = val(I ∨ ), that is to say:

v ∈ val(I ∨ ) ⇐⇒ ∆(γ -v -1, I) = ∅.
Remark 2.21. Another consequence of the equality V = val(I ∨ ) is that the number is equal to the dimension of I ∨ /t γ-λ O D . Therefore, the inequality in Proposition 2.19 is in fact an equality. Moreover, since for all w ∈ Z p , there exist λ , ν ∈ Z p such that γ -λ + N p ⊆ val(I ∨ ) ⊆ γ -ν + N p and γ -ν w γ -λ , it also has the following consequence:

Λ i (w, val(I ∨ )) = ∅ ⇐⇒ Λ i (γ -w -e i , val(I)) = ∅ (10)

Poincaré series of a fractional ideal

This section follows a suggestion of Antonio Campillo. Let

(D = D 1 ∪ • • • ∪ D p , 0)
be the germ of a reduced reducible curve, and I ⊂ Q(O D ) be a fractional ideal.

The following definitions are inspired by [START_REF] Campillo | The Alexander polynomial of a plane curve singularity via the ring of functions on it[END_REF]. We recall that

I v = {g ∈ I; val(g) v}.
We consider the set of formal Laurent series

L = Z[[t -1 1 , . . . , t -1 p , t 1 , . . . , t p ]]. This set is not a ring, but it is a Z[t -1 1 , . . . , t -1 p , t 1 , . . . , t p ]-module. We set: L I (t 1 , . . . , t p ) = v∈Z p c I (v)t v (11)
with c I (v) = dim C I v /I v+1 and

P I (t) = L I (t) p i=1 (t i -1) (12)
Remark 2.22. In [START_REF] Campillo | The Alexander polynomial of a plane curve singularity via the ring of functions on it[END_REF], the authors study the case I = O D with D a plane curve. They prove that P O D is in fact a polynomial, and for plane curves with at least two components,

P O D (t) t 1 • • • t p -1 is the Alexander polynomial of the curve (see [CDGZ03, Theorem 1]).
Our purpose here is to deduce from Theorem 2.4 a relation between P I (t) and P I ∨ (t).

The following lemma is a direct consequence of the definition (12) of P I :

Lemma 2.23. We define for v ∈ Z p ,

α I (v) = J⊆{1,...,p} (-1) Card(J c ) c I (v -e J ) (13)
where we denote for

J = {j 1 , • • • , j k }, e J = e j 1 + • • • + e j k . Then P I (t) = v∈Z p α I (v)t v
We use the previous lemma to prove the following property:

Lemma 2.24. The formal Laurent series P I (t) is a polynomial.

Proof. Let λ, ν ∈ Z p be such that ν + N p ⊆ val(I) ⊆ λ + N p . The only possibly non-zero α I (v) are those such that λ v ν. Indeed, let us assume for example that v p < λ p or v p > ν p . We can then prove thanks to Corollary 2.15 that for all J ⊂ {1, . . . , p} such that p / ∈ J, c I (v-e J∪{p} ) = c I (v-e J ). By the definition (13), it gives us the result.

The symmetry of Theorem 2.4 has the following consequence: Proposition 2.25. With the same notations,

P I ∨ (t) = (-1) p+1 t γ P I 1 t 1 , . . . , 1 t p (14)
Proof. The property ( 14) is in fact equivalent to the following property:

∀v ∈ Z p , α I ∨ (v) = (-1) p+1 α I (γ -v) ( 15
)
This property is obvious if v / ∈ {ω ∈ Z p ; γ -ν w γ -λ} since both α I ∨ (v) and α I (γ -v) are zero.

By (13), it is sufficient to prove that for all v ∈ Z p , c I ∨ (v) = p -c I (γ -v -1). We have:

c I ∨ (v) = Card{i ∈ {1, . . . , p} ; Λ i (v + e 1 + • • • + e i-1 , val(I ∨ )) = ∅} c I (γ -v -1) = Card{i ∈ {1, . . . , p} ; Λ i (γ -v -e 1 -• • • -e i , val(I)) = ∅}
The result follows from the equivalence (10).

3 On the structure of the set of values of logarithmic residues

In this part we give several properties of the module of logarithmic residues along plane curves or complete intersection curves. We first recall some definitions. We then study the structure of the set of values of logarithmic residues along plane curves, and give the relation with the set of values of Kähler differentials, which is used in the analytic classification of branches proposed in [START_REF] Hefez | The analytic classification of plane branches[END_REF], and [START_REF] Hefez | The analytic classification of plane curves with two branches[END_REF] for two branches. We extend some of these properties to the set of values of multi-residues along complete intersection curves.

Preliminaries on logarithmic residues

We recall here some definitions and results about logarithmic vector fields, logarithmic differential forms and their residues, which can be found in [START_REF] Saito | Theory of logarithmic differential forms and logarithmic vector fields[END_REF].

Let us consider a reduced hypersurface germ (D, 0) ⊂ (C n , 0) defined by f ∈ C {x 1 , . . . , x n }. We denote by Θ n the module of germs of holomorphic vector fields on (C n , 0), and C{x} = C {x 1 , . . . , x n }. Definition 3.1. A germ of vector field δ ∈ Θ n is called logarithmic along D if δ(f ) = αf with α ∈ C{x}. We denote by Der(-log D) the module of logarithmic vector fields along (D, 0).

A germ of meromorphic q-form ω ∈ 1 f Ω q with simple poles along D is called logarithmic if f dω is holomorphic. We denote by Ω q (log D) the module of logarithmic q-forms on (D, 0). Moreover, these two modules are reflexive and each is the dual C{x}-module of the other (see [Sai80, Lemma 1.6]). Definition 3.2. If Der(-log D) (or equivalently Ω 1 (log D)) is a free C{x}-module, we call (D, 0) a germ of free divisor.

In particular, plane curves are free divisors (see [START_REF] Saito | Theory of logarithmic differential forms and logarithmic vector fields[END_REF]1.7]).

Proposition 3.3 (Saito criterion). The germ (D, 0) is a free divisor at 0 if and only if there exists (δ 1 , . . . , δ n ) in Der(-log D) such that δ j = a ij ∂ x i with det ((a ij ) 1 i,j n ) = uf , where u is invertible.

To define the notion of logarithmic residues, we need the following characterization of logarithmic differential forms: Proposition 3.4. A meromorphic q-form ω with simple poles along D is logarithmic if and only if there exist g ∈ C{x}, which does not induce a zero divisor in O D = C{x}/(f ), a holomorphic (q -1)-form ξ and a holomorphic q-form η such that:

gω = df f ∧ ξ + η (16)
Definition 3.5. The residue res q (ω) of ω ∈ Ω q (log D) is defined by

res q (ω) := ξ g ∈ Q(O D ) ⊗ O D Ω q-1 D
where ξ and g are given by (16), and

Ω q-1 D = Ω q-1 C n df ∧ Ω q-2 C n + f Ω q-1 C n
is the module of Kähler differentials on D.

If q = 1, we write res(ω) instead of res 1 (ω). We define We denote by J D ⊆ O D the Jacobian ideal of D, that is to say the ideal of O D generated by the partial derivatives of f .

R D := res(ω); ω ∈ Ω 1 (log D) ⊆ Q(O D ),
The following result gives the relation between the module of logarithmic residues and the Jacobian ideal: Proposition 3.6 ([GS14, Proposition 3.4]). Let (D, 0) be the germ of a reduced divisor. Then

J ∨ D = R D . If moreover D is free, R ∨ D = J D .

Some basic properties of the values of logarithmic residues

We give here some properties of the set of values of logarithmic residues and of the Jacobian ideal of plane curves.

Since plane curves are free divisors, the module Der(-log D) is free. Let us assume that

δ i = α i ∂ x + β i ∂ y , i = 1, 2 is a basis of Der(-log D).
Then by duality, a basis of Ω 1 (log D) is:

ω 1 = β 2 dx -α 2 dy f ω 2 = -β 1 dx + α 1 dy f If g = c 1 • f x + c 2 • f y with c 1 , c 2 ∈ C induces a non zero divisor in O D , then the module of residues is generated by res(ω 1 ) = c 1 • β 2 -c 2 • α 2 g and res(ω 2 ) = -c 1 • β 1 + c 2 α 1 g
. Thus, the module of logarithmic residues can be generated by two elements.

Let D = D 1 ∪ • • • ∪ D p be the germ of a reduced plane curve defined by f = f 1 • • • f p where for all i ∈ {1, . . . , p}, f i is irreducible. We first want to prove that the negative values of R D determine all the values of R D . Since we will need a similar result for the Jacobian ideal, we state it for an arbitrary fractional ideal.

We recall that for g ∈ Q(O D ), val i (g) = ∞ means that the restriction of g on D i is zero.

The following proposition shows that the values of the zero divisors are determined by the faces of the negative quadrant with origin ν. Proposition 3.7. Let I ⊂ Q(O D ) be a fractional ideal and let ν ∈ Z p be such that t ν O D ⊆ val(I). The values of the zero divisors of I are determined by

M I = p q=1 σ∈Sq w ∈ val(I); ∀i ∈ {1, . . . , q} , w σ(i) = ν σ(i)
where S q is the group of permutations of q elements. More precisely, the values of the zero divisors are exactly the values obtained by replacing for v ∈ M I the coordinates which satisfies v j = ν j by ∞.

Proof. Let g ∈ I be a zero divisor. Then by Proposition 2.11, inf(val(g), ν) ∈ val(I). It is then obvious that inf(val(g), ν) ∈ M I .

Conversely, let v ∈ M I . There exists q ∈ {1, . . . , p} and σ ∈ S q such that for all i ∈ {1, . . . , p}, v i = ν i . Let h ∈ I be such that val(h) = v. Since t ν O D ⊆ I, there exists g ∈ I satisfying for all i ∈ {1, . . . , q}, g| D σ(i) = h| D σ(i) and for all j / ∈ σ ({1, . . . , q}), val j (g) > v j . Then h -g is a zero divisor whose value satisfies for i ∈ {1, . . . , q}, val σ(i) (h -g) = ∞ and for all j / ∈ σ ({1, . . . , q}),

val j (h -g) = v j . Corollary 3.8. Let I ⊂ Q(O D ) be a fractional ideal and ν ∈ Z p . Assume that ν + N p ⊆ val(I). Let v ∈ Z p . Then v ∈ val(I) ⇐⇒ inf(v, ν) ∈ val(I)
In particular, it means that the set val(I) ∩ {v ∈ Z p ; v ν} determines the set val(I).

Proof. The implication ⇒ comes from Proposition 2.11. For the implication ⇐, let v ∈ Z p be such that inf(v, ν) ∈ val(I). If v ν, then v = inf(v, ν) ∈ val(I). If there exists j such that v j > ν j , then inf(v, ν) ∈ M I where M I is defined in Proposition 3.7. Therefore, since t ν O D ⊆ I, there exists a zero divisor g ∈ I, q ∈ {1, . . . , p} and σ ∈ S q which satisfy for all i ∈ {1, . . . , q}, val

σ(i) (g) = ∞ if v σ(i) = ν σ(i)
and for all j / ∈ σ ({1, . . . , q}), val j (g) = v j < ν i . Let w = max(v, ν). Then w ∈ val(I) and v = inf(w, val(g)) ∈ val(I).

Remark 3.9. By Propositions 3.7 and 3.8, the set val(I) ∩ {v ∈ Z p ; v ν} also determines val(I). More precisely, we have:

val(R D ) = {v ∈ val(R D ); v 0} ∪ {v ∈ Z; inf(v, 0) ∈ val(R D )}
Let us determine the values of R D which come from the branches or union of branches. Proposition 3.11. Let q ∈ {1, . . . , p} and D = D i 1 ∪ • • • ∪ D iq be the union of q branches of D.

Then Ω 1 (log D ) ⊆ Ω 1 (log D)
Moreover, v ∈ w ∈ val(R D ); ∀j ∈ {1, . . . , q} , v i j = 0 if and only if there exists a logarithmic 1-form ω ∈ Ω 1 (log D ) such that for all j / ∈ {i 1 , . . . , i q }, val j (res(ω)) = v j , and for all j ∈ {1, . . . , q}, val i j (res(ω)) = ∞.

Proof. For the first part of the statement, we set F 1 the equation of D . Let ω be a logarithmic 1-form along D . Then, F 1 ω and F 1 dω are holomorphic, so that f ω and f dω are holomorphic.

The second part of the statement comes from this inclusion and Proposition 3.7.

In particular, the logarithmic residues of the irreducible components satisfy the following property:

Corollary 3.12. We have the following inclusion: [START_REF] Schulze | Saito's normal crossing condition[END_REF], it is a splayed divisor, and in fact it is even a normal crossing plane curve.

R D 1 ⊕ • • • ⊕ R Dp → R D Therefore, val 1 (R D 1 ) × • • • × val p (R Dp ) ⊆ val(R D ). Remark 3.13. If D = D 1 ∪ D 2 is a plane curve satisfying R D = R D 1 ⊕ R D 2 , then by
We now study the set of values of the dual of R D , namely, the Jacobian ideal J D . We show that the modules of logarithmic vector fields Der(-log D i ) for i ∈ {1, . . . , p} give informations on the structure of the set of values of the Jacobian ideal. ∈ {i 1 , . . . , i q }, val j (δ(f j )) = v ji =j val j (f i ). In particular, the set of zero divisors of J D are determined by the family of modules i∈I Der(-log D i ) for I ⊂ {1, . . . , p}.

Proof. We first notice that for all g ∈ J D , there exists δ ∈ Θ 2 such that δ(f

) = g in O D . Moreover, δ(f ) induces in O D = p i=1 O D i the element: δ(f ) = (f 2 • • • f p δ(f 1 ), . . . , f 1 • • • f p-1 δ(f p )) By Proposition 3.7, v ∈ M J D if and only if there is δ ∈ Θ 2 such that val(δ(f )) = (∞, . . . , ∞, v k , . . . , v p )
which is equivalent to:

∀i ∈ {1, . . . , k -1}, δ(f i ) ∈ (f i ) and ∀i ∈ {k, . . . , p}, val i (δ(f )) = v i
Hence the result.

The relation with Kähler differentials

We give here the relation between logarithmic residues and Kähler differentials. More precisely, we show that the values of the Jacobian ideal can be obtained from the values of the Kähler differentials, and the relation with the values of logarithmic residues comes from Theorem 2.4.

We first recall the definition of the module of Kähler differentials:

Ω 1 D = Ω 1 C 1 O C 2 df + f Ω 1 C 2 Definition 3.15. Let D = D 1 ∪ • • • ∪ D p
be a plane curve germ with irreducible components D i , and ϕ i (t i ) = (x i (t i ), y i (t i )) be a parametrization of D i . Let ω = adx + bdy ∈ Ω 1 D . We denote by ϕ * i (ω) the 1-form defined on D i by:

ϕ * i (ω) = a • ϕ i (t i )x i (t i ) + b • ϕ i (t i )y i (t i ) dt i
We denote ϕ * (ω) = ϕ * 1 (ω), . . . , ϕ * p (ω) . We then define the valuation of ω along D i by:

val i (ω) = val i (ϕ * i (ω)) = val i a • ϕ i (t i )x i (t i ) + b • ϕ i (t i )y i (t i ) + 1
The value of ω is then val(ω) = (val 1 (ω), . . . , val p (ω)).

Proposition 3.16. We have the following equality: val(J D ) = γ +val(Ω 1 D )-1. To be more precise, there exists g ∈ C D of value γ such that

J D = g • ϕ * (Ω 1 D ) dt
, where we denote by

ϕ * (Ω 1 D ) dt
the fractional ideal of O D generated by (x 1 (t 1 ), . . . , x p (t p )) and (y 1 (t 1 ), . . . , y p (t p )).

To prove this equality, we need the following propositions: Proposition 3.17 ([DdlM87, Theorem 2.7]). Let f = f 1 • • • f p be a reduced equation of a plane curve germ. We assume that for all i ∈ {1, . . . , p}, f i is irreducible. We denote by D i the branch defined by f i , and by c i its conductor. The conductor of D is given by

γ = c 1 + p i=2 val 1 (f i ), . . . , c p + p-1 i=1 val p (f i ) Lemma 3.18. Let D = D 1 ∪ • • • ∪ D p
be a plane curve germ defined by a reduced equation f ∈ C{x, y}. Then:

val(f x ) = γ + val(y) -1 val(f y ) = γ + val(x) -1
Proof. If f is irreducible, the result is given by Teissier's lemma (see [START_REF] Cassou-Noguès | Invariants of plane curve singularities and Newton diagrams[END_REF]lemma 2.3]). If f is reducible, we use Proposition 3.17, the equality val j ∂f ∂x = i =j val j (f i )+val j ∂f j ∂x and Teissier's lemma to obtain the result.

Proof of Proposition 3.16. For lack of reference, we give here the proof. Let i ∈ {1, . . . , p}. Then, since ϕ i is a parametrization of D i , we have

f x (ϕ i (t i )) • x i (t i ) + f y (ϕ i (t i )) • y i (t i ) = 0. We first assume that x i (t i ) • y i (t i ) = 0. Then, there exists g i (t i ) ∈ C {t i } such that f x (ϕ i (t i )) y (t i ) = - f y (ϕ i (t i )) x (t i ) = g i (t i )
By lemma 3.18 we have val i (f x ) = γ i + val i (y) -1, so that val i (g i ) = γ i . Since γ is the conductor, there exists g ∈ C{x, y} such that for all i ∈ {1, . . . , p}, g i (t i ) = g(ϕ i (t i )), which gives us the result.

If for example x i (t i ) = 0, then y i (t i ) = 0 and f y (t i ) = 0. We set again g

i (t i ) = f x (ϕ i (t i )) y (t i ) , which is of valuation γ i . The relation f y (t i ) = g i (t i ) • x i (t i ) is of course true.
The end of the proof is the same as for the case x i (t i ) • y i (t i ) = 0.

Corollary 3.19. We have the following equivalence:

v ∈ val(R D ) ⇐⇒ ∆(-v, val(Ω 1 D )) = ∅ Moreover, R D = 1 g • ϕ * (Ω 1 D ) dt ∨ Proof.
It is a consequence of both Proposition 3.16 and Theorem 2.4.

Remark 3.20. The latter corollary gives also the relation between meromorphic regular forms as defined in [START_REF] Barlet | Le faisceau ω • X sur un espace analytique X de dimension pure[END_REF] and Kähler differentials. Indeed, by [Ale90, §4], the module R D of logarithmic residues is isomorphic to the module of regular meromorphic forms ω D , which can be defined as

ω D = Ext 1 O S Ω 1 D , Ω 2 S . In particular, ω D 1 g • ϕ * (Ω 1 D ) dt ∨ .
Remark 3.21. Another consequence of Proposition 3.16 is the following inclusion:

γ + val(O D )\{0} -1 ⊆ val(J D ) Indeed, if h ∈ mO D , with m the maximal ideal of O D , then val(dh) = val(h), which gives us the inclusion val(O D )\ {0} ⊆ val(Ω 1 D ).
We mention here the relation between logarithmic forms and the torsion of Kähler differentials. It leads to a determination of the dimension of the torsion of Ω 1 D when D is a plane curve which is slightly different from the proofs of O.Zariski (see [START_REF] Zariski | Characterization of plane algebroid curves whose module of differentials has maximum torsion[END_REF]) and R.Michler (see [START_REF] Michler | Torsion of differentials of hypersurfaces with isolated singularities[END_REF]).

We first assume that (D, 0) is a germ of reduced hypersurface of (C n , 0). The following property has been proved by A.G.Aleksandrov: Proposition 3.22 ([Ale05, 3.1]). For all 1 q n, the following map:

Ω q (log D) df f ∧ Ω q-1 C n + Ω q C n → Tors(Ω q D ) [ω] → [f ω] is an isomorphism of O D -modules.
Proof. It is a consequence of the characterization (16) of logarithmic forms.

Corollary 3.23. The map res q induces an isomorphism of O D -modules:

res q (Ω q (log D) Ω q-1 D Tors(Ω q D )
Proof. We have the following exact sequences:

0 → Ω q C n → Ω q (log D) → res q (Ω q (log D)) → 0 0 → Ω q C n → df f ∧ Ω q-1 C n + Ω q C n res q --→ Ω q-1 D → 0
Therefore, res q (Ω q (log D)

Ω q-1 D Ω q (log D) df f ∧Ω q-1 C n +Ω q C n
Tors(Ω q D ).

Corollary 3.24. Let (D, 0) ⊆ (C 2 , 0) be a plane curve germ. We denote by τ

= dim C O D /J D the Tjurina number. Then R D O D Tors(Ω 1 D ) and dim C Tors(Ω 1 D ) = τ .
Proof. We use Propositions 2.6 and 3.6 to prove that

dim C R D /O D = dim C O D /J D = τ .

Complete intersection curves

This section is devoted to the study of complete intersection curves, which are a particular case of Gorenstein curves.

We begin with the definition of multi-logarithmic forms along a reduced complete intersection C defined by a regular sequence (h 1 , . . . , h k ) in (C m , 0) given in [START_REF] Aleksandrov | Multidimensional residue theory and the logarithmic de Rham complex[END_REF]. We then focus on the case of complete intersection curves for which we have natural extensions of several results of the plane curve case.

Definition 3.25 ([Ale12]). Let ω ∈ 1 h 1 ...h k Ω q with q ∈ N. Then ω is called a multi-logarithmic differential q-form along the complete intersection C if ∀i ∈ {1, . . . , k} , dh i ∧ ω ∈ k j=1 1 h j Ω q+1
We denote by Ω q (log C) the C{x}-module of multi-logarithmic q-forms along C.

To simplify the notations, we set

Ω q := k j=1 1 h j Ω q .
If k = 1, the definition of multi-logarithmic forms coincides with the definition of logarithmic forms 3.1.

Then we have the following characterization which should be compared with [Sai80, 1.1]:

Theorem 3.26 ([Ale12, §3, Theorem 1]). Let ω ∈ 1 h 1 •••h k Ω q
, with q k. Then ω ∈ Ω q (log C) if and only if there exist a holomorphic function g ∈ C{x} which does not induce a zero-divisor in O C , a holomorphic differential form ξ ∈ Ω q-k and a meromorphic q-form η ∈ Ω q such that:

gω = ξ ∧ dh 1 ∧ • • • ∧ dh k h 1 • • • h k + η (17)
Definition 3.27. Let ω ∈ Ω q (log C), q k. Let us assume that g, ξ, η satisfy the properties of Theorem 3.26. Then the multi-residue of ω is:

res q C (ω) := ξ g ∈ Q(O C ) ⊗ O C Ω q-k C = Q(O C ) ⊗ O C Ω q-k C We define R q-k C := res q C (Ω q (log C)). In particular, if q = k, res k C (ω) ∈ Q(O C
), and we denote

R C := res k C Ω k (log C) .
It is proved in [START_REF] Aleksandrov | Multidimensional residue theory and the logarithmic de Rham complex[END_REF] that for ω ∈ Ω q (log C) the multi-residue res q C (ω) is well-defined with respect to the choices of ξ, g and η in (17).

Proposition 3.28 ([Sch13, Proposition 4.1]). Let J C ⊆ O C be the Jacobian ideal, that is to say the ideal of O C generated by the k × k minors of the Jacobian matrix. Then:

J ∨ C = R C
Remark 3.29. In a forthcoming paper, we will give a more direct proof of this duality, which does not rely on the isomorphism between the module of logarithmic multi-residues and the module of regular meromorphic forms given in [AT01, Theorem 3.1].

From now on, we assume that C is a reduced complete intersection curve defined by a regular sequence (h 1 , . . . , h m-1 ).

We recall that C C denotes the conductor ideal of C. The following proposition is a generalization of Proposition 3.16. Proposition 3.30. Let C = C 1 ∪• • •∪C p be a reduced complete intersection curve defined by a regular sequence (h 1 , . . . , h m-1 ). Then there exists g ∈ C C with val(g) = γ such that

J C = g • ϕ * (Ω 1 C ) dt . In particular, val(J C ) = γ + val(Ω 1 C ) -1. Proof. Let ϕ i (t i ) = (x i,1 (t i ), . . . , x i,m (t i )) be a parametrization of C i . Let J i denote the k × k minor of Jac(h 1 , . . . , h m-1
) obtained by removing the column i. Let i ∈ {1, . . . , p}. Then, for all j ∈ {1, . . . , m -1} we have h j • ϕ i (t i ) = 0, thus:

Jac(h 1 , . . . , h k ) • ϕ i (t i ) x i,1 (t i ), . . . , x i,m (t i ) t = 0, . . . , 0 t
We multiply on the left by the adjoint of the matrix obtained by removing the last column of Jac(h 1 , . . . , h k ) • ϕ i (t i ), which gives the needed relations: for all j ∈ {1, . . . , m -1},

J m • ϕ i (t i ) • x i,j (t i ) + (-1) m-(j-1) J j • ϕ i (t i ) • x i,m (t i ) = 0
We assume for example that x i,m (t i ) = 0.

By setting g

i (t i ) = J m • ϕ i (t i ) x i,m (t i )
one obtains for all ∈ {1, . . . , m},

g i (t i ) • x i, (t i ) = (-1) m-J • ϕ i (t i ) (18)
It remains to prove that val i (g) = γ i . Let us denote by Π C the ramification ideal of the curve C, which is the O C -module generated by

x i,1 (t 1 ), . . . , x i,p (t p ) 1 i m
By [Pie79, Corollary 1, Proposition 1], one has:

C C Π C = J C O C
Thus, we have the equality inf(val(Π C )) + γ = inf(val(J C )).

The equalities (18) imply that for all i ∈ {1, . . . , p}, and j ∈ {1, . . . , m} we have

val i (J j ) = inf(val i (J C )) if and only if val i (x i,j ) = inf(val i (Π C )).
Therefore, if j is such that val i (J j ) = inf(val i (J C )), then val i (J j ) = γ i + val i (x i,j ), which gives us val i (g i ) = γ i . Definition 3.31. Let ω = a j dx j ∈ Ω 1 C . We set

val i (ω) = val i (ϕ * i (ω)) = 1 + val i   m j=1 (a j • ϕ i )(t i ) • x i,j (t i )  
Corollary 3.32. The values of J C and Ω 1 C satisfy:

val(J C ) = γ + val(Ω 1 C ) -(1, . . . , 1)
Moreover, with the notation of part 2, for all v ∈ Z p , we have the following equivalence:

v ∈ val(R C ) ⇐⇒ ∆(-v, val(Ω 1 C )) = ∅ and R C = 1 g • ϕ * (Ω 1 C ) dt ∨
, where g is given by Proposition 3.30.

Proof. The statement for J C is a direct consequence of Proposition 3.30. The second statement is a consequence of the symmetry Theorem 2.4 together with Proposition 3.28.

Remark 3.33. As for plane curves, the isomorphism between multi-residues and regular meromorphic differential forms given in [START_REF] Aleksandr | Tsikh : Théorie des résidus de Leray et formes de Barlet sur une intersection complète singulière[END_REF] gives the relation between regular meromorphic forms and Kähler differentials.

Equisingular deformations of plane curves and the stratification by logarithmic residues

The purpose of this last section is to study the behaviour of the values of logarithmic residues in an equisingular deformation of a plane curve. By the results of part 3, the stratification by the values of logarithmic residues is the same as the stratification by the values of Kähler differentials which is an essential ingredient of the analytic classification of plane curves described in [START_REF] Hefez | The analytic classification of plane branches[END_REF] and [START_REF] Hefez | The analytic classification of plane curves with two branches[END_REF] respectively for irreducible curves and for reducible curves with two branches. They prove that in a stratum of the stratification by the values of Kähler differentials, the analytic classification can be expressed in terms of a normal parametrization of the curve. The analytic equivalence is then represented by the group action of the roots of the unity of a certain order determined by val(Ω 1 D ) (see [HH11, Theorem 2.1]). We end this section with several algorithms which can be used to compute the set of values of R D , inspired by [START_REF] Briançon | Le nombre de modules du germe de courbe plane x a + y b = 0[END_REF] and [START_REF] Hefez | Standard bases for local rings of branches and their modules of differentials[END_REF].

We first recall some results on equisingular and admissible deformations, and we then study some properties of the stratification by the values of logarithmic residues.

Let D be a plane curve defined in a neighbourhood U of the origin of C 2 by a reduced equation f ∈ O C 2 (U ). We consider a deformation F of f with base space (S, 0) = (C k , 0) with ring O S . It means that F (x, y, s)

∈ O C 2 ⊗O S satisfies F (x, y, 0) = f (x, y). Let X = U × S, O X = O C 2 ⊗O S , W = F -1 (0) ⊆ U × S.
We assume F (0, 0, s) = 0 for all s. For s ∈ S, we set D s = W ∩ C 2 × {s} and m S,s the maximal ideal of O S,s , and F s = F (., s). In particular, D 0 = D and F 0 = f .

Equisingular and admissible deformations of plane curves

The following numbers are classical invariants of plane curves: Definition 4.1. Let D be a reduced plane curve defined by f ∈ C{x, y}.

• The Milnor number is µ = C{x, y}/(f x , f y )

• The Tjurina number is τ = C{x, y}/(f x , f y , f )

• The delta-invariant is δ = dim C O D /O D
The following proposition gives the relation between µ and δ: Proposition 4.2 [START_REF] Milnor | Singular points of complex hypersurfaces[END_REF]). We have the following relation:

µ = 2δ -p + 1
where p is the number of irreducible components of (D, 0).

Let us assume that F is an equisingular deformation of f (i.e. for all s ∈ S, µ(F s ) = µ(f )). From the equisingularity Theorem for plane curves (see [Tei77, §3.7]), it implies that a parametrization ϕ of (D, 0) gives rise to a deformation ϕ s of the parametrization. We denote by val Ds (g) the value of g ∈ Q(O Ds ) along D s . Another consequence of the equisingularity Theorem for plane curves is: Corollary 4.3. With the same notations, if F is an equisingular deformation of f :

1. All fibers D s have the same conductor γ.

2. Let x(t, s) = (x 1 (t 1 , s), . . . , x p (t p , s)), y(t, s) = (y 1 (t 1 , s), . . . , y p (t p , s)) be a parametrization of D s . For all s ∈ S, inf val Ds (x(t, s)), val Ds (y(t, s)) = inf val D (x(t, 0)), val D (y(t, 0)) = (m (1) , . . . , m (p) )

where m (j) is the multiplicity of the component D j of D.

Proof.

1. It comes from Theorem 3.17, since by the equisingularity Theorem (see [Tei77, §3.7, (10)]), the conductors and the intersection multiplicities (D i .D j ) = val i (f j ) do not depend on s.

2. For all j ∈ {1, . . . , p}, inf val D s,j (x j (t j , s), y j (t j , s)) is the multiplicity of D s,j , which does not depend on s by the equisingularity Theorem.

The following proposition will be used in the next section, since it gives a common denominator for the logarithmic residues with interesting properties. Proposition 4.4. There exists α, β ∈ C such that for all s in a neighbourhood of 0, val(αF x (s) + βF y (s)) = γ + m (1) , . . . , m (p) -1. In particular, αF x (s) + βF y (s) induces a non zero divisor in O D whose value does not depend on s.

Proof. One can prove that thanks to the equisingularity Theorem, there exists a linear change of coordinates (u, v) such that for all s in a neighbourhood of 0 ∈ S, val Ds (u) = m (1) , . . . , m (p) . The conclusion follows from Corollary 4.3 and Lemma 3.18.

We want now to understand the behaviour of a generating family of the module of residues. We recall that plane curves are free divisors. Moreover, they are the only free divisors with isolated singularities, since by [START_REF] Aleksandrov | Nonisolated Saito singularities[END_REF], the singular locus of a free divisor is of codimension one in the hypersurface. The equisingularity assumption is not sufficient to obtain a deformation (ρ 1 (s), ρ 2 (s)) of a generating family of R D such that (ρ 1 (s), ρ 2 (s)) generate R Ds : equisingularity is not the "good" functor of deformation for free divisors. A functor of deformation adapted to free divisors is suggested by M.Torielli in [START_REF] Torielli | Deformations of free and linear free divisors[END_REF].

The following definition is equivalent to the definition of M.Torielli (see [Tor13, Definition 3.1]) thanks to both [Tor13, Proposition 3.7] and [GLS07, Theorem 1.91]: Definition 4.5. Let D be a free divisor defined in a neighbourhood of 0 ∈ C n by a reduced equation f . An admissible deformation X of D with base space S is a deformation of D such that the module O C n ×S,0 /(F, F x , F y ) is a flat O S,0 -module.

The following proposition describes an admissible deformation of a plane curve thanks to the Tjurina number.

Proposition 4.6. Let F be a deformation of f ∈ O C 2 (U ) with base space S such that for all s ∈ S,

x j ∈Sing(Ds) τ x j = τ 0 where τ 0 is the Tjurina number of D 0 . Such a deformation is an admissible deformation.

Proof. We use Theorem 1.81 of [START_REF] Greuel | Introduction to singularities and deformations[END_REF] for

F = p * O C 2 ×S /(F, F x , F y ) with p : C 2 × S → S the canonical epimorphism (which is finite). The map s → dim C F s /m S,s F s is then constant, thus, F 0 is a flat O S,0 -module. Since F 0 = O C 2 ×S,0 /(F, F x , F y ), the result follows.
Proposition 4.7 ([Tor13, Lemma 3.22]). Let F (x, y, s) be an equisingular and admissible deformation of the plane curve defined by f with base space S. Let (δ 1 , δ 2 ) be a basis of the module of logarithmic vector fields along D. It induces relations between f, f x , f y . By flatness, we can extend them to obtain relative logarithmic vector fields δ 1 , δ 2 ∈ (Θ U ×S/S )/(m S Θ U ×S/S ) of F . Then, for s in a neighbourhood of 0 ∈ S, δ 1 (s), δ 2 (s) is a basis of Der(-log D s ).

Corollary 4.8. If δ i = A i (x, y, s)∂ x +B i (x, y, s)∂ y , by the duality between the modules Der(-log D) and Ω 1 (log D) we find residues:

         res( ω 1 )(s) = -βA 2 (s) + αB 2 (s) αF x (s) + βF y (s) res( ω 2 )(s) = βA 1 (s) -αB 1 (s) αF x (s) + βF y (s)
such that they generate the module of residues for all s in a neighbourhood of 0 ∈ S, where α, β ∈ C are given by Proposition 4.4.

Properties of the stratification by logarithmic residues

We consider an equisingular deformation F of f with base space (S, 0) (C k , 0) for a k ∈ N. We denote by R s the module of logarithmic residues of D s . The purpose of this section is to study some properties of the following partition of S: Definition 4.9. Let F (x, y, s) be an equisingular deformation of a reduced plane curve D defined by f ∈ C{x, y}, with base space S. The stratification by logarithmic residues is the partition S = V ⊆Z p S V where s ∈ S V if and only if val(R s ) = V .

The following proposition will be useful to give the relation between the stratification by logarithmic residues and the stratification by the Tjurina number.

Proposition 4.10. Let (D, 0) be a plane curve germ. Then:

dim C R D /O D = τ -δ (19)
Proof. Thanks to Propositions 2.6 and 3.6 we have:

dim C R D /O D = dim C R D /O D -dim C O D /O D = dim C O ∨ D /R ∨ D -δ = dim C O D /J D -δ = τ -δ s ∈ S V ⇐⇒ s ∈ v∈V   1 r M   V F r A v-1 (s) ∩ 1 k p V F r (A v-1+e k (s) c    
where F r (A) denotes the ideal generated by the r×r minors of the matrix A and M = min(q, v +1). We notice that the elements v / ∈ V can not be reached since otherwise the dimension of R s /O Ds will be strictly greater than τ -δ, which can be proved by an argument similar to the first two steps of the proof of Theorem 2.4.

Hence the result for reducible curves.

Let us assume now that D is irreducible. In this case, the rank of the matrix increases exactly by 1 when a valuation is reached. We set V = {v 1 < . . . < v L } = (V + u) ∩ {0, . . . , u}. Then:

s ∈ S V ⇐⇒ s ∈ L =1 v -1 j=v -1 +1 V (F (A j (s)) ∩ (V (F (A v ) c
Therefore the stratum S V is locally closed.

We recall here the examples of [START_REF] Pol | Logarithmic residues along plane curves[END_REF] with more details. The first example shows that the stratification by logarithmic residues may be strictly finer than the stratification by the Tjurina number, whereas the second shows that the stratification by logarithmic residues does not satisfy the frontier condition.

Example 4.13. We consider f (x, y) = x 5 -y 6 and the equisingular deformation of f given by F (x, y, s 1 , s 2 , s 3 ) = x 5 -y 6 + s 1 x 2 y 4 + s 2 x 3 y 3 + s 3 x 3 y 4 . The stratification by τ is composed of three strata, S 20 = {0}, S 19 = {(0, 0, s 3 ), s 3 = 0} and S 18 = {(s 1 , s 2 , s 3 ), (s 1 , s 2 ) = (0, 0)}. The computation of the values of J Ds is quite easy in this case and gives thanks to Theorem 2.4, where S 18 = {(s 1 , s 2 , s 3 ), s 1 = 0} and S 18 = {(0, s 2 , s 3 ), s 2 = 0}:

Stratum dim C R Ds /O Ds negative values S 20 10 -1, -2, -3, -4, -7, -8, -9, -13, -14, -19 S 19 9 -1, -2, -3, -4, -7, -8, -9, -13, -14 S 18 8 -1, -2, -3, -4, -7, -8, -9, -14 S 18 8 -1, -2, -3, -4, -7, -8, -9, -13

The stratum S 18 divides into two strata for the values of R Ds , therefore, the stratification by logarithmic residues is strictly finer than the stratification by τ .

Example 4.14. Let us consider the deformation F (x, y, s 1 , s 2 ) = x 10 + y 8 + s 1 x 5 y 4 + s 2 x 3 y 6 for s 1 , s 2 in a neighbourhood of 0 so that the deformation is equisingular. It is given in [START_REF] Briançon | Déformation d'une singularité isolée d'hypersurface et polynômes de Bernstein[END_REF], as an example of the stratification by the b-function not satisfying the frontier condition.

A stratification S = α S α satisfies the frontier condition if for α = β, S α ∩ S β = ∅ implies S α ⊆ S β , with S β the closure of S β . Contrary to the previous example, this curve is not irreducible. We first give a property of quasi-homogeneous curves: Lemma 4.15. Let D be a quasi-homogeneous plane curve germ with p branches. Then:

γ -1 + (val(O D )\{0}) = val(J D )
Proof. The inclusion ⊆ is given by Remark 3.21. For the other inclusion, we notice that Remark 3.21 implies t 2γ-1 O D ⊆ J D ⊆ C D . We have the following equality: Let us come back to our example. We notice that F (x, y, s 1 , 0) is quasi-homogeneous. Therefore, the previous lemma shows that the values of the Jacobian ideal along the quasi-homogeneous stratum does not change. Therefore, the quasi-homogeneous stratum is a stratum of the stratification by logarithmic residues.

Moreover, one can check that there are three strata for the stratification by the Tjurina number: the quasi-homogeneous stratum S 1 defined by s 2 = 0 for which τ = 63, the stratum S 2 defined by s 1 = 0 and s 2 = 0 for which τ = 54 and the stratum S 3 defined by s 1 s 2 = 0 for which τ = 53. Therefore, the stratification by logarithmic residues does not satisfy the frontier condition, since there is a stratum S ⊆ S 2 which contains the origin in its closure, but not the whole quasi-homogeneous stratum.

Algorithms to compute the logarithmic residues along plane curves with one or two components

We suggest here several methods which can be used to compute the values of logarithmic residues. Thanks to the symmetry Theorem 2.4, computing the values of J D is equivalent to the computation of the values of R D .

Irreducible semi-quasi homogeneous polynomials

This algorithm is useful to study the equisingular deformation of a quasi-homogeneous polynomial of the form x a -y b , with gcd(a, b) = 1, and is inspired by [START_REF] Briançon | Le nombre de modules du germe de courbe plane x a + y b = 0[END_REF].

We consider the following equation of an irreducible curve, with s ij ∈ C and gcd(a, b) = 1:

F (x, y) = x a -y b + 1 i<a-1 1 j<b-1 ib+ja>ab s ij x i y j (20)
A parametrization of the curve is given by x(t) = t b + g(t), y(t) = t a + h(t) where g, h ∈ C{t} with val(g) > b, val(h) > a.

We set for i, j ∈ N 2 , ρ(i, j) = ib + ja. We define a monomial ordering by: (i, j) < (i , j ) if and only if ρ(i, j) < ρ(i , j ) or ρ(i, j) = ρ(i , j ) and i < i . If H = i,j a i,j x i y j ∈ C{x, y} is non zero, we set exp(H) = min ((i, j), a i,j = 0) and ρ(H) := ρ(exp(H)).

Polynomials of the form (20) are studied in [START_REF] Briançon | Le nombre de modules du germe de courbe plane x a + y b = 0[END_REF]. The authors give an algorithm to compute the "escalier " of the curve, which is by definition the complement in N 2 of the set E = {exp(g); g ∈ (F, F x , F y ) ⊆ C{x, y}}

  and we call R D the module of logarithmic residues of D. In particular, R D is a finite type O Dmodule. Moreover, we have the inclusion O D ⊆ R D (see [Sai80, Lemma 2.8]).

  The inclusion O D ⊆ R D gives the following corollary: Corollary 3.10. The set of values of R D is determined by the set {v ∈ val(R D ); v 0}

Proposition 3. 14 .

 14 Let v ∈ Z p . With the notation of Proposition 3.7, v ∈ M J D if and only if there exists D = D i 1 ∪ • • • ∪ D iq and δ ∈ Der(-log D ) = Der(-log D i 1 ) ∩ • • • ∩ Der(-log D iq ) such that for j /

  dim C C D /t 2γ-1 O D = dim C C D /J D + dim C J D /t 2γ-1 O DBy Propositions 4.10 and 2.6, we have dim C C D /J D = τ -δ. Since D is quasi-homogeneous, we have τ = µ so that by Proposition 4.2 we have dimC C D /J D = δ -p + 1. Moreover, since δ = dim C O D /O D , we have dim C O D /C D = 2δ, thus dim C C D /t 2γ-1 O D = 2δ -p. Therefore: dim C J D /t 2γ-1 O D = δ -1 Let m be the maximal ideal of O D . Then val(m) = val(O D )\{0} and the quotient m/C D has dimension δ -1. If val(J D ) = γ -1 + (val(O D )\{0}), by an argument similar to the one of the two first steps of the proof of Theorem 2.4, we have dim C J D /t 2γ-1 O D > δ -1, which is a contradiction. Hence the result.

Proposition 4.11. The stratification by logarithmic residues satisfies the following properties: 1. If s, s do not belong to the same stratum of the stratification by τ , they do not belong to the same stratum for the stratification by logarithmic residues. In other words, the stratification by logarithmic residues is finer than the stratification by τ .

2. The stratification by logarithmic residues is finite.

Proof. The first claim is a direct consequence of Proposition 4.10, since the equisingularity condition ensures that δ does not depend on the fiber, and the dimension of the quotient R s /O Ds can be computed from the values by Corollary 2.15. The second claim comes from both Proposition 4.4, which gives a lower bound u of the set of values of logarithmic residues which do not depend on s, and Corollary 3.10. As a consequence, the values of R s are determined by the values v of R s satisfying u v 0.

The hypothesis of D being irreducible was forgotten in [Pol15, Proposition 4.2]:

Proposition 4.12. Each stratum S V of the stratification by logarithmic residues is constructible. If moreover D is irreducible, then each stratum is locally closed.

Proof. For lack of reference, we suggest a proof. By the appendix by Teissier in [START_REF] Zariski | Le problème des modules pour les branches planes[END_REF], the strata of the stratification by the Tjurina number are locally analytic and locally closed. It is therefore sufficient to consider the behaviour of logarithmic residues in a τ -constant stratum S τ . For the sake of simplicity, we denote S = S τ . By Corollary 4.8, for all s, the O S -module R s is generated by

where α, β ∈ C are given by Proposition 4.4. The value of the common denominator αF x (s)+βF y (s) does not depend on s, so that it is sufficient to consider the values of the numerators.

We denote by N 1 and N 2 the numerators of ρ 1 (s) and ρ 2 (s). We recall that the values v of R s satisfying v 0 are sufficient to determine val(R s ), so that it is sufficient to consider the set {X 1 , . . . , X q } := x i y j N k ; val(x i y j N k ) u , where u = val(αF x (s) + βF y (s)).

For all i ∈ {1, . . . , q}, we have

We use the rank of the matrices A v (s) to characterize the property v ∈ val(R s ) for s ∈ S:

Indeed, if the conditions of the right-hand side are satisfied, then for all k ∈ {1, . . . , p}, there exists a linear combination

Therefore, for a given V ⊆ Z p for which S V ∩ S = ∅ and V := (V + u) ∩ {w ∈ Z p ; 0 w u}:

More precisely, they give the explicit computation of a finite family (A j ) -1 j K of points of N 2 such that E = K j=-1 A j + N 2 , and none of the A j 's can be removed. Then it is possible to prove:

Proposition 4.16. We have the following equality:

Irreducible plane curve

In [START_REF] Hefez | Standard bases for local rings of branches and their modules of differentials[END_REF], an algorithm is proposed to compute the set of values of Kähler differentials of an irreducible plane curve. By Proposition 3.19, it gives also the values of R D . In fact, one can see that the algorithm of [START_REF] Briançon | Le nombre de modules du germe de courbe plane x a + y b = 0[END_REF] corresponds to the algorithm of [START_REF] Hefez | Standard bases for local rings of branches and their modules of differentials[END_REF] by Proposition 3.16. Moreover, if a generating family of R D is known, the algorithm of [HH07, Theorem 2.4] can be used directly on this family to compute the values of R D .

Plane curves with two branches

Let D = D 1 ∪ D 2 be a plane curve germ with two irreducible components. We suggest here an algorithm to compute the set of negative values of R D . It is more technical than in the irreducible case, and cannot be generalized to plane curves with three or more branches. It can be compared to the fact that the analytic classification proposed in [START_REF] Hefez | The analytic classification of plane curves with two branches[END_REF] for two branches is also more complicated than in the irreducible case, and can not be easily extended to plane curves with three or more branches.

The algorithm in [START_REF] Hefez | Standard bases for local rings of branches and their modules of differentials[END_REF] is given for irreducible curves, for which the set of valuations is totally ordered, so that we cannot apply it directly to reducible plane curves. Nevertheless, we can use it if we consider only one of the components.

First step

First of all, we set g ∈ (f x , f y ) a non-zero divisor of O D , and we fix it as the common denominator of all residues of D 1 , D 2 and D, so that we can consider only the numerators to compute the set of values in each case.

Let i ∈ {1, 2}. We consider only the branch D i . By applying the algorithm of [HH07, Theorem 2.4] we compute a standard basis G i of O D i . We use it to compute also a standard basis R i of R D i , thanks to a generating family of R D i . As in [START_REF] Hefez | Standard bases for local rings of branches and their modules of differentials[END_REF], a G i -product is an element of the form q j=1 h α j j where h j ∈ G i , α j ∈ N and q ∈ N. To determine the missing values, we first compute the projection val 1 (R D ) of val(R D ). To do this, we apply the algorithm of [HH07, Theorem 2.4] to a generating family of R D , but by considering only the valuation along D 1 . We deduce a family R of elements of R D such that for all

Second step

We set M 0 = {(0, v 2 ); v 2 ∈ val 2 (R D 2 ) ∩ Z 0 } and H 0 = R 2 . By Proposition 3.7, it gives all the values of val(R D ) ∩ ({0} × Z 0 ).

Let us assume that for a k ∈ N * we have constructed sets M k-1 and

and for all v 2 ∈ val 2 (M k-1 ), there exists ρ ∈ H k-1 and a G 2 -product h with val 2 (h

Otherwise, there are several cases to consider.

First case: -k ∈ val 1 (R D ) ∩ val 1 (R D 1 ). By Proposition 2.11, one can see that

). Therefore the inclusion in ( 21) is an equality and

). There exists ρ 0 ∈ R such that val 1 (ρ 0 ) = -k. Let w 2 = val 2 (ρ 0 ). We may assume by Proposition 2.11 that w 2 0 since 0 ∈ val(R D ).

First sub-case: w 2 / ∈ val 2 (M k-1 ). Thanks to Propositions 2.11 and 2.13, one can check that:

) and v 2 w 2 } and H k = H k-1 ∪ {ρ 0 } Second sub-case: w 2 ∈ val 2 (M k-1 ). Thanks to propositions 2.11 and 2.13, one can check that by a convenient linear combination of ρ 0 and elements of form h • ρ with h a G 2 -product and ρ ∈ H k-1 , there exists ρ 0 ∈ R D with val(ρ 0 ) = (-k, w 2 ) and w 2 / ∈ val 2 (M k-1 ). We then recognize the previous sub-case, and we have H k = H k-1 ∪ {ρ 0 }.

We can stop when the minimal value -q of val 1 (R D ) is reached. Then, by Proposition 3.10: