Daniel Barlet
email: daniel.barlet@univ-lorraine.fr

Barlet Daniel

Algebraic differential equations of periods integrals

Keywords: AMS Classification, 32 S 25-32 S 40 Key words, Period-integral -Hermitian period -Formal Brieskorn Module -Geometric (a,b)-module -Fresco -Bernstein polynomial

We explain that in the study of the asymptotic expansion at the origin of a period-integral like γz ω/df or of a hermitian period like f =s ρ.(ω/df ) ∧ (ω /df ) the computation of the Bernstein polynomial of the "fresco" (filtered differential equation) associated to the pair of germs (f, ω) gives a better control than the computation of the Bernstein polynomial of the full Brieskorn module of the germ of f at the origin. Moreover, it is easier to compute as it has a better functoriality and smaller degree. We illustrate this in the case where f ∈ C[x 0 , . . . , x n ] has n + 2 monomials and is not quasi-homogeneous, by giving an explicit simple algorithm to produce a multiple of the Bernstein polynomial when ω is a monomial holomorphic volume form. Several concrete examples are given.

Introduction

This article simplifies and improves two unpublished papers, see [START_REF] Barlet | Algebraic differential equations associated to some polynomials[END_REF] and [START_REF] Barlet | A note on some fiber-integrals math[END_REF] on the computation of the Bernstein polynomial associated to a period-integral or to a hermitian period.

Let me explain the goal of this paper. We consider a holomorphic function f : U → C on an open neighborhood U of the origin in C n+1 which is singular at 0. The singularity is not assumed to be isolated, but we choose U small enough in order that f (0) = 0 is the only critical value of f on U 1 . We are interested, for instance, in the meromorphic extension of the holomorphic functions for (λ)

|h|:

1 Γ(λ) U |f | 2λ . f h .ρ.ω ∧ ω
where ω and ω are given holomorphic (n + 1)-forms on U , h is in Z and ρ is a C ∞ c (U ) function identically 1 near 0. We shall assume that for a given integer q ≥ 1 and for a given ξ ∈ Q the meromorphic extension of 1 Γ(λ) U |f | 2λ .ϕ (H(ξ, q))

1 recall that for any holomorphic function f singular at the origin, such a U always exists.

have poles at points in ξ + Z which are of order at most q -1, for any (n + 1, n + 1) differential form ϕ ∈ C ∞ c (U \ {0}) (so ϕ ≡ 0 near 0). For instance, if the singularity of f is isolated at the origin, this hypothesis will always be true for q = 1 and any ξ ∈ Q. When the singularity of f is not isolated, this condition will be satisfied for q = 1, for instance as soon as the local monodromy of f acting on the reduced cohomology of the Milnor' fibre at each point nearby 0 but distinct of 0 does not present the eigenvalue exp(2iπ.ξ). In general, this assumption means that any pole of order ≥ q at a point in ξ + Z comes form the germ of our situation at the origin. But note that even if the eigenvalue exp(2iπ.ξ) of the local monodromy of f acting on the reduced cohomology of the Milnor' fibre at any point nearby 0 (including 0) is simple, we may find a pole of order 2 for such an integral because of the phenomenon of "entanglement of consecutive strata" (see [START_REF] Barlet | Interaction de strates consécutives pour les cycles évanescents[END_REF] for a topological description and [START_REF] Barlet | Sur certaines singularités d'hypersurfaces II[END_REF] for a description in term of "Brieskorn modules" of this phenomenon) may appear. We shall give in the theorem 3.2.1 and corollary 3.2.2 some necessary numerical conditions which control the order of poles in ξ+Z for a given holomorphic (n+1)-form which is much more precise than the "classical condition" (which is in fact valid for such an integral when we replace ρ.ω∧ω by any differential form ϕ ∈ C ∞ c (U ) n+1,n+1 ) asking that the Bernstein polynomial of f at 0 has at most (q -1) roots (counting multiplicities) in the set ξ + Z. The precise result for the Mellin transforms of hermitian periods is given in the theorem 3.2.1 (the remark following the proof of the corollary 3.2.2 indicates also a variant which can be obtained by the same method. ) The examples given at the end of this paper show not only that the Bernstein polynomial of the fresco associated to the pair (f, ω) is much easier to compute than the full Bernstein polynomial of f , but also that it has, in general, a much smaller number of roots.

The main tool around this kind of technic will be the following generalization of the use of a Bernstein identity to control the poles of the Mellin transform of a "hermitian period" of the form

F (λ) := 1 Γ(λ) . U |f | 2λ . f h .ω ∧ ψ ( 1
)
where ω is a holomorphic form in Ω n+1 (U ), where ψ ∈ C ∞,(0,n+1) c

(U ) is d-closed near 0 and where h is in Z.

Theorem 1.0.1 Let f : U → C be a holomorphic function on a polydisc U with center 0 in C n+1 and assume that f (0) = 0 is the only critical value of f on U . For ξ given in Q, assume that the hypothesis H(ξ, 1) is satisfied (see above). Let ω be a (n+1) holomorphic differential form on U . Assume that the class induced by ω in E n+1 is annihilated by the element2 P := (a -λ 1 .b).S 1 (b)...(a -λ k .b).S k (b) in Ã, where for each j ∈ [1, k], S j ∈ C [[b]] satisfies S j (0) = 1. Note that this assumption depends only of the germs at the origin of f and of ω. Now fix ψ ∈ C ∞,(0,n+1) c (U ) which is d-closed near 0 and assume that for some h ∈ Z the meromorphic extension of

F ψ h (λ)[ω] := 1 Γ(λ) . U |f | 2λ . f .ω ∧ ψ
has a pole of order d ≥ 1 at some point in ξ + Z. Then there exists at least d values of j ∈ [1, k] such that λ j is in ξ + Z.

Note that the hypothesis of the existence of such a P is always true. But in practice (see the section 4) we may have such a P but we dont know that its initial form in (a,b) corresponds to the Bernstein polynomial of the fresco associated to ω. It is only a member of the (principal) left ideal which annihilates the class of ω in E n+1 . So, under the hypothesis of this theorem, the Bernstein polynomial of the fresco F ω associated to the pair (f, ω) (which is the geometric (a,b)-module generated by [ω] in E n+1 ) divides the polynomial

B(λ) := k j=1 (λ + λ j + j -k).
In the more precise statement given in section 3 (see theorem 3.1.2) we precise the values of these roots of the Bernstein polynomial of F ω from the "jumps" of the orders of poles in ξ + Z.

For the definition of the geometric (a,b)-module E n+1 and the notion of the Bernstein polynomial of a fresco, see the section 2 below.

We shall give a more precise result in the theorem 3.1.2 and some interesting variants using the hypothesis H(ξ, q) in the theorem 3.2.1 and corollary 3.2.2 in section 3.

Remark.

Notice that here we use in fact only a "one variable" differential equation (in fact multiplication by the variable in C and integration relative to this variable) instead of partial differential operators on C n+1 as in the Bernstein identity for f at the origin. This is precisely one of the interest of using an (a,b)-module structure in this setting.

So we begin this article by a short overview on geometric (a,b)-modules and frescos intended for the reader which is not familiar with the use of Brieskorn modules in the study of the singularities of a holomorphic function on a complex manifold.

In opposition with the preprint [START_REF] Barlet | A note on some fiber-integrals math[END_REF] cited above, we let aside the global point of view, that is to say the study of the global fresco associated to a period-integral in the case of a proper holomorphic function on a complex manifold, because it uses more heavy tools and very often the local study presented here would be enough to obtain good informations, using a partition of unity. The reader interested in this global setting may consult the preprint [START_REF] Barlet | A note on some fiber-integrals math[END_REF] mentioned above and also the preprint [START_REF] Barlet | A finiteness theorem for S-relative formal Brieskorn module[END_REF].

It is important to notice that we are dealing here with general singularities of a holomorphic function (not only the isolated singularity case as in the classical use of the Brieskorn module) and our illustration in the case of a (not quasi-homogeneous) polynomial in C[x 0 , . . . , x n ] with n + 2 monomials does not assume also that the singularity is isolated.

2 A short overview on (a, b)-modules and frescos The initial motivation comes from the study of germs of isolated singularities of holomorphic functions (f, 0) : (C n+1 , 0) → (C, 0) initiated at the end of the sixties by Milnor [START_REF] Milnor | Singular Points of Complex Hypersurfaces[END_REF], Brieskorn [START_REF] Brieskorn | Die Monodromie der isolierten Singularitäten von Hyperflächen[END_REF], Deligne [START_REF] Deligne | Equations différentielles à points singuliers réguliers[END_REF], Malgrange, [START_REF] Malgrange | Intégrale asymptotique et monodromie[END_REF], Varchenko [START_REF] Varchenko | Asymptotic behavior of holomorphic forms determines a mixed Hodge structure[END_REF], Saito Kyoji [START_REF] Saito | Kyoji Period mapping associated to a primitive form[END_REF], Saito Morihiko [START_REF] Saito | Morihiko On the structure of Brieskorn lattice[END_REF], ... and many others. To my knowledge the first who introduced the "operator" ∂ -1 z was Kyoji Saito in the beginning of the eighties (see [START_REF] Saito | Kyoji Period mapping associated to a primitive form[END_REF]). The main reason comes from the fact that, looking at period-integrals of the type z → γz ω/df where (f, 0) : (C n+1 , 0) → (C, 0) is a germ of holomorphic function with an isolated singularity, ω ∈ Ω n+1 C n+1 ,0 is a germ of holomorphic volume form and γ z , z ∈ H is a horizontal family of compact n-cycles in the fibers {f = z} of f , H → D * been the universal cover of a small punctured disc D * with center 0. The derivation ∂ z of such an integral is given by the following formula

∂ z γz ω/df = γz du/df (2)
where u ∈ Ω n C n+1 ,0 satisfies ω = df ∧ u. But in general it is not possible to find such a u ∈ Ω n C n+1 ,0 because writing ω = g(x).dx, the holomorphic germ g is not in the Jacobian ideal of f . Nevertheless, as the coherent sheaf Ω n+1 df ∧ Ω n has support inside {f = 0} near 0, the Nullstellensatz gives a positive integer p such that f p annihilates this sheaf near 0 and we may find u ∈ f -p .Ω n C n+1 ,0 such that ω = df ∧ u and (2) holds. But, of course, this implies that in formula (2) the derivation in z needs a denominator which is a power of z. Thanks to the positivity theorem of Malgrange (see [START_REF] Malgrange | Intégrale asymptotique et monodromie[END_REF]) we may write the formula (2) as follows:

z 0 γt du/df .dt = γz ω/df = γz u (3)
If we begin with ω := du ∈ Ω n+1 C n+1 ,0 we see that formula (3) does not need any denominator in z. Moreover the surjectivity of the de Rham differential d : Ω n → Ω n+1 shows that du may be any germ in Ω n+1 C n+1 ,0 and we may write

b γz du/df = γz df ∧ du/df = γz u (4)
so that the action of b only needs to solve the equation du = ω, and this is always possible with u ∈ Ω n C n+1 ,0 without introducing a denominator in f (so no denominator in z downstairs). As a is given by the formula :

a γz du/df = z. γz du/df = γz f.du/df (5)
because the n-cycle γ z is in the fiber f -1 (z) we see that the A-module structure on the quotient3 E f := Ω n+1 C n+1 ,0 d Ker(df ) n 0 does not need to consider meromorphic (n + 1)-differential forms with poles along the fibre {f = 0}. Note that in the case of an isolated singularity for f we have the equality

Ker(df ) n 0 = df ∧ Ω n-1 0
because the partial derivatives of f define a regular sequence at the origin. Also in this case we find that E f bE f is equal to the finite dimensional vector space O C n+1,0 J(f ) 0 where J(f ) is the Jacobian ideal of f . So we find the classical "Brieskorn lattice". But why is this presentation interesting if, at the end, we are compelled to introduce denominators in f (or in z working downstairs) to reach an ordinary differential system (or a differential equation ) ?

The answer comes from the following considerations: If you keep a module structure over the algebra A as a substitute for a differential system you have a richer structure (so more precise informations) than a structure of module over the localized Weyl algebra C z, z -1 , ∂ z associated to your differential system. This comes from the fact that the commutation relation a.b -b.a = b 2 is homogeneous of degree 2 in (a, b) and implies the existence of the decreasing sequence of two-sided ideals in A given by b m .A = A.b m , ∀m ∈ N. So any A-module E is endowed with a "natural filtration" (b m .E) m∈N by sub-A-modules. For instance Varchenkho [START_REF] Varchenko | Asymptotic behavior of holomorphic forms determines a mixed Hodge structure[END_REF] proves that in the case of an isolated singularity this filtration defines the Hodge filtration of the mixed Hodge structure on the cohomology of the Milnor's fibre of f . Another simple remark may also help to convince the reader that a module structure over A is interesting: The proof of this lemma is easily deduced from the following formula which is an easy consequence of the exercice above: In fact, the presence of the filtration by the two-sided ideals b m .A of the algebra A and the lemma above leads to the following considerations • The "fundamental" operation4 in the action of A is b !

• It seems convenient, as we are interested in the asymptotic expansions of the period-integrals γz ω/df when z → 0, to complete the algebra A for the uniform structure defined by the filtration b m .A, m ∈ N.

Note that for the "obvious" action of A on formal power series in z this filtration is associated to the valuation in z (see the remark following the exercise above) .

This means that we shall work with the algebra

à := { ν≥0 P ν (a).b ν , P ν ∈ C[a] ∀ν ∈ N}. (6)
The initial idea of Kyoji Saito was to add some convergence conditions in order that such series acts on convergent (multivalued) series likes

r∈R,j∈[0,N ] C{z}.z r .(Log z) j
where R is a finite subset in Q and N is a non negative integer, which are the kind of asymptotic expansions which are valid for our period-integrals. But thanks to the regularity of the Gauss-Manin connection, we dont loose any information by staying at the formal series level and this avoids a lot of painful (standard) estimates ! Remark also that the construction given in the lemma above is also valid for the algebra à and, moreover, that a module E over à without b-torsion is of finite type over C [[b]] if and only if the complex vector space E/b.E is finite dimensional. So, our definition of an (a, b)-module will be

• An (a,b)-module is a left Ã-module which is a free and finite type module over the (commutative

) sub-algebra C[[b]] ⊂ Ã.
Examples.

1. Let (f, 0) : (C n+1 , 0) → (C, 0) be a germ of holomorphic function with an isolated singularity. Let Ωp 0 be the formal completion at the origin of Ω p C n+1 ,0 . The quotient Êf := Ωn+1 0 df ∧ d( Ωn 0 ) endowed with the actions of a := ×f and b := df ∧ d -1 is an (a,b)-module (Note that the absence of b-torsion is a theorem; see [START_REF] Sebastiani | Preuve d'une conjecture de Brieskorn[END_REF] or [START_REF] Barlet | Sur certaines singularités non isolées d'hypersurfaces I[END_REF]).

Let

E := C[[b]]e 0 ⊕ C[[b]].

Geometric (a,b)-modules

The (a,b)-modules which appear in singularity theory of a function are special. They correspond to regular differential system and the notion of regularity is easy to define for an (a,b)-module: First we say that the (a,b)-module E has a simple pole when a.E ⊂ b.E. When it is the case, -b -1 .a acts on the (finite dimensional) vector space E/b.E and its minimal polynomial is called the Bernstein polynomial of E. We say in this case that E is regular. This is equivalent to the fact that E can be embedded in an (a,b)-module having a simple pole. Then we defined the Bernstein polynomial of a regular (a,b)-module E as the Bernstein polynomial of its saturation Ẽ by b -1 .a.

There is one more specific property for the (regular) (a,b)-modules coming from the singularity of a function f , which reflects the fact that the monodromy of f is quasi-unipotent and the positivity theorem of Malgrange: the fact that the roots of the Bernstein polynomial are negative rational numbers (compare with the famous theorem of Kashiwara [START_REF] Kashiwara | b-function and holonomic systems, rationality of roots of bfunctions[END_REF]). So we call geometric a regular (a,b)-module whose Bernstein polynomial have negative rational roots.

Example. In the previous example 2 the (a,b)-module has a simple pole and its Bernstein polynomial is, by definition, the minimal polynomial of the matrix -1 -1 0 -1 so its Bernstein polynomial is (λ + 1) 2 . Compare with the Bernstein type identity

(2λ + 1).∂ z -z.∂ 2 z (z λ+1 .Log z) = (λ + 1) 2 .z λ .Log z.
The following easy proposition will be needed in the sequel. Although it is rather standard, we shall sketch the proof for the convenience of the reader ]. This gives the regularity of F . The last points to prove is the fact that the Bernstein polynomial of F has negative rational roots (i.e. F is geometric) and the fact that when F is normal E/F is also geometric. We shall argue by induction on the rank of F . In the rank 1 case let e be a generator of

-(λ -ν) divides B Q = B E , or we have B E [x] = (x + (λ -ν)).B Q [x].
In both cases, as E is geometric, we obtain that -(λ -ν) is a negative rational number, and so is -λ. Moreover, in both cases, Q is also geometric. The induction step follows easily by considering a rank 1 normal sub-module G in F , using the following lemma and the fact that a quotient of a geometric (a,b)-module by a normal rank 1 sub-module is again geometric already proved above.

For a proof a the following lemma see for instance the remark 1.2 following the proposition 1.3 in [START_REF] Barlet | Théorie des (a,b)-modules I, Complex Analysis and Geometry[END_REF]. As we want to consider, in the non isolated singularity case, a sheaf of geometric (a,b)-module along the singular set {df = 0} of the zero set {f = 0} of a holomorphic function on a complex manifold M , we have to replace the completion used in the classical case of an isolated singularity case by a f -completion which is in fact the z-completion downstairs. This will not change seriously the considerations above, thanks to the following easy proposition which implies that any geometric (a,b)module is in fact a module over the algebra  := { p,q≥0 c p,q .a p .b q } which contains both Note that the hypothesis "geometric" insure that a is injective (this is not the case if we assume only the regularity). But the condition that there exists k ∈ N * such that

C[[b]] and C[[a]].
a k .E ⊂ b.E is enough to see that C[[a]
] acts on the (a,b)-module E. This condition is clearly satisfied by any regular (a,b)-module (see [START_REF] Barlet | Théorie des (a,b)-modules I, Complex Analysis and Geometry[END_REF]).

Frescos

We have seen that the (a,b)-module structure may be an interesting way to study the differential system associated to period-integrals for a germ of holomorphic function (f, 0) : (C n+1 , 0) → (C, 0). In fact, the Brieskorn module, or the (a,b)-module Êf which is the completion of the Brieskorn module Ω n+1 0 df ∧ dΩ n-1 0 in the isolated singularity case. It gives in fact a filtered version of the differential system satisfied by all the period-integrals associated to the germ (f, 0). But if we are interested by the period-integrals corresponding to a specific holomorphic differential form, it is clear that such a differential system, that is to say the all (a,b)-module Êf , does not give very precise informations. In term of differential system, we would prefer to have a specific differential equation satisfied by the integral-periods γz ω/df for our choice of ω than the differential system satisfied by all period-integrals, so associated to all choices of ω ∈ Ω n+1 0 . The analog of the differential equation in term of (a,b)-modules is the notion of "fresco". A fresco is a, by definition, a geometric (a,b)-module which is generated, as a Ã-module, by one generator. For instance, in the previous situation, we shall consider the fresco given by Ã.

[ω] ⊂ Êf and we shall call it the fresco of the pair (f, ω) at the origin. The following structure theorem describes in a very simple way such a fresco (see [START_REF] Barlet | Périodes évanescentes et (a,b)-modules monogènes[END_REF]).

Theorem 2.3.1 Any rank k fresco6 F with generator e is isomorphic (as an Ã-module) to a quotient à Ã.Π, the isomorphism F → à Ã.Π being defined by sending the generator e of F to the class of 1. We may choose Π having the following form

Π := (a -λ 1 .b).S 1 .(a -λ 2 .b).S 2 . . . S k-1 .(a -λ k .b).S k (7)
where the numbers -(λ j + j -k) are the roots of the Bernstein polynomial of F and where S j are in C[b] and satisfy S j (0) = 1 (so each

S j is invertible in C[[b]]).
Note that the initial form in (a,b) of Π is

P F := (a -λ 1 .b) . . . (a -λ k .b). It is
B G (x) = B F (x + rk(H)).B H (x)
between the Bernstein polynomials.

Recall that any element P in the algebra A which is homogeneous of degree k in (a,b) and monic in a may be written P = (a -r 1 .b) . . . (a -r k .b) where r 1 , . . . , r k are complex numbers. This equality is not unique but the sequence r j + j -k, j ∈ [1, k] depends only on P (see [START_REF] Barlet | Périodes évanescentes et (a,b)-modules monogènes[END_REF] proposition 2.0.2).

Our next proposition will be useful in computations of examples.

Proposition 2.3.2 Let F be a rank k fresco with generator e. Assume that Q ∈ Â has the following properties:

i) The initial form Q of Q in (a, b) has degree d and is monic in a. ii) Q[e] = 0 in F .
Then Q is a left multiple in A of P F , the Bernstein element of F . If moreover we have d = k, then Q is the Bernstein element of F .

Proof. Using the structure theorem of [START_REF] Barlet | Périodes évanescentes et (a,b)-modules monogènes[END_REF] recalled in the theorem 2.

A general existence theorem

Now consider a germ f : (C n+1 , 0) → (C, 0) such that {f = 0} is reduced. Let Ω• the formal f -completion of the sheaf of holomorphic differential forms on (C n+1 , 0) and let Ker df • be the kernel of the map

∧df : Ω• -→ Ω•+1 .
Then for any p ≥ 2 the p-th cohomology sheaf7 of the complex ( Ker df • , d • ) has a natural structure of left Â-module, where the action of a is given by multiplication by f and the action of b is (locally) given by df ∧ d -1 .

The following result is known (see [START_REF] Barlet | Sur certaines singularités non isolées d'hypersurfaces I[END_REF], [START_REF] Barlet | Brieskorn modules and Gauss-Manin systems for non isolated hypersurface singularities Bull[END_REF] and [START_REF] Barlet | Sur certaines singularités d'hypersurfaces II[END_REF]) ii) E p is b-separated and b-complete (so also a-complete). Then it is a Ã-module (and also a Â-module).

iii) There exists an integer m ≥ 1 such that We shall study several examples in section 4.

a m .E p ⊂ b.E p . iv) We have B(E p ) = A(E p ) = Ã(E p
3 Mellin transform of hermitian periods

The main result

We consider now a holomorphic function on an open polydisc U centered at the origin in C n+1 such that f (0) = 0 is the only critical value of f on U . Let ω be a holomorphic (n + 1)-differential form on U . Then let ψ be a C ∞ differential form with compact support in the polydisc U of type (0, n+1) which satisfies dψ ≡ 0 near 0. For any h ∈ Z define, at least for (λ) large enough, the holomorphic function

λ → F ψ h (λ)[ω] = 1 Γ(λ) . U |f | 2λ . f h .ω ∧ ψ. (9)
Note that the existence of a Bernstein identity for f in a neighborhood of 0 insures that for U small enough and any differential form ϕ ∈ C ∞ c (U ) (n+1,n+1) the holomorphic function defined by U |f | 2λ .ϕ has a meromorphic extension to the all complex plane with a finite series of poles of order at most n + 1 at points of the form ξ j -N where ξ j are negative rational numbers (see [START_REF] Kashiwara | b-function and holonomic systems, rationality of roots of bfunctions[END_REF]).

We shall make the following hypothesis, which we shall denote by H(ξ, 1):

• For ξ ∈ Q the local monodromy of f acting on the reduced cohomology of the Milnor's fibre at any point distinct from 0 does not admit the eigenvalue exp(2iπ.ξ).

Proposition 3.1.1 We assume the hypothesis H(ξ, 1). Let ω be a holomorphic (n + 1)-differential form on U and fix ψ be a C ∞ differential form with compact support in U of type (0, n + 1) which satisfies dψ = 0 near 0. We have the following formulas in E n+1 (recall that the geometric (a,b)-module E n+1 is defined in the theorem 2.4.1 above and that the action of a and b are defined by: a.

[ω] = [f.ω], b.[ω] = [df ∧ u] where u ∈ Ω p (U ) satisfies du = ω 8 ).
i) If there exists v ∈ Ω n (U ) satisfying df ∧ v ≡ 0 and dv = ω on U , then F ψ h [ω] has no pole in ξ + Z for any h and any ψ.

ii) F ψ h (λ)[a.ω] -(λ + 1).F ψ h-1 (λ + 1)[ω] has no pole in ξ + Z. iii) F ψ h (λ)[b.ω] + F ψ h-1 (λ + 1)[ω] has no pole in ξ + Z.
iv) So for any µ ∈ C the meromorphic function

F ψ h (λ)[(a -µ.b).ω] -(λ + µ + 1).F ψ h-1 (λ + 1)[ω]
has no pole in ξ + Z, combining ii) and iii).

Of course, the simplest example of such a ψ is given by ψ := ρ.ω where ρ is a C ∞ function with compact support such that ρ ≡ 1 in a neighborhood of 0 and where ω is a holomorphic (n + 1)-differential form in U .

proof. Write ω = du on U with u ∈ Ω n (U ). This is always possible as U is a polydisc so Stein and contractible. Then for (λ) > 1 + |h| the differential form α := |f | 2λ . f h .u ∧ ψ is C 1 in U and has compact support. So as we have:

dα = |f | 2λ . f h .du ∧ ψ + λ.|f | 2(λ-1) . f h+1 .df ∧ u ∧ ψ + (-1) n .|f | 2λ . f h .u ∧ dψ.
Stokes formula gives, as dψ vanishes near 0,

Γ(λ)

.

U dα = 0 = F ψ h (λ)[ω] + F ψ h+1 (λ -1)[b.ω] + G(λ)
where G(λ) is a meromorphic function on C which has no pole in ξ +Z thanks to our hypothesis H(ξ, 1). This implies i) and iii) using Γ(λ) = λ.Γ(λ -1). The formula ii) is easy and left to the reader.

Remark. The point i) of the previous proposition shows that F ψ h (λ)[ω] has no pole in ξ + Z when ω induces the zero class in E n+1 = H n+1 ( Ker df • 0 , d • ), and the point iii) implies the same conclusion when the class of ω in E n+1 is of b-torsion. So the polar part of F ψ h (λ)[ω] in ξ + Z depends only of the class induced by ω in E n+1 = E n+1 b-torsion for h and ψ fixed. This remark will be crucial in the sequel.

We shall give the following more precise result than the theorem 1.0.1 stated in the introduction. Let me recall the situation. Let f be a holomorphic function on a polydisc U with center 0 in C n+1 and assume that f (0) = 0 is the only critical value of f on U . Let ω be a (n + 1) holomorphic differential form on U and let ψ be a C ∞ differential form with compact support in U of type (0, n + 1) which satisfies dψ ≡ 0 near 0. Then define as above

F ψ h (λ)[ω] := 1 Γ(λ) . U |f | 2λ . f h .ω ∧ ψ (9)
Theorem 3.1.2 Fix ξ ∈ Q and an integer d ≥ 1. Assume that the hypothesis H(ξ, 1) is satisfied. Then assume that, for a given ψ, the meromorphic extension of F ψ h (λ)[ω] has no pole of order ≥ d + 1 at any point in ξ + Z, for any choice of h ∈ Z, but that there exists a point in ξ + Z where, for some h, this meromorphic extension has a pole order d. For each integer s ∈ [1, d] let ξ s be the biggest element in ξ + Z for which there exists h ∈ Z such that F ψ h (λ)[ω] has a pole of order at least equal to s at ξ s . Then each ξ s for s ∈ [1, d] is a root of the Bernstein polynomial of the the fresco F ω . Moreover, if ξ s = ξ s+1 = • • • = ξ s+p then ξ s is a root of the Bernstein polynomial of F ω with multiplicity at least equal to p + 1.

Remark that the theorem implies that the Bernstein polynomial of the fresco F ω is a multiple of d s=1 (λ -ξ s ) and that we have

ξ d ≤ ξ d-1 ≤ • • • ≤ ξ 1 < 0 by definition.
The proof of this theorem needs some lemmata. proof. Write S(b

) := 1 + ∞ m=1 s m .b m . As F ψ h (λ)[b m .ω] = (-1) m .F ψ h-m (λ + m)[ω], for each m ≥ 1, thanks to point iii) in the proposition 3.1.1, the meromorphic ex- tension of F ψ h (λ)[b m
.ω] cannot have a pole of order ≥ s at the point ξ s for any choice of h. So the pole of order ≥ s given by the initial term (i.e. m = 0) for a suitable value of h, stays maximal. This proves 1. Because we assume µ + ξ s = 0, the point iv) in the proposition 3.1.1 shows that the pole at ξ s -

1 of F ψ h (λ)[(a -µ.b
)ω] has the same order than the pole at ξ s for F ψ h+1 (λ) [ω]. Also, the same formula shows that for any integer p ≥ 0 the order of the pole ξ s + p for F ψ h (λ)[(a -µ.b)ω] is less or equal to the order of the pole

ξ s + p + 1 ≥ ξ s + 1 for F ψ h-1 (λ)[ω]
which is at most s -1 by definition of ξ s . This allows to conclude. proof. Using the point iv) of the proposition 3.1.1 we obtain that, for some suitable choice of h, the meromorphic extension of F ψ h (λ)[(a + ξ s+1 .b).ω] has a pole of order ≥ s at the point ξ s+1 -1 = ξ s -1. Assume that for some integer p ≥ 0 and some h ∈ Z the meromorphic extension of F ψ h (λ)[(a + ξ s+1 .b).ω] has a pole of order ≥ s at ξ s + p. Then using again the formula of the point iv) in the proposition 3.1.1 we find that (λ -ξ s + 1).F ψ h-1 (λ + 1)[ω] has, for a suitable choice of h ∈ Z, a pole of order ≥ s at λ = ξ s + p. But ξ s + p -ξ s + 1 = p + 1 = 0 so we find a pole of order ≥ s at the point ξ s + p + 1 for F h-1 (λ) [ω]. As p + 1 ≥ 1 this contradicts the definition of ξ s , so ξ s -1 is the biggest pole of order ≥ s in ξ + Z when h varies in Z, for the meromorphic extension of

F ψ h (λ)[(a + ξ s .b).ω].
Lemma 3.1.5 In the situation of the theorem 3.1.2, there exists a minimal integer

j d ∈ [0, k -1] such that ξ d -j d = -λ k-j d and for each s ∈ [1, d -1] there exists a minimal integer j s ∈ [j s+1 + 1, k -1] such that ξ s -j s = -λ k-js . Moreover the meromorphic extension of F ψ h (λ)[P k-js-1
.ω] has a pole of order ≥ s -1 at the point ξ s -j s -1, where we define P r := (a -λ 1 .b).S 1 (b) . . . S r-1 .(a -λ r .b).S r for r ∈ [1, k] where P k := Π is given by applying the theorem 2.3.1 to the fresco F ω and satisfies moreover the condition that the sequence (λ j + j), j ∈ [1, k] is non decreasing.

proof. The fact that we may assume that the sequence (λ j + j), j ∈ [1, k] is non decreasing is consequence of the existence of a principal Jordan-Hölder sequence for a fresco (see [START_REF] Barlet | Asymptotics of a vanishing period : characterization of semisimplicity[END_REF] theorem 1.2.5 or [9] corollary 3.5.4). Note also that the roots of the Bernstein polynomial of F ω are given by the numbers -(λ j + j -k) for j ∈ [1, k] . Now the first point is to prove that there exists an integer j ∈ [0, k -1] such that ξ d -j = -λ k-j , because we may then define j d as the minimal such integer. So assume that no such j exists. Then applying the lemma 3.1.3 we will obtain that the meromorphic extension of F ψ h (λ)(Π.ω) has a pole of order at least equal to d ≥ 1 at the point ξ d -k. But points i) and iii) in the proposition 3.1.1 implies that there is no pole in ξ + Z in the meromorphic extension of F ψ h (λ)[Π.ω] as [Π.ω] = 0 in E n+1 . Contradiction. So such a j exists and j d is well defined. The same argument than for s = d shows that for each s ∈ [1, d -1] there exists at least one j ∈ [1, k] such that ξ s -j = -λ k-j . Now we have

ξ s+1 = -λ k-j s+1 + j s+1 ≤ ξ s = -λ k-j + j.
The non decreasing property of the sequence λ j +j implies then that j ≥ j s+1 . If the inequality is strict, then we obtain j s := j and the point iv) in the proposition 3.1.1 implies that F ψ h (λ)[P k-js-1

.ω] has a pole of order ≥ s -1 at the point ξ s -j s -1. If we have ξ s+1 = ξ s then the lemma 3.1.4 shows that ξ s -j s+1 -1 is still in this case the biggest pole of order ≥ s in ξ + Z for the meromorphic extension of F ψ h (λ)[(a + ξ s .b).P k-j s+1 -1 .ω] when h varies in Z. So we can continue to apply S k-j s+1 -2 and then a -λ k-j s+1 -2 .b, etc... until we reach another j < j s+1 such that ξ s -j = -λ j , and then we conclude using the same argument as above.

Remarks.

1. The sequence j s , s ∈ [1, d] is strictly decreasing, so the sequence ξ s -j s is strictly decreasing and there are exactly d rational numbers

ξ s = -(λ k-js -j s ) = -(λ k-js + k -j s -k)
counting multiplicities (remark that the multiplicities correspond to equalities λ k-js -λ k-j s+1 = j s -j s+1 ).

2. As long as r ≤ j s the rational number ξ s stays the biggest pole of order ≥ s in ξ + Z for the meromorphic extension of

F ψ h (λ)[Q r .ω] when h describes Z, where Q r := S r .(a -λ r+1 .b).S r+1 ...(a -λ k .b).S k (@)
Proof of the theorem 3.1.2. We shall prove first, by induction on d ≥ 1, that there exists at least d values of j ∈ [1, k] such that -λ j belongs to ξ + Z. So assume that either d = 1 or that d ≥ 2 and that our claim is proved for d -1.

Then consider the poles of the meromorphic extension of F h (λ)[Q k-j d .ω] (where Q r is defined in the formula (@) above) and where the integer j d is defined in the lemma 3.1.5. Using the lemma 3.1.3 applied to ξ d , we obtain that it has a maximal pole of order d at the point ξ d -j d for a suitable choice of h and, applying the lemma 3.1.5 we conclude that the meromorphic extension of

F h (λ)[(a -λ k-j d .b).Q k-j d .ω]
has a pole of order at least equal to d -1 at the point

ξ d -j d -1. But the form ω := (a -λ k-j d .b).Q k-j d .ω is killed in E n+1 by P k-j d -1 , so the induction hypothesis gives at least d -1 values of j ∈ [1, k -j d -1] such that -λ j is in ξ + Z. As -λ k-j d = ξ d -j d belongs to ξ + Z

Some variants

The following variant of the previous result is also useful.

Assume now that we fix ξ + Z, and integer q ≥ 1 and that we make the following hypothesis:

• For any differential form ϕ ∈ C ∞,(n+1,n+1) c (U \{0}) the meromorphic extension of 1 Γ(λ) . U |f | 2λ .ϕ (H(ξ, q))
has never a pole of order at least equal to q at some point in ξ + Z.

Note that we require that ϕ ≡ 0 near 0.

Theorem 3.2.1 Let ω ∈ Ω n+1 (U ) and fix a differential form ψ ∈ C ∞,(0,n+1) c (U ) which is d-closed near 0.
Assume that the condition H(ξ, q) is satisfied and that for some integer h ∈ Z the meromorphic extension of

F ψ h (λ[ω]) := 1 Γ(λ) . U |f | 2λ . f h .ω ∧ ψ (10)
has a pole of order q + d -1 at some point in ξ + Z with d ≥ 1, maximal in N * . Let ξ 0 be maximal in ξ + Z such there exists h ∈ Z with the property that F ψ h (λ)[ω] has a pole of order q + d -1 at ξ 0 . Then ξ 0 is a root of the Bernstein polynomial B ω of the fresco F ω associated to the germs of f and ω at the origin, and B ω admits at least d roots in ξ 0 + N (counting multiplicities).

proof. The argument is analog to the one given in the proof of the theorem 3.1.2. The change to make in the proof is that we must take in account here only the polar parts of order at least equal to q of the poles at points in ξ + Z. So in the proposition 3.1.1 and in the lemmata 3.1.3, 3.1.4 and 3.1.5 we have to replace "no poles in ξ + Z" by "no pole of order ≥ q in ξ + Z" under the hypothesis H(ξ, q). Also we define ξ s as the maximal pole of order ≥ q + s -1 for s ∈ [1, d].

The other difference in the argument lies in the fact that in the Stokes formula the extra term given by the differential of ψ equal to:

G(λ) := (-1) n Γ(λ) . U |f | 2λ . f h .u ∧ dψ
has no pole of order ≥ q at a point in ξ + Z because we assumed dψ ≡ 0 near 0 and we may apply our hypothesis H(ξ, q) with ϕ := f h .u ∧ dψ for h ≥ 0 or with ϕ := f -h .u ∧ dψ for h ≤ 0.

Let specialize now the form ψ ∈ C ∞,(0,n+1) c

(U ) such that dψ ≡ 0 near the origin by defining ψ := ρ.ω where ω is a fixed holomorphic (n + 1)-differential form on U and where ρ is a function in C ∞ c (U ) which is identically equal to 1 near the origin (so d(ρ.ω ) = d ρ ∧ ω vanishes identically near the origin). Then we consider the Mellin transform of the hermitian period

z → 1 (2iπ) n . f =z ρ.(ω/df ) ∧ (ω /df ).
In the following corollary we use the hermitian symmetry between ω and ω in order to obtain a better control of the poles of the Mellin transform using the Bernstein polynomials of the frescos associated to ω and ω at the origin. Corollary 3.2.2 Let f : (C n+1 , 0) → (C, 0) be a non constant holomorphic germ. Fix ξ ∈ Q and a positive integer q. Assume that the hypothesis H(ξ, q) holds for f and consider ω and ω two germs of (n+1)-holomorphic forms. Let q +d-1, d ≥ 1, be the maximal order of pole9 for

(2iπ) n+1 .F h ω,ω (λ) := 1 Γ(λ) . U |f | 2λ . f h .ρ.ω ∧ ω
for any choice of a point in ξ + Z and for any choice of h ∈ Z. Let ξ 0 be maximal in ξ + Z such that there exists some h ∈ Z and a pole of order q + d -1 at ξ 0 ∈ ξ + Z for F h ω,ω (λ). Then ξ 0 is a root of the Bernstein polynomial of the fresco F ω and there exists at least d roots of the Bernstein polynomial of F ω in (ξ 0 + N) ∩ [ξ 0 , 0[ counting multiplicities. Moreover, under our hypothesis, there exists ξ 1 ∈ ξ + Z such that for some h ∈ Z, F h ω ,ω (λ) has a pole of order q + d -1 at ξ 1 . Let ξ 1 be maximal in ξ + Z such that this happens. Then ξ 1 is a root of the Bernstein polynomial of the fresco F ω and there exists at least d roots of the Bernstein polynomial of

F ω in (ξ 1 + N) ∩ [ξ 1 , 0[ counting multiplicities.
proof. The first statement is a special case of the previous theorem. We shall deduce the second statement by using complex conjugaison. Let ξ 0 ∈ ξ + Z and h 0 ∈ Z such that F (λ) := (2iπ) n+1 .F h 0 ω,ω (λ) has a pole of order q + d -1 at ξ 0 . As F (λ) has only real poles, the poles of F ( λ) are the same than the poles of F (λ) with the same orders. Moreover we may assume that the function ρ is real, so F ( λ) is given by

(2iπ) -(n+1) Γ(λ) . U |f | 2(λ+h 0 ) . f -h 0 .ρ.ω ∧ ω = (2iπ) -(n+1) Γ(µ -h 0 ) . U |f | 2µ . f -h 0 .ρ.ω ∧ ω.
where µ = λ + h 0 . But F (λ) is holomorphic when (λ) ≥ 0 and (λ + h 0 ) ≥ 0 so we may replace Γ(µ -h 0 ) by Γ(µ) in the right hand-side without changing the poles and their orders. We conclude that F -h 0 ω ,ω has a pole of order q + d -1 at ξ 0 + h 0 and applying the first statement gives the conclusion.

Remark. Of course, with the same method, we can obtain an analog result than in the theorem 3.2.1 for the asymptotic expansion at the origin of a period-integral of the type

s → γs ω/df
where ω is a holomorphic (n + 1)-form on U and where (γ s ) s∈H is a horizontal family of compact n-cycles in the fibers of f : Assuming the hypothesis H(ξ, q) for q ≥ 1 and the existence of a non zero term like s m-ξ .(Log s) q+d-2 with d ≥ 1 (or s m .(Log s) q+d-1 for ξ = 0), in such an expansion will imply that the Bernstein polynomial of the fresco F ω will have at least d roots (counting multiplicities) in the set ξ + Z. This gives a numerical criterion to insure that such a term will not appear in the expansion we are interested in. [START_REF] Barlet | Théorie des (a,b)-modules I, Complex Analysis and Geometry[END_REF] The case of a polynomial with (n + 1) variables and (n + 2) monomials

The purpose of this section is to give a general algorithm in order to obtain an "estimate" of the Bernstein polynomial of the fresco associated to (f, ω) for any polynomial f ∈ C[x 0 , . . . , x n ] with (n + 2) monomials and for any monomial holomorphic differential form ω = x β .dx of degree n + 1, where β is in N n+1 (we exclude the quasi-homogeneous case for f which is obvious). Using the results of the previous sections we obtain a rather precise information of the exponents of the asymptotic expansions of the period integrals γz ω/df where (γ z ) z∈H is a horizontal family of n-cycles in the fibers of f . This gives also rather precise informations on the poles of the meromorphic extensions of the Mellin transform of the hermitian periods z → f =z ρ.(ω/df ) ∧ (ω /df ):

1 Γ(λ) . C n+1 |f | 2λ . f h .ρ.ω ∧ ω where ρ ∈ C ∞ c (C n+1
) satisfies ρ ≡ 1 near 0, where ω, ω are monomial holomorphic differential forms of degree n + 1 and where h is in Z. We shall illustrate the result by several examples.

Our setting

We consider a polynomial f ∈ C[x 0 , . . . , x n ] which is the sum of n + 2 monomials

f = n+2 j=1 m j
where m j := σ j .x α j , with σ j ∈ C * and α j ∈ N n+1 are not 0. Define the matrix with (n + 1) lines and (n + 2) columns M = (α i,j ) and let M be the square (n + 2, n + 2) matrix obtained from M by adding a first line equal to (1, . . . , 1). We shall assume the following conditions:

(C1) α 1 , . . . , α n+1 is a Q-basis of Q n+1 . (C2) The rank of M is n + 2.

Remarks.

1. Only the condition (C2) is restrictive on f : when (C2) is fulfilled the condition (C1) may always be satisfied without changing f by a suitable ordering of the n + 2 monomials.

The condition (C2

) is equivalent to the fact that f is not quasi-homogeneous.

A diagonal linear change of variables allows to reduce the study to the case where

f (x) = n+1 j=1 x α j + λ.x α n+2 (a)
for some λ ∈ C * . So in what follows, we shall assume that m j = x α j for j ∈ [1, n+1] and m n+2 = λ.x α n+2 where λ ∈ C * is a parameter.

Then we shall write, using our hypothesis (C1):

α n+2 = n+1 j=1 ρ j .α j (b)
where ρ j are rational numbers. We define

H := {j ∈ [1, n + 1] / ρ j = 0}; J + := {j ∈ [1, n + 1] / ρ j > 0}; J -:= {j ∈ [1, n + 1] / ρ j < 0}.
Let |r| be the smallest positive integer such that |r|.ρ j := p j is an integer for each j ∈ [1, n + 1]. Write now the relation above as

|r|.α n+2 + j∈J - (-p j ).α j = j∈J + p j .α j . (c)
Now define d + h and d as respectively the supremum and infimum of the two numbers |r| + j∈J -(-p j ) and j∈J + p j . Then d and h are positive : -the non vanishing of d is consequence of the fact that |r| ≥ 1 and that at least one p j is positive.

-the non vanishing of h is consequence of the fact that the equality of these two integers would imply that the first line in M satisfies the same linear relation (b) than all the other lines in M , contradicting our hypothesis (C2). The relation (e) above gives the following equality between the monomials (m j ) j∈ [1,n+2] :

m |r| n+2 . j∈J - m -p j j = λ |r| . j∈J + m p j j (d)
and we shall write it

m ∆ = λ r .m δ (e)
where ∆ and δ are in N n+2 of respective length d + h and d.

Remark that ∆ j and δ j are zero for each j ∈ H. Note that the relation (e) defines the sign of r which is in Z * .

We shall also use the following observation later on :

Lemma 4.1.1 The j-th element of the first column of the matrix M -1 is zero if and only if j is in H.

Proof. The co-factor of the element (1, j) in M is the (n + 1, n + 1) determinant of the matrix with columns α 1 , . . . , αj , . . . , α n+2 . This matrix has rank at most n if and only if α n+2 is a linear combination of α 1 , . . . , αj , . . . , α n+1 . This is the case if and only if ρ j = 0, thanks to our hypothesis (C1).

The result

Let Ω p be the C[x 0 , . . . , x n ]-module of algebraic p-differential forms on C n+1 and fix a polynomial f ∈ C[x 0 , . . . , x n ] with (n + 2) monomials

f := n+2 j=1 m j (11)
where m j := x α j for j ∈ [1, n + 1] and m n+2 := λ.x α n+1 , with λ ∈ C * , satisfying the conditions (C1) and (C2) above.

We define df p : Ω p → Ω p+1 the C[x 0 , . . . , x n ]-linear map given by exterior product by df and we note Ker(df ) p its kernel. Let

m η → λ η n+2 . n i=0 x n+2 j=1 α i,j .η j i where η ∈ N n+2 , is a sub-A-module of E f [λ].
Proof. We want to prove that this image is stable by the action of a and b. The stability by a is obvious because a is given by the multiplication by n+2 j=1 m j . Let γ ∈ N n+1 and compute b using a primitive in x i , for any i ∈

[0, n]: b(x γ+β .dx) = 1 γ i + β i + 1 .x i .x γ+β . ∂f ∂x i .dx.
Remark now that x i . ∂f ∂x i .dx = n+2 j=1 α i,j .m j so the previous computation gives, with γ := M.η for some η ∈ N n+2 and then γ

i := n+2 j=1 α i,j .η j , Γ i (η, β).b(m η .x β .dx) = n+2 j=1 α i,j .m j .m η .x β .dx ∀i ∈ [0, n], (@ i )
where Γ i (η, β) := 1 + β i + n+2 j=1 α i,j .η j . Note that we have also

a(m η .x β .dx) = n+2 j=1 m j .m η .x β .dx (@ n+1 )
The formulas (@ i ) for i ∈ [0, n + 1] are enough to conclude the proof.

Corollary 4.2.2 Fix β ∈ N n+1 . For each η ∈ N n+2 there exists an element P β,η (a, b) in A, homogeneous of degree q := |η| := n+2 j=1 η j in (a,b) such that:

1. There exists c(β, η) ∈ Q * such that P β,η (a, b)[x β .dx] = c(β, η).m η .x β .dx in E f .

2. Assuming that η satisfies η j = 0 for each j ∈ H, there exists rational numbers (depending on β and η) r 1 , . . . , r q such that P β,η (a, b) = q h=1 (a -r h .b) in A.

Proof. Let first show by induction on q ≥ 0 that such a P β,η (a, b) satisfying 1. exists and that it satisfies 2. when η has no component on H. As for q = 0 the assertion is clear with P ≡ 1, assume that our assertion is proved for any η with |η| = q -1 with q ≥ 1.

Then it is enough to prove the assertion for m j .m η for each j ∈ [1, n + 2] and each η with |η| = q -1.

Then consider the equations (@ i ) for i ∈ [0, n + 1] as a square Q-linear system of size (n + 2, n + 2) with unknown the elements m j .m η .x β .dx in E f . The matrix of this system is in Gl(Q, n + 2) thanks to our hypothesis (C2), and so there exists rationals numbers u j and v j such that we have , for each j ∈ [1, n + 2] m j .m η .x β .dx = (u j .a + v j .b)(m η .xβ.dx).

With our induction hypothesis this gives that the homogeneous degree q element in A defined by P β,η+1 j (a, b) := (u j .a + v j .b).P β,η (a, b) satisfies 1.

Assuming that η + 1 j has no component on H we obtain that P β,η (a, b) is monic in a (up to a non zero rational number) and then it is enough to show that u j is not zero to conclude the induction. This is given by the lemma 4.1.1.

Proof. The previous corollary applied to both sides of the equality in E f [λ] m ∆ .x β .dx = λ r .m δ .x β .dx deduced from (e) and the corollary 4.2.2 allow to conclude because we know that ∆ j = 0 and δ j = 0 for each j ∈ H.

This theorem has the following corollary. This shows that the computation of P d (a, b) gives rather precise informations on the asymptotic expansion at 0 of such a integral-period. We leave the corresponding statement for the poles of the Mellin transform of the hermitian period-integrals corresponding to such an f and monomials holomorphic differential forms ω and ω to the reader. Of course, by conjugaison, the Bernstein polynomial of the fresco (f, ω ) also gives constraints on the possible poles of this Mellin transform, as in the corollary 3.2.2.

Examples

The control of the Bernstein polynomial of a fresco will use the theorem 4.2.3 and the proposition 2.3.2. We assume that λ is a non zero complex number. Then 0 is the only singular point of the hypersurface {f = 0} : as on the set Σ := {df = 0} ⊂ C 3 we have f (x, y, z) = 1 5 λ.x.y.z, we easily deduced that Σ ∩ {f = 0} = {0}. Now using the method explained above we obtain, after some elementary computations, for each monomial form ω below a degree 4 polynomial multiple of the Bernstein polynomial of the fresco F ω . Note that such a fresco has rank at most equal to 4 and if the rank is equal to 4 then we obtain the Bernstein polynomial itself. Of course, the reader interested by more monomials can easily complete this list, where | means "divides" :

• ω = dx ∧ dy ∧ dz B 1 (ξ) | (ξ + 7 10 )(ξ +

2. 1

 1 Why to use an (a, b)-module structure instead of a differential system ? Note first that in "(a, b)" a is the multiplication by the variable z and b is the primitive vanishing at z = 0, so b(f )(z) := z 0 f (t).dt where f is, for instance, a holomorphic multivalued function with an eventual ramification point at z = 0. So we are working with the non commutative algebra A generated by a and b with the commutation relation a.b -b.a = b 2 as unique relation. This relation corresponds to the usual commutation relation ∂ z .z -z.∂ z = 1 in the Weyl algebra C z, ∂ z . Then why not to use the usual Weyl algebra ?

exercice.

  Show that a.b m = b m .a + m.b m+1 , ∀m ∈ N is consequence of the commutation relation corresponding to m = 1. Note that this relation implies that a and b m commute modulo b m+1 .A. Also, looking at the "natural action" of A on C[[z]] which is given by a(z m ) = z m+1 and b(z m ) = z m+1 /(m + 1), you will see that b m .C[[z]] = z m .C[[z]], m ∈ N is the filtration by the valuation.

Lemma 2 . 1 . 1

 211 Let E := ⊕ k j=1 C[b].e j be a free C[b]-module with basis e 1 , . . . , e k and let x 1 , . . . , x k be any given collection of elements in E. Then there exists an unique A-module structure on E such that a) The action of a is defined by a.e j = x j for each j ∈ [1, k]. b) The action of b is given by the C[b]-structure of E.

  a.(S(b).e j ) = S(b).x j + b 2 .S (b).e j ∀j ∈ [1, k] where S (b) is the "usual" derivative of the polynomial S ∈ C[b].

  e 1 be the Ã-module defined by a.e 0 := b.e 0 and a.e 1 = b.e 1 -b.e 0 . Then it is an easy exercice to show that E is isomorphic toC[[z]] ⊕ C[[z]].Log zwhere a := ×z and b := z 0 . Determine the filtration (b m .E) m∈N in this example and compare it with the filtration by the (a m .E) m∈N . Compute the module over the Weyl algebra generated by Log z and compare with E.

  For a general (a,b)-module the saturation Ẽ of E by the action of b -1 .a is not always a finite type C[[b]]-module. When Ẽ is of finite type over C[[b]], Ẽ is an (a,b)-module (with simple pole) with the same rank over C[[b]] than the rank of E.

Proposition 2 . 2 . 1

 221 Let E be a geometric (a,b)-module and F any sub-Ã-module in E. Then F is a geometric (a,b)-module. When F is normal, the quotient E/F is again a geometric (a,b)-module. proof. From the regularity of E we may assume that E is a simple pole module (i.e. a.E ⊂ b.E). Then the Bernstein polynomial of E is the minimal polynomial of the action of -b -1 .a on the finite dimensional vector space E/b.E. As F is a C[[b]] sub-module of E which is free and finite rank on C[[b]], F is also free and finite rank on C[[b]] and stable by a. So F is an (a,b)-module. Its saturation by b -1 .a is again contained in E and so it is also free of finite type on C[[b]

F

  over C[[b]] such that a.e = λ.b.e (see the classification of rank 1 regular (a,b)-module in [B.93], lemma 2.4). Let ν in N be maximal such that b -ν .e lies in E. Then C[[b]].b -ν .e = b -ν .F is a normal 5 sub-module of E and we have an exact sequence of simple poles (a,b)-modules 0 → b -ν .F → E → Q → 0 and also an exact sequence of (-b -1 .a) finite dimensional vector spaces 0 → C.b -ν .e → E/b.E → Q/b.Q → 0. Then the minimal polynomial B E of the action of -b -1 .a on E/b.E is either equal to the minimal polynomial B Q of the action of -b -1 .a on Q/b.Q, and in this case

Lemma 2 . 2 . 2

 222 Let E be a regular (a,b)-module and let F ⊂ E be a sub-(a,b)-module. Assume that λ is a root of the Bernstein polynomial B F of F . Then there exists λ ∈ λ + N such that λ is a root of the Bernstein polynomial B E of E.

Proposition 2 . 2 . 3

 223 Any geometric (a,b)-module is complete for the decreasing filtration by the C[a]-sub-modules (not stable by b in general) (a m .E) m∈N .

8 )

 8 called the Bernstein element of the fresco F . It does not depend of the choice of the generator of F over à (choice which determines Π) and is related to the Bernstein polynomial B F ∈ C[λ] of F by the relation in the ring A[b -1 ]: (-b) k .B F (-b -1 .a) = P F , where k := rk(F ). (In the case of a fresco F the Bernstein polynomial B F is equal to the characteristic polynomial of the action of -b -1 .a on F /b. F where F is the saturation of F by b -1 .a (see [9] theorem 3.2.1). This makes the computation of the Bernstein polynomial of a fresco easier than for a general geometric (a,b)-module, for instance by the use of the following remark: If 0 → F → G → H → 0 is an exact sequence of frescos we have the relation P G = P F .P H (product in A) between the Bernstein elements and this gives the relation (see [9] proposition 3.4.4):

  3.1 above, we have an isomorphism F Ã Ã.Π where the initial form in (a,b) P F of Π is the Bernstein element of F . As F is a Â-module (see the proposition 2.2.3) we have also an isomorphism F Â Â.Π of Â-modules and our hypothesis ii) implies that there exists Z ∈ Â such that Q = Z.Π. This gives Q = in(Z).P where in(Z) is the initial form in (a,b) of Z. This already implies that d ≥ k and that in(Z) is of degree d -k. In the case d = k we have In(Z) = 1 and Q = P .

Theorem 2 . 4 . 1

 241 For each integer p the germ at 0 of the p-th cohomology sheaf of the complex ( Ker df • , d • ) (modified for p = 1, see the footnote), denoted by E p , satisfies the following properties: i) We have in E p the commutation relation a.b -b.a = b 2 .

  ) and there exists an integer N ≥ 1 such that a N .A(E p ) = 0 and b2N .B(E p ) = 0. v) The quotient E p := E p B(E p ) is a geometric (a,b)-module. Recall that B(E) is the b-torsion in E, Ã(E) the a-torsion of E and A(E) the C[b]-module generated by Ã(E) in E.We shall mainly use this result in the case where ω is an (n + 1)-holomorphic differential form in an open neighborhood U of the origin of 0 ∈ C n+1 ; note that the condition df ∧ ω = dω = 0 is automatic in this case. So it defines a class [ω] in E n+1 and generates the frescoF ω := Ã.[ω] ⊂ E n+1 0 thanks tothe proposition 2.2.1 and property v) of the previous theorem. Definition 2.4.2 We shall denote B ω ∈ C[x] and P ω ∈ A respectively the Bernstein polynomial and the Bernstein element of the fresco F ω .

Lemma 3 . 1 . 3

 313 In the situation of the theorem 3.1.2, let S ∈ C[[b]] which satisfies S(0) = 1 and let µ ∈ C such that µ = -ξ s for a given s ∈ [1, d]. Then 1. ξ s is still the biggest pole of order ≥ s in ξ + Z for the meromorphic extension of F ψ h (λ)[S(b)ω] when h varies in Z. 2. ξ s -1 is the biggest pole of order ≥ s in ξ + Z for the polar part of the meromorphic extension of F ψ h (λ)[(a -µ.b)ω] when h varies in Z.

Lemma 3 . 1 . 4

 314 In the situation of the theorem 3.1.2, assume that ξ s+1 = ξ s , for some s ∈ [1, d -1]. Then ξ s -1 is the biggest pole of order ≥ s in ξ + Z for the meromorphic extension of F ψ h (λ)[(a + ξ s .b)ω] when h varies in Z.

  this completes the proof of our induction. So we obtain that at least d roots (counting multiplicities) of the Bernstein polynomial of F ω are among the ξ s for s ∈[1, d].

Lemma 4 . 2 . 1

 421 E f := Ω n+1 d(Ker(df ) n endowed with its natural structure of module over the C-algebra A := C a, b where the variables a and b satisfy the commutation relation a.b -b.a = b 2 . Recall that on E f the action of a is the multiplication by f and the action of b is given by df ∧ d -1 where d is the de Rham differential (which is surjective on Ω n+1 ). We extend this structure of (left) A-module to E f [λ] := E f ⊗ C C[λ] by asking that a and b are C[λ]-linear. For any monomial x β the image of C[m 1 , . . . , m n+2 ].x β via the map defined by

Theorem 4 . 2 . 3

 423 Assume that f ∈ C[x 0 , . . . , x n ] has (n + 2) monomials and satisfies the conditions (C1) and (C2) described above. Let d and h the positive integers defined after (c) and r ∈ Z * defined in (c) and (e). For each β ∈ N n+1 there exists homogeneous elements P d+h and P d of respective degrees d + h and d which are products of homogeneous factors of degree 1 of the form a -ξ.b where ξ ∈ Q such thatP d+h (a, b) -c.λ r .P d (a, b) [x β .dx] = 0 in E f [λ] where c ∈ Q *

Corollary 4 . 2 . 4 13 ) 0 .

 424130 Let f ∈ C[x 0 , . . . , x n ] as in the previous theorem and choose any monomial x β . Define for any horizontal family of n-cycles (γ s ) s∈S over a simply connected open set S in C * avoiding the critical values of f and having 0 in its boundary, the integral-period: ϕ β (s) := γs x β .dx df (Then ϕ β is solution on S of the differential equation (which is regular singular at 0) obtained from (11) by letting a = ×s and b := s Proof. Thanks to the proposition 2.3.2 the Bernstein element of the fresco generated by [x β .dx] in E f ⊗ A Ã is a left multiple of P d (a, b) as c and λ are not 0. This is enough to conclude.

4. 3 . 1

 31 f λ := x 5 + y 5 + z 5 + λ.x.y.z 2

Compare this hypothesis with the theorem 2.3.1 which is recalled in section 2.

This quotient allows to define b[ω] = [df ∧ u] independently on the choice of u ∈ Ω n 0 such that ω = du because when ω is in d(Ker(df ) n 0 the period-integral is identically 0 near z = 0.

This is psychologically the most difficult fact to accept after a standard education in maths.

G ⊂ E is a normal sub-module when bx ∈ G for x ∈ E implies x ∈ G.

Recall that we consider the rank as a C[[b]]-module, where C[[b]] ⊂ Ã.

For p = 1 we have to replace Ker df 1 by a quotient ; see[START_REF] Barlet | Sur certaines singularités d'hypersurfaces II[END_REF].

Such a u always exists.

Note that if we have several ψ which are d-closed near 0 and for which the meromorphic extension of F ψ h [ω] presents poles in ξ + Z, we may obtain more roots in ξ + Z for the Bernstein polynomial of F ω from this result.

Note that the polar parts of order ≥ q of the poles in ξ + Z are independent of the choice of ρ because of our hypothesis H(ξ, q).

• ω = x.dx ∧ dy ∧ dz B x (ξ) | (ξ + 9 10 )(ξ + 1)(ξ + 6 5 )(ξ + 7 5 ).

• ω = z.dx ∧ dy ∧ dz B z (ξ) | (ξ + 1) 3 (x + 3 2 ). • ω = z 2 .dx ∧ dy ∧ dz B z 2 (ξ) | (ξ + 6 5 ) 2 (ξ + 13 10 )(ξ + 9 5 ).

• ω = x.y.dx ∧ dy ∧ dz B x.y (ξ) | (ξ + 11 10 )(ξ + 7 5 ) 2 (ξ + 8 5 ).

• ω = x 2 .dx ∧ dy ∧ dz B x 2 (ξ) | (ξ + 6 5 )(ξ + 8 5 ) 2 (ξ + 11 10 ). Remark that the global computation for these forms gives here the same frescos than in the affine case because f λ has an isolated singularity at the origin.

The singularity of the hypersurface {f = 0} is the origin : It is easy to see that any monomial of f is a linear combination of f and x. ∂f ∂x , y. ∂f ∂y , z. ∂f ∂z , so that each monomial in f has to vanish on the singular set of {f = 0}. Then this implies easily our claim. Again using the theorem 4.2.3 and the proposition 2.3.2 allows, after some elementary computations, to find for each monomial form ω below a degree 3 polynomial dividing the Bernstein polynomial of the fresco F ω . [START_REF] Barlet | Sur certaines singularités d'hypersurfaces II[END_REF] )(ξ + 11 7 )(ξ + 15 7 ).

• ω = x.y.dx ∧ dy ∧ dz B x.y (ξ) | (ξ + 10 7 )(ξ + 12 7 )(ξ + 13 7 ).

4.3.3 f := x.y 2 .z 3 + y.z 2 .t 3 + z.t 2 .x 3 + t.x 2 .y 3 + λ.x.y.z.t

In this case the singularity is not isolated : the singular of {f = 0} is the union of the lines {x = y = z = 0}, {y = z = t = 0}, {z = t = x = 0}, {t = x = y = 0}. The estimate for the Bernstein polynomial associated to the monomial 1 (so of the fresco 4 . So we may have a maximal unipotent monodromy.


domain: chim sde sdv

text:
Nicolas Chouliaras

Theophanis Gemtos

Vinasse (Alcohol Industry Waste) Recycling in Cotton Crop

est destinée au dépôt


Davergne Thomas
email: thomas.davergne@gmail.com

Bailly Florian

Violaine Foltz

Lambert Anne

Fautrel Bruno

Gossec Laure

Effectiveness of a Short Functional Restoration Program for Patients with Chronic Low Back Pain: a cohort study of 193 Patients

Keywords: Low back pain, functional restoration program, disability

Background: Functional restoration programs (FRPs) are integrative programs to improve function in chronic low back pain (cLBP). They are costly and time-consuming. The aim was to assess the effectiveness of a condensed FRP (CFRP) for patients with cLBP in professional activity.

Methods: Longitudinal 3 months study of patients with cLBP in one tertiary care hospital, participating in a CFRP over 4 separate days. The primary outcome was the Oswestry Disability Index (ODI). Secondary outcomes included pain, quality of life (EQ5D), patient acceptable symptom state, presenteeism, absenteeism and psychological distress.

Outcomes were compared using paired sample Student's t-test or Chi2 between baseline and last follow-up. Logistic regression was used to identify factors associated with better response (improvement of ODI higher than 12.8).

Results: In all, 193 patients were analysed, mean age 44.6 (standard deviation (SD) 10.4) years, mean cLBP duration 9.0 (SD 8.8) years. A small improvement was observed for ODI (mean difference -5.9, 95% confidence interval: -7.6, -4.1), as well as most other outcomes.

Multivariate analysis showed an association between ODI improvement and higher duration of low back pain (odds ratio for 5 years: 1.41 (1.06,1.88)) and lower baseline back strength (Sorensen, odds ratio for 1 minute: 0.54 (0.29,0.99)).

Conclusion: this CFRP showed small effect to improve function, pain and other quality of life, in cLBP. Four-day programs may be an interesting option in cLBP patients still in professional activity for whom a long 1-month FRP is difficultly manageable. Further studies with randomized controlled designs are needed to confirm the benefits.

Introduction

Low back pain (LBP) is a major problem throughout the world: leading cause of years lived with disability in both developed and developing countries and the sixth cause of overall disease burden (disability-adjusted life-years) [START_REF] Gbd | Disease and Injury Incidence and Prevalence Collaborators. Global, regional, and national incidence, prevalence, and years lived with disability for 354 diseases and injuries for 195 countries and territories, 1990-2017: a systematic analysis for the Global Burden of Disease Study 2017[END_REF]. The lifetime prevalence of low back pain is up to 84% [START_REF] Airaksinen | Chapter 4 European guidelines for the management of chronic nonspecific low back pain[END_REF] with a negative social and psychological impact [START_REF] Bailly | The impact of chronic low back pain is partly related to loss of social role: A qualitative study[END_REF].

LBP has a certain risk of recurrence and can become chronic [START_REF] Maher | Non-specific low back pain[END_REF]. Multidisciplinary rehabilitation, also known as functional restoration programs (FRPs), appears to be one of the most effective therapies to improve back pain and decrease disability and long-term work impairment in chronic LBP (cLBP) [START_REF] Maher | Non-specific low back pain[END_REF][START_REF] Kamper | Multidisciplinary biopsychosocial rehabilitation for chronic low back pain[END_REF] and is recommended by national guidelines [START_REF] Chou | Nonpharmacologic therapies for acute and chronic low back pain: a review of the evidence for an American Pain Society/American College of Physicians clinical practice guideline[END_REF][START_REF]Non-specific low back pain and sciatica: management. NICE guideline: short version[END_REF][START_REF] Kce | Low back pain and radicular pain: assessment and management[END_REF].

Key concepts of these programs are based on a physical component (such as specific exercise modalities) and at least one other element from a biopsychosocial approach (psychological, social, occupational or educational) and are delivered by health professionals from different disciplines [START_REF] Kamper | Multidisciplinary biopsychosocial rehabilitation for chronic low back pain[END_REF]. However, time and resources required for full-time FRPs are a limit to their use. Furthermore, FRPs concern patients with long-term disabling cLBP with a prolonged sick leave and aim to allow patients to get back to work [START_REF] Poulain | Long-term return to work after a functional restoration program for chronic low-back pain patients: a prospective study[END_REF]. Such programs are not well adapted for patients still at work, since they require full time participation over several weeks (usually 2-6 weeks), while the number of persons with LBP and still working is very high [START_REF] Andersson | Epidemiological features of chronic low-back pain[END_REF][START_REF] Salathé | Treatment Efficacy, Clinical Utility, and Cost-Effectiveness of Multidisciplinary Biopsychosocial Rehabilitation Treatments for Persistent Low Back Pain: A Systematic Review[END_REF].

Targeting patients with cLBP in professional activity may be useful since they are at high risk of work consequences. Thus, a condensed program of the same type as a full-time FRP may be a more feasible alternative for patients with persistent disability who are still at work. Previous studies have shown that such condensed programs have the potential to improve function, prevent sick leave, improve presenteeism and pain and appear to be well accepted [START_REF] Demoulin | Effectiveness of a semi-intensive multidisciplinary outpatient rehabilitation program in chronic low back pain[END_REF][START_REF] Albaladejo | The efficacy of a short education program and a short physiotherapy program for treating low back pain in primary care: a cluster randomized trial[END_REF][START_REF] Van Hooff | Daily functioning and self-management in patients with chronic low back pain after an intensive cognitive behavioral programme for pain management[END_REF][START_REF] Garofoli | Feasibility of a short multidisciplinary education and exercise therapy program for patients with non-specific low back pain: A 3-month retrospective open pilot study[END_REF].

The objective was to assess the effectiveness of a CFRP for patients with cLBP in professional activity.

Patients and Methods

Study design

In this observational cohort study, data were prospectively collected in a single tertiary care center in Paris, France. This department provides pluridisciplinary (physicians and health professionals) services for patients with cLBP in an ambulatory setting [START_REF] Poulain | Long-term return to work after a functional restoration program for chronic low-back pain patients: a prospective study[END_REF]. Outcomes were measured at baseline, at the end of the 4-week program and at the 3-month follow-up and the main analysis was based on the latest available data (i.e. 3 months when available, else 1 month).

This study was in the context of usual care. No approval was required as stated by the ethics committee in charge or this study (ethical committee Sud-Est VI, Lyon ,Groupement Hospitalier Est, Bâtiment Pinel, 59 Boulevard Pinel, 69 500 BRON, FRANCE). Consent was obtained for experimentation with human subjects. This study was reported in accordance with the STrengthening the Reporting of OBservational studies in Epidemiology (STROBE) statement (Supplementary material A) [START_REF] Von Elm | The Strengthening the Reporting of Observational Studies in Epidemiology (STROBE) statement: guidelines for reporting observational studies[END_REF].

Population

Patients included in the program met the following criteria: nonspecific cLBP (>3 months as defined in the European guidelines for the management of chronic [START_REF] Airaksinen | Chapter 4 European guidelines for the management of chronic nonspecific low back pain[END_REF]), not currently in prolonged sick leave (more than one week), no current indication for surgery (e.g. pain radiating to a precise and entire leg dermatome with neurologic signs [START_REF] Abenhaim | The Role of Activity in the Therapeutic Management of Back Pain: Report of the International Paris Task Force on Back Pain[END_REF], fluent in French, and difficulties to maintain daily physical activity due to LBP despite outpatient care including physiotherapy. Nonspecific LBP was defined as LBP not attributable to a recognizable, known specific pathology (e.g., osteoporosis, structural deformity, infection, tumour, fracture, inflammatory disorder, radicular syndrome, or cauda equina syndrome) [START_REF] Maher | Non-specific low back pain[END_REF]. Exclusion criteria were specific LBP and medical contraindication to physical exercise.

Every patient who participated in the CFRP between April 2012 and April 2018 was analysed if the Oswestry Disability Index and lumbar and radicular pain at baseline and at the end of the intervention or 3-month follow-up was available.

Intervention

The CFRP is a complex multidisciplinary intervention proposed as a usual care, according to different national guidelines, where FRP is defined as intervention that combines and coordinates physical, vocational, and behavioral components and is provided by multiple health care professionals with different clinical backgrounds [START_REF] Chou | Nonpharmacologic therapies for acute and chronic low back pain: a review of the evidence for an American Pain Society/American College of Physicians clinical practice guideline[END_REF][START_REF]Non-specific low back pain and sciatica: management. NICE guideline: short version[END_REF][START_REF] Kce | Low back pain and radicular pain: assessment and management[END_REF]. The program takes place in our department of rheumatology in Pitié-Salpêtrière Hospital, 6 hours per day, 1 day a week, for 4 weeks. Patients are included during consultation or hospitalization and are in groups of 6 maximum per CFRP. The CRFP was developed on the basis of therapeutic patient education programs and physical exercises [START_REF] Kce | Low back pain and radicular pain: assessment and management[END_REF][START_REF] Desthieux | Functional restoration programs for chronic low back pain patients are not just a fitness classchange in muscle mass assessed through DXA in 95 patients: a monocentric longitudinal study[END_REF]. The interdisciplinary team involved and the different components of intervention are detailed in Table 1. A few hours per day are devoted to physical exercises (17% to 67%), led by a physical activity specialist, physiotherapist or yoga teacher which are divided into three periods: warm-up, stretching, and muscle strengthening. Specific exercises adapted to work circumstances (like handling: e.g, lift a weight from the ground) are taught by an occupational therapist. Muscle strengthening is based on isotonic contraction on fitness devices. Patient education is given over on 1 hour per week and focuses on back pain, lumbar ergonomics, cognitive therapy and information on medication. Educational booklets, containing general information about back pain such as possible causes, tips to stay active or treatment, are also given to the patients. Patients were present each day of the program for the same duration (9 am to 4 pm) with a varying duration of lunch break. Although physical activity and exercises were promoted during patient education sessions, no prescription of exercises was made between each session and exercises were not mandatory, or measured.

Data collection

The outcomes were collected prospectively in a paper form. The primary outcome was the Oswestry Disability Index, which gives a subjective percentage score of level of function (disability) on a 0 to 100 scale, 100 being the highest disability state [START_REF] Fairbank | The Oswestry Disability Index[END_REF]. Secondary outcomes included a visual analogue scale of lumbar pain and radicular pain (0-100 scale), EQ-5D-3L for quality of life [START_REF] Group | EuroQol--a new facility for the measurement of health-related quality of life[END_REF] converted into a global index score with 1 indicating no restriction of quality of life and -1 the worse quality of life possible [START_REF]EQ-5D Value Sets: Inventory, Comparative Review and User Guide[END_REF], patient acceptable symptom state where patients reported their satisfaction on state of symptoms by picking "acceptable" or "non-acceptable" [START_REF] Tubach | Minimum clinically important improvement and patient acceptable symptom state in pain and function in rheumatoid arthritis, ankylosing spondylitis, chronic back pain, hand osteoarthritis, and hip and knee osteoarthritis: Results from a prospective multinational study[END_REF]. We also collected the validated World Health Organization Health and Work Performance Questionnaire for presenteeism (reflecting the patient's overall performance while at work during the past 4 weeks, score from 0 to100, 100 being the highest level of performance) and absenteeism (proportion of hours lost with regards to the hours expected to work per month, score from 0 to100, 100 indicates the higher amount of absenteeism) [START_REF] Kessler | Using the World Health Organization Health and Work Performance Questionnaire (HPQ) to evaluate the indirect workplace costs of illness[END_REF] and the Hospital Anxiety and Depression scale (HAD) with a probable anxiety or depression disorder corresponding to a score >11 [START_REF] Zigmond | The hospital anxiety and depression scale[END_REF]. Physical examination assessed Shirado and Sorensen for strength to measure the endurance (in seconds) of the trunk flexor and extensor muscles, Schober test and fingertip-to-floor test for flexibility.

Other variables collected at baseline were age, gender, body mass index, duration of LBP, a measure of stress at work using the Karasek questionnaire (psychological demand, social latitude, social support), professional activity (yes or no) and socio-professional category (high being defined as higher managerial, administrative and professional occupations) [START_REF] Niedhammer | Propriétés psychométriques de la version française des échelles de la demande psychologique, de la latitude décisionnelle et du soutien social du « Job Content Questionnaire » de Karasek : résultats de l'enquête nationale SUMER[END_REF].

Statistical methods

Before-after comparison used data at last follow-up (3 months follow-up or at the end of intervention if not available) and was performed using paired sample Student's t-test or Mac Nemar's Chi2 test on complete data only. Analyses were also conducted separately with data at end of treatment and with at 3 months for sensitivity analysis (Supplementary material B). Mean differences and standardized mean differences were computed. To interpret the standardized mean differences, the following norms were used: small effect (<0.3), moderate effect (0.3-0.8) or large effect (>0.8) [START_REF] Cohen | Statistical Power Analysis for the Behavioral Sciences[END_REF]. Changes above the minimal clinically important difference were computed for Oswestry (12.8 out of 100), pain (20 out of 100), anxiety and depression scores (both 1.5) [START_REF] Johnsen | Comparison of the SF6D, the EQ5D, and the oswestry disability index in patients with chronic low back pain and degenerative disc disease[END_REF][START_REF] Ostelo | Clinically important outcomes in low back pain[END_REF][START_REF] Puhan | The minimal important difference of the hospital anxiety and depression scale in patients with chronic obstructive pulmonary disease[END_REF]. Subgroup analysis was performed to compare effectiveness between patients with high level of symptoms at baseline (Oswestry >20/100 and pain > 20/100) versus low level of symptoms.

The normality of each variable was assessed visually and since this condition was met, parametric tests were used. Factors associated with response, defined as an increase in Oswestry Disability Index higher than the minimal clinically important difference, were analysed using Student, Chi2 and multivariate logistic regression. Explanatory variables in the logistic regression model were entered from a set of a priori variable if statistically associated in univariate analysis (p<0.2), then selected based on missing data and correlation (Supplementary material C). Variables finally entered in the multivariate model were: duration of low back pain, Sorensen (back strength), baseline anxiety (HAD), baseline patient acceptable symptom state, baseline lumbar pain, socio-professional category, Karasek social support and the multivariate analysis concerned 129 patents with full data (i.e. data at 3 months follow-up or at the end of intervention if not available). P values smaller than 0.05 were considered significant. Analyses were carried out using R version 3.5.1.

Results

Patients

In all, of the 274 patients who followed the program during the evaluation period, 193 patients were analysed since they had follow-up data (85 patients at the end of the intervention and 108 at 3 months, figure 1). The 85 patients who did not return for the visit at 3 months presented the same characteristics at baseline except for Shirado score which was lower (data not shown).

Baseline characteristics are reported in table 2. Mean age was 44.6 (standard deviation (SD) 10.4) years and 76 (39%) were male, mean LBP duration was 9.0 (SD 8.8) years (median = 5.0 years ; minimum = 8.0 months). The mean body mass index was in overweight values: 26.6 (SD 8.8) kg/m 2 . Among the 162 (84%) with a paid work, 78 (48%) were in high socioprofessional category, mean psychological demand was high (>21) and mean social support and social latitude were low (<24 and <70 respectively) according to Karasek score. At baseline, mean intensity of lumbar pain was 38.2 (SD 23.4), the mean Oswestry Disability Index was 29.7 (SD 14.0) and symptom state was acceptable for 79 (46%) of patients.

Effectiveness of the CFRP

Comparisons between baseline and last follow-up (i.e. data at 3 months follow-up or at the end of intervention if not available) showed a small improvement in the Oswestry Disability Index (mean difference -5.9, 95%, confidence interval (CI): -4.1, -7.6), as well as lumbar pain, EQ-5D quality of life, patient acceptable symptom state, Shirado and Sorensen, fingertip to floor test, anxiety, depression and absolute presenteeism (Table 3). This improvement was also noted for patients only at one month or 3 months follow-up: Oswestry Disability Index (mean difference -3.9, 95%, CI: -1.2, -5.6 at one month and mean difference -6.8, 95% CI : -4.0, -9.6 at 3 months follow-up), VAS of lumbar pain (mean difference -5.9, 95%, CI: -2.2, -9.6 at one month and mean difference -6.9, 95% CI : -1.7, -10.1 at 3 months follow-up) (Supplementary material B). The change was higher than the minimal clinically important difference for 47 (24%) patients for the Oswestry disability Index, 60 (31%) for lumbar pain, 80 (44%) for anxiety and 68 (38%) for depression. Furthermore, 42 (24%) changed status positively for the patient acceptable symptom state. Standardised mean differences of quantitative variables (figure 2) indicated higher effect for trunk extensor muscles and Oswestry Disability Index (0.55 and 0.4 respectively). Before-after comparisons based on data only at the end of the intervention (n=85) or at 3 months follow-up (n=108) showed similar results (Supplementary material B). Subgroup analysis showed a significantly higher improvement in patients with higher symptoms at baseline (mean difference -8.3) than those with lower symptoms at baseline (mean difference -3.2), 95%CI -8.8, -1.4.

Variables associated to better response to CFR Among the 193 participants, 47 (24%) had a clinically significant improvement of their Oswestry Disability Index at last follow-up and were classified as good response. In univariate analyses, a significant association was observed between good response at last follow-up and baseline characteristics: low functional status or high physical impairment (Oswestry Disability Index: p<0001), low back strength (Sorensen: p=0.04), high absolute presenteeism (World Health Organization Health and Work Performance Questionnaire: p=0.01) and substantial anxiety (HAD, p=0.03). Multivariate analysis showed a significant association between good response and high duration of low back pain (odds ratio of 5 years: 1.41 (95% CI: 1.06,1.88)) and low baseline back strength according to Sorensen (odds ratio for 1 minute: 0.54 (95% CI: 0.29,0.99)) (table 4).

Analyses based on data at 3 months follow-up were overall confirmatory (Supplementary material D).

Discussion

This study explored an innovative condensed program applicable to patients with cLBP without sick leave. It constitutes an interesting alternative easy to implement for patients still working for whom a long FRP requires a long work interruption. Currently, FRPs are being used to improve the outcomes of CLBP, however, they are long, time-consuming and costly [START_REF] Chou | Nonpharmacologic therapies for acute and chronic low back pain: a review of the evidence for an American Pain Society/American College of Physicians clinical practice guideline[END_REF][START_REF]Non-specific low back pain and sciatica: management. NICE guideline: short version[END_REF][START_REF] Kce | Low back pain and radicular pain: assessment and management[END_REF]. The option proposed here is a four-days intervention spread over one month, including both physical activity and behavioral components, which is compatible with maintaining professional activity. In this pilot observational study, we were able to demonstrate that the CFRP led to a statistically significant but not clinically significant improvement in the key outcomes of CLBP which are disability and pain, as well as improvements in most of the other outcomes analyzed including work outcomes, psychological status, strength and flexibility, both at the end of intervention and 3 months follow-up. Furthermore, it appeared that patients with the best response to the CFRP were those with a higher duration of low back pain and a lower baseline back-strength at baseline. At baseline, the Oswestry Disability Index, pain levels and anxiety and depression level were moderate in this young to middle aged population. These levels of pain and disability were lower than those described in previous studies where the Oswestry score is usually around 40 and pain around between 50 and 80 [START_REF] Poulain | Long-term return to work after a functional restoration program for chronic low-back pain patients: a prospective study[END_REF][START_REF] Demoulin | Effectiveness of a semi-intensive multidisciplinary outpatient rehabilitation program in chronic low back pain[END_REF][START_REF] Albaladejo | The efficacy of a short education program and a short physiotherapy program for treating low back pain in primary care: a cluster randomized trial[END_REF][START_REF] Van Hooff | Daily functioning and self-management in patients with chronic low back pain after an intensive cognitive behavioral programme for pain management[END_REF][START_REF] Garofoli | Feasibility of a short multidisciplinary education and exercise therapy program for patients with non-specific low back pain: A 3-month retrospective open pilot study[END_REF][START_REF] O'sullivan | Cognitive Functional Therapy for Disabling Nonspecific Chronic Low Back Pain: Multiple Case-Cohort Study[END_REF]. This difference in level of symptoms could be explained because patients included in this CFRP had chronic but moderately impactful symptoms and were still at work. Thus, it is logical that the symptom level is lower than in patients starting a full time FRP [START_REF] Poulain | Long-term return to work after a functional restoration program for chronic low-back pain patients: a prospective study[END_REF]. Surprisingly, the level of lumbar pain (38.2 (23.4)) at baseline was lower that the level of radicular pain (45.5 (19.9)) (Table 2). This result could be explained by the fact that mean radicular pain was calculated for patients with radicular pain only (i.e. >1).

Comparison between baseline and last follow-up showed a mean difference of 5.9 in the Oswestry Disability Index. This improvement was also noted for patients only at one month (mean difference -3.9) or 3 months follow-up (mean difference -6.8). This significant effect on the Oswestry was of a small magnitude; furthermore only 47 (24%) patients had a clinically relevant improvement of the Oswestry Disability Index. This small effect may be partly explained by the moderate disability at baseline. Subgroup analysis showed a significantly higher improvement in patients with higher symptoms at baseline than those with lower symptoms at baseline. In other studies, larger effects were often observed (between 10 and 20 points) [START_REF] Van Hooff | Daily functioning and self-management in patients with chronic low back pain after an intensive cognitive behavioral programme for pain management[END_REF][START_REF] O'sullivan | Cognitive Functional Therapy for Disabling Nonspecific Chronic Low Back Pain: Multiple Case-Cohort Study[END_REF]. However, this effect was observed after an intensive functional therapy with cognitive treatment or after a 2 week program, including cognitive behavioral training, physical activities and education, which was much more time-consuming and resource-consuming than our CRFP [START_REF] Van Hooff | Daily functioning and self-management in patients with chronic low back pain after an intensive cognitive behavioral programme for pain management[END_REF][START_REF] O'sullivan | Cognitive Functional Therapy for Disabling Nonspecific Chronic Low Back Pain: Multiple Case-Cohort Study[END_REF].

Change in levels of pain was of small amplitude (mean difference -7.7/100). This change was lower than previous study [START_REF] Albaladejo | The efficacy of a short education program and a short physiotherapy program for treating low back pain in primary care: a cluster randomized trial[END_REF][START_REF] O'sullivan | Cognitive Functional Therapy for Disabling Nonspecific Chronic Low Back Pain: Multiple Case-Cohort Study[END_REF]. which could be explained by a less intensive cognitive treatment for pain and lower baseline level of pain.

In this study, CFRP was effective to improve pain, quality of life, patient acceptable symptom state, strength, flexibility, anxiety, depression and presenteeism. Effectiveness on presenteeism is important since a main aim of a CFRP is to improve work outcomes. No significant difference was observed on leg pain and relative absenteeism, probably due to lack of power and small follow-up. Depression did not significantly change, but levels at baseline were low. A significant association was observed between response and higher duration of low back pain and lower baseline back strength. This could be explained because patients with higher duration of low back pain and lower back strength may have also the worst state of function, though this should be explored further.

This study has strengths and weaknesses. The design was a cohort study which does not control for other sources of effects (such as contextual factors) but allows to reflect clinical practice [START_REF] Gamble | An Introduction to the Fundamentals of Cohort and Case-Control Studies[END_REF]. The results of this study should be confirmed by randomized controlled trials. However, the positive effect observed across different outcomes is obtained from a large population (N=193) and pleads in favor of the efficacy of the CFRP. In terms of population, the CFRP is mostly indicated for patients currently in professional activity. However, 16% of patients were included even though they were not currently working, mostly because they were on short-term sick leave. The outcomes chosen for the study reflect both physical, psychological and professional activity. It could have been useful to analyse factors associated with absenteeism. However, because this variable was not available in all the patients, we analysed the Oswestry disability index as primary outcome: this is a well validated and widely used outcome in LBP [START_REF] Fairbank | The Oswestry Disability Index[END_REF]. We did not collected the number of recurrences features of chronical pain states of an single episode to characterise chronicity. However, since mean duration was 9.0 year, we have classified patients as "chronic". The follow-up chosen for this study (3 months) is short to evaluate the impact of an intervention with an average duration of the disease of 9 years. A longer follow-up should be considered in a more in-depth study. Finally, a high lack of follow up at the 3 month time point was observed (36%). This could be explained by a high rate of patient still working who had difficulties to reach the hospital during a working day just for an assessment. Another explanation may be that there is no need to return to the hospital due to better health. This drop out ratio didn't allow to carry out an intent to treat analysis. However, no differences were found when comparing their characteristics (except for strength at baseline) and the efficacy of the CFRP was similar at the end of intervention and 3 months follow-up.

In conclusion, a 4-day interventions for chronic LBP in professional activity showed a small effect in improving physical function and avoiding professional consequences of cLBP while maximising feasibility. These results, if confirmed by other studies, may encourage other centres to implement a condensed functional program, and may help to choose who to orient towards this CFRP. Further studies are needed to assess the benefits on work outcomes such as sick leave with longer follow-up.

  Disclosures: Davergne Thomas, None; Bailly Florian, None, Foltz Violaine, None, Lambert Anne, None, Fautrel Bruno, None, Gossec Laure, None. Funding: This usual care study was performed in Assistance Public des Hôpitaux de Paris (AP-HP). Thomas Davergne received funding from Groupe d'Etudes et de Recherche sur les Pathologies de l'Appareil Locomoteur (GERPAL).

Figure:

  Figure:

Figure 1 :

 1 Figure 1: Flow chart of patients with chronic low back pain participating in a condensed functional restoration program between April 2012 and April 2018

Figure 2 :

 2 Figure 2: Standardised mean differences of quantitative variables measured before and after the CFRP for patients with cLBP.

Table 1 :

 1 Description of the condensed functional restoration program for patient with chronic low back pain.

				Week
	Therapist	Activity	Duration	number
	Physiotherapist trained in
	therapeutic education of the patient Assessment	2h	1, 4
	Physical activity specialist	Stretching and Fitness	2h	1
	Pain Physician	Neurophysiology of pain information	1.5h	1,2
		Fitness, Proprioception, Sitting gymnastics
	Physiotherapist	and Gym ball	2h	2,3
	Occupational therapist	Lumbar ergonomy (theory and practice)	1h	2,3
	Yoga teacher	Yoga lesson	1h	2,4
	Physiotherapist trained in cognitive-
	behavioural therapy	Cognitive-behavioural therapy	1.5h	3
	Nurse	Information on medication	1.5h	3
	Occupational therapist and
	Adapted physical activity specialist Sport and sexuality workshop	1.5h	4
	Dietitian	Lifestyle workshop	1.25h	4

Table 2 :

 2 Baseline characteristics of 193 patients with chronic low back pain attending a condensed functional restoration program

	Characteristics

£

High socio-professional category was being defined as higher managerial, administrative and professional occupations $ Mean radicular pain calculated on patients with radicular pain, n=96

Table 4 :

 4 Factors associated to response to the CFRP in 193 patients at last follow-up: Univariate and multivariate logistic regression.

	Variable	Univariate	Odds Ratio (95%CI)	P value
		analysis
		(P value)
	Duration of low back pain, for 5	0.18	1.41 (1.06,1.88)	0.02
	years
	Sorensen (back strength), for 1	0.03	0.54 (0.29,0.99)	0.05
	minute
	Baseline anxiety (HAD)	0.04	1.09 (0.97,1.22)	0.15
	Baseline Patient acceptable	0.18	0.75 (0.3,1.91)	0.55
	symptom state
	Baseline lumbar pain	0.24	0.99 (0.97,1.01)	0.86
	Socio-professional category	0.01	0.73 (0.31,1.76)	0.49
	Karasek social support	0 .27	1.01 (0.93,1.08)	0.81

CI: Confidence interval HAD: Hospital Anxiety and Depression scale