Nicolas Dutertre
email: dutertre@cmi.univ-mrs.fr

ON THE REAL MILNOR FIBRE OF SOME MAPS FROM R n TO R 2

We consider a real analytic map-germ (f, g) : (R n , 0) → (R 2 , 0). Under some conditions, we establish degree formulas for the following quantities :

, where (α, δ) is a regular value of (f, g) and 0 < |(α, δ)| ≪ ε ≪ 1.

Introduction

Let F = (f 1 , . . . , f k ) : (K n , 0) → (K k , 0), with 0 < k < n and K = R or C, be an analytic map-germ such that 0 is an isolated singularity of F -1 (0), i.e rank[DF (x)] = k at every x ∈ F -1 (0) \ {0} close to the origin. Let g : (K n , 0) → (K, 0) be a function-germ. We are interested in studying topological invariants associated with the mappings F and (F, g).

Let B ε ⊂ K n be a closed ball centered at the origin of radius ε. For any regular value δ ∈ K k close to the origin, the Milnor fibre of F associated with δ is the set F -1 (δ) ∩ B ε , where 0 < |δ| ≪ ε ≪ 1. We will denote it by W ε F -δ .

In the complex case, the topology of the Milnor fibre is well known. Milnor [Mi] (in the case k = 1) and Hamm [Ha] (in the case k > 1) proved that W ε F -δ has the homotopy type of a bouquet of µ(F ) spheres of dimension nk. This number µ(F ) is called the Milnor number of F and the Euler-Poincaré characteristic of the Milnor fibre is given by χ(W ε F -δ ) = 1 + (-1) n-k µ(F ). Minor also showed that, when k = 1, µ(F ) is equal to the topological degree of the mapping ∇F |∇F | : ∂B ε → S 2n-1 1 . This gives the following algebraic characterization of the Milnor number :

µ(F ) = dim C O C n ,0 ∂f ∂x 1 , . . . , ∂f ∂xn ,
where O C n ,0 is the algebra of holomorphic function-germs at the origin. This last formula was extended to the case k > 1 by Lê [Le] and Greuel [Gr], who obtained the following formula : where F ′ = (f 1 , . . . , f k-1 ) and I is the ideal generated by f 1 , . . . , f k-1 and the minors ∂(f 1 ,...,f k ) ∂(x i 1 ,...,x i k ) . Hence, proceeding by induction, one can get a formula for µ(F ) in terms of dimensions of appropriate local algebras.

µ(F ′ ) + µ(F ) = dim C O C n ,
In the real case, the topology of the Milnor fibre depends on δ and can not be as well described as in the complex case. Nevertheless, there exist formulas similar to the ones mentionned above relating its Euler-Poincaré characteristic to topological degrees of mapping defined in terms of F . For instance, if k = 1, the Khimshiashvili formula [Kh] states that :

χ(W ε F -δ ) = 1 -sign(-δ) n • deg 0 ∇F,
where deg 0 ∇F is the topological degree of the mapping ∇F |∇F | : ∂B ε → S n-1 1 . We proved in [START_REF] Dutertre | On the Milnor fibre of a real map-germ[END_REF] that if (δ, α) is a regular value of (F, g) with 0 < |α| ≪ |δ| ≪ ε ≪ 1 then :

χ(W ε F -δ ∩ {g ≥ α}) -χ(W ε F -δ ∩ {g ≤ α}) ≡ χ(W ε (F -δ,g-α) ) + χ(W ε F -δ ) ≡ dim R O R n ,0 /I mod (2)
, where O R n ,0 is the ring of real analytic function-germs at the origin and I is the ideal generated by f 1 , . . . , f k and the minors ∂(g,f 1 ,...,f k ) ∂(x i 1 ,...,x i k+1 ) . This formula generalized the case g = x 2 1 + • • • + x 2 n already shown by Duzinski et al. [DLNS]. This a mod 2 formula and one may ask if, as in the complex case, it is possible to express the following quantities :

χ(W ε F -δ ∩ {g ≥ α}) -χ(W ε F -δ ∩ {g ≤ α}), and
χ(W ε F -δ ), in terms of topological degrees of mappings defined in terms of F and g.

When k = n -1 and g = x 2 1 + • • • + x 2 n , Aoki et al. ([AFN1], [AFS]) proved that : χ(W ε F -δ ) = deg 0 H and 2 × deg 0 H is the number of half-branches of F -1 (0), where H = ∂(g,f 1 ,...,f n-1 ) ∂(x 1 ,...,xn) , f 1 , . . . , f n-1 . They extended this result to the case g = x n in [START_REF] Aoki | An algebraic formula for the topological types of one parameter bifurcation diagrams[END_REF] and Szafraniec extended it to any g in [START_REF] Szafraniec | On the number of branches of a 1-dimensional semi-analytic set[END_REF].

When k = 1 and g = x 1 , Fukui showed in [Fu] that :

χ(W ε F -δ ∩ {x 1 ≥ 0}) -χ(W ε F -δ ∩ {x 1 ≤ 0}) = -sign(-δ) n • deg 0 H
, where H = (F, ∂F ∂x 1 , . . . , ∂F ∂xn ). In [START_REF] Dutertre | Degree formulas for a topological invariant of bifurcations of function germs[END_REF], we proved that when n = 2, 4 or 8 and k = 1, it is possible to construct a mapping H : (R n , 0) → (R n , 0) such that :

χ(W ε F -δ ∩ {g ≥ α}) -χ(W ε F -δ ∩ {g ≤ α}) = -deg 0 H.
This last result was generalized by Fukui and Khovanskii [FK]. In that paper, the authors consider an analytic function-germ g : (R n , 0) → (R, 0) that satisfies the following condition (P ) : there exits C ∞ -vector fields v 2 , . . . , v n which span the tangent space at x to g -1 (g(x)), whenever x is a regular point of g, and ∇g, v 2 , . . . , v n agree with the orientation of R n . They define a mapping H : (R n , 0) → (R n , 0) by H = (F, v 2 F, . . . , v n F ) and they prove that if 0 is isolated in H -1 (0), if the set of critical points of g does not intersect W ε F -δ and if (δ, 0) is a regular value of (F, g) then :

χ(W ε F -δ ∩ {g ≥ 0}) -χ(W ε F -δ ∩ {g ≤ 0}) = sign(-δ) n • deg 0 H.
In this paper, we continue this work of computing Euler-Poincaré characteristics of real Milnor fibres, especially for mappings with two components. In Section 2, we give generalizations of Khimshiashvili's formula. We consider an analytic function-germ f : (R n , 0) → (R, 0), with an isolated critical point at 0, that satisfies the condition (P ) described above. Let g : (R n , 0) → (R, 0) be an other function-germ. We define a mapping k(f, g) : (R n , 0) → (R n , 0) in terms of f and g and we assume that it has an isolated zero at the origin. We prove (Theorem 2.1) that :

if n is even : χ(W ε (f,g-δ) ) = 1 -deg 0 ∇f + sign(δ) • deg 0 k(f, g), if n is odd : χ(W ε (f,g-δ) ) = 1 -deg 0 k(f, g).
We also show that if n is even :

χ(W ε f -α ∩ {g ≥ δ}) -χ(W ε f -α ∩ {g ≤ δ}) = deg 0 k(f, g
), where (α, δ) is an appropriate regular value of (f, g). Then we assume that g has an isolated critical point at the origin as well and we define an other mapping l(f, g). If it has an isolated zero at the origin, then we have (Theorem 2.9) :

if n is even : χ(W ε (f -δ,g) ) = 1 -deg 0 ∇g -sign(δ) • deg 0 l(f, g), if n is odd : χ(W ε f -δ ∩ {g ≥ 0}) -χ(W ε f -δ ∩ {g ≤ 0}) = deg 0 ∇g + sign(δ) • deg 0 l(f, g),
where 0 < |δ| ≪ ε ≪ 1.

In Section 3, we give a generalization of the formula of Fukui mentionned above. We work in R 1+n equipped with the coordinate system (x 0 , x 1 , . . . , x n ) and we consider a function-germ F : (R 1+n , 0) → (R, 0) with an isolated critical point at 0. We assume that F satisfies the following condition (Q) : there exists C ∞ vector fields V 2 , . . . , V n on R 1+n such that V 2 (p), . . . , V n (p) span the tangent space at p to F -1 (F (p))∩x -1 0 (x 0 (p)) whenever p is a regular point of (F, x 0 ) and such that (e 0 , ∇F (p), V 2 (p), . . . , V n (p)) agrees with the orientation of R 1+n . Here e 0 is the vector (1, 0, . . . , 0). Let G : (R 1+n , 0) → (R, 0) be an other function-germ. We define three mappings H(F, G), I(F, G) and J(F, G) : (R 1+n , 0) → (R 1+n , 0). We prove that if 0 is isolated in J(F, G) -1 (0) and (0, δ, 0) is a regular value of (F, G, x 0 ) then (Theorem 3.1) :

deg 0 H(F, G) = sign(-δ) n • χ W ε (F,G-δ) ∩ {x 0 ≥ 0} -χ W ε (F,G-δ) ∩ {x 0 ≤ 0} , deg 0 J(F, G) = sign(-δ) n • χ W ε (F,G-δ) -χ W ε (G-δ,x 0 )
, where 0 < |δ| ≪ ε ≪ 1. Then we apply these formulas to the case where F and G are one-parameter deformations of two function-germs f and g : (R n , 0) → (R, 0). Denoting by f t and g t the deformations given by f t (x) = F (t, x) and g t (x) = G(t, x) and applying a deformation argument as Fukui did in [Fu], we obtain degree formulas for χ(W ε (ft,gt) ) and

χ(W ε ft ∩ {g t ≥ 0}) -χ(W ε ft ∩ {g t ≤ 0}
), where 0 < |t| ≪ ε ≪ 1 (Corollary 3.15). When the deformations of f and g are of the form :

F (t, x) = f (x) -γ 1 (t) and G(t, x) = g(x) -γ 2 (t), where γ = (γ 1 , γ 2 ) : (R, 0) → (R 2 , 0) is an analytic arc such that γ(t) = 0 if t = 0, γ ′ 1 (t) = 0 if t = 0
, and the image of γ consists of regular values of (f, g) (except the origin in R 2 of course), we get formulas for

χ(W ε (f -γ 1 (t),g-γ 2 (t)) ) and χ(W ε f -γ 1 (t) ∩ {g ≥ γ 2 (t)}) -χ(W ε f -γ 1 (t) ∩ {g ≤ γ 2 (t)}), where 0 < |t| ≪ ε ≪ 1 (Corollary 3.17).
In Section 4, we present different cases where we can apply the results of the previous sections. There are two cases : when n = 2, 4 or 8 and when ∂f ∂x 1 ≥ 0 and ∂F ∂x 1 ≥ 0. In this last situation, we explain how the results concerning the one-parameter deformations can be simplified with the aid of Theorem 2.1 (see Corollaries 4.1 and 4.2).

We will use the following notations. Some of them have already appeared in this introduction :

(1) if H : (R n , 0) → (R n , 0) is a mapping with an isolated zero at the origin then deg 0 H is the topological degree of

H |H| : S n-1 ε → S n-1 1 where S n-1 ε
is the sphere of radius ε centered at the origin, (2) if F : R n → R k is a mapping then W ε F denotes the set F -1 (0) ∩ B n ε , where B n ε is the ball of radius ε centered at the origin and

∂W ε F is F -1 (0) ∩ S n-1 ε , (3) if f : R n → R is a smooth function then f x i denotes the partial derivative ∂f ∂x i and ∇f is the gradient of f , (4) if F = (F 1 , . . . , F k ) : R n → R k , 0 < k ≤ n, is a smooth mapping then DF (x) is its Jacobian matrix at x and ∂(F 1 ,...,F k ) ∂(x i 1 ,...,x i k ) is the determinant of the following k × k minors of DF (x) :    F 1x i 1 • • • F 1x i k . . . . . . . . . F k x i 1 • • • F kx i k    .

Generalizations of Khimshiashvili's formula

In this section, we prove formulas similar to Khimshiashvili's one for the fibre of a function on a hypersurface with an isolated singularity. We need to put the same conditon (P ) as Fukui and Khovanskii's condition (P ) [FK], either on the function or on the function defining the hypersurface.

Let (x 1 , . . . , x n ) be a coordinate system in R n and let f : (R n , 0) → (R, 0) be an analytic function-germ with an isolated critical point at the origin. We assume that f satisfies the following condition (P ) : there exists C ∞ vector fields v 2 , . . . , v n on R n such that v 2 (x), . . . , v n (x) span the tangent space at x to f -1 (f (x)), whenever x is a regular point of f , and such that the orientation of (∇f (x), v 2 (x), . . . , v n (x)) agrees with the orientation of R n .

Let g : (R n , 0) → (R, 0) be an other analytic function-germ. We define a mapping k(f, g) : (R n , 0) → (R, 0) in the following way :

k(f, g) = (f, v 2 g, . . . , v n g).
We will prove the following theorem :

Theorem 2.1. If 0 is an isolated critical point of f and is isolated in k(f, g) -1 (0), then we have :

if n is even : χ(W ε (f,g-δ) ) = 1 -deg 0 ∇f + sign(δ) • deg 0 k(f, g), if n is odd : χ(W ε (f,g-δ) ) = 1 -deg 0 k(f, g), where 0 < |δ| ≪ ε ≪ 1.
Furthermore, if n is even, we also have :

χ(W ε f -α ∩ {g ≥ δ}) -χ(W ε f -α ∩ {g ≤ δ}) = deg 0 k(f, g), where 0 ≤ |δ| ≪ |α| ≪ ε ≪ 1 and (α, δ) is a regular value of (f, g).
To establish this theorem, we need a series of lemmas. From now on, we will assume that the hypothesis of Theorem 2.1 are fullfilled. For all (i, j) ∈ {1, . . . , n} 2 , we will set m ij = ∂(g,f )

∂(x i ,x j ) .

Lemma 2.2. For δ = 0 sufficiently small, (0, δ) is a regular value of (f, g).

Proof. Since f has an isolated critical point, f -1 (0) \ {0} is smooth (or empty). By the Curve Selection Lemma, the critical points of g |f -1 (0)\{0} lie in g -1 (0). Lemma 2.3. Let p be a regular point of f . The function g |f -1 (f (p)) has a critical point at p if and only if v i g(p) = 0 for all i ∈ {2, . . . , n}.

Proof. If p is a regular point of f then v 2 (p), . . . , v n (p) span the tangent space at f -1 (f (p)). Therefore g |f -1 (f (p)) has a critical point at p if and only if v i (p), ∇g(p) = 0 for all i ∈ {2, . . . , n}.

Lemma 2.4. The origin is an isolated singularity of f -1 (0) ∩ g -1 (0) if and only if 0 is isolated in k(f, g) -1 (0).

Proof. A point p, distinct from the origin, is in k(f, g) -1 (0) if and only if g |f -1 (0)\{0} has a critical point at p. But, as noticed above, such a point lies in g -1 (0). Lemma 2.5. Let α = 0 be a small regular value of f . Let p be a point in f -1 (α). The function g |f -1 (α) has a non-degenerate critical point at p if and only if k(f, g)(p) = (α, 0, . . . , 0) and detDk(f, g)(p) = 0. Furthermore if λ(p) is the Morse index of g |f -1 (α) at p then one has :

(-1) λ(p) = sign [detDk(f, g)(p)] .
Proof. Since α is a regular value of f , there exists j such that f x j (p) = 0. Assume that j = 1. From [START_REF] Szafraniec | A formula for the Euler characteristic of a real algebraic manifold[END_REF], p is a non-degenerate critical point of g |f -1 (α) if and only if :

det ∇f (p) ∇m 1i (p) 2≤i≤n = 0.
Furthermore, we have :

(-1) λ(p) = (-1) n-1 • sign(f x 1 (p)) n • det ∇f (p) ∇m 1i (p) .
We have to relate det ∇f (p)

∇m 1i (p) to det ∇f (p) ∇v i g(p)
. For i ∈ {2, . . . , n}, let u i (p) be the vector (f x i (p), 0, . . . , 0, -f x 1 (p), 0, . . . , 0), where -

f x 1 (p) is the i-th coordinate. Then (u 2 (p), . . . , u n (p)) is a basis of T p f -1 (α) and it is not difficult to see that : det(∇f (p), u 2 (p), . . . , u n (p)) = (-1) n-1 • f x 1 (p) n-2 • ( n i=1 f 2 x i (p)).
Hence there exists a (n -1) × (n -1) matrix B(p) such that :

∇f (p) u i (p) = 1 0 0 B(p) • ∇f (p) v i (p) , with sign[detB(p)] = (-1) n-1 sign[f x 1 (p) n-2 ].
Hence, for i ∈ {2, . . . , n} :

u i (p) = n j=2 B ij (p) • v j (p),
and :

m 1i (p) = u i g(p) = n j=2 B ij (p) • v j g(p).
Since v j g(p) = 0, we have :

∇m 1i (p) = n j=2 B ij (p) • ∇v j g(p),
and :

∇f (p) ∇m 1i (p) = 1 0 0 B(p) • ∇f (p) ∇v i g(p) .
With this equality, it is easy to conclude.

To prove Theorem 2.1, we will use Morse theory for manifolds with corners. The reader may refer to [Du4,Section 2] for a brief description of this theory. The following lemma deals with the critical points of g |∂W ε f -α . Lemma 2.6. For all α and ε such that 0 < |α| ≪ ε ≪ 1, one has :

• at all correct critical points of

g |∂W ε f -α with g > 0, ∇g |f -1 (α) points outwards, • at all correct critical points of g |∂W ε f -α with g < 0, ∇g |f -1 (α) points inwards, • there are no correct critical points of g |∂W ε f -α in g -1 (0). Proof.
The proof is the same as [START_REF] Dutertre | Degree formulas for a topological invariant of bifurcations of function germs[END_REF], Lemma 4.1.

Lemma 2.7. We can choose α small enough and we can perturbe g into g in such a way that g|W ε f -α has only Morse critical points. Proof. Let (x; t) = (x 1 , . . . , x n ; t 1 , . . . , t n ) be a coordinate system of R 2n and let ḡ(x, t) = g(x)

+ t 1 x 1 + • • • t n x n . For (i, j) ∈ {1, . . . , n} 2 , we define M ij (x, t) by M ij (x, t) = ∂(f,ḡ) ∂(x i ,x j ) (x, t).
Notice that :

M ij (x, t) = m ij (x, t) + f x i (x)t j -t i f x j (x).
Let Γ be defined by :

Γ = (x, t) ∈ R 2n | M ij (x, t) = 0 for (i, j) ∈ {1, . . . , n} 2 .
At a point p, if f does not vanish then there exists i ∈ {1, . . . , n} such that f x i (p) = 0. This implies that Γ \ {f = 0} is a smooth manifold (or empty) of dimension n + 1. Actually if p belongs to Γ \ {f = 0}, then one can assume that f x 1 (p) = 0. In this case, around p, Γ is defined by the vanishing of M 12 , . . . , M 1n and the gradient vector fields of these functions are linearly independent. Let π be the following mapping :

π : Γ \ {f = 0} → R 1+n (x, t) → (f (x), t).
By the Bertini-Sard theorem, we can choose (α, s) close to 0 in R 1+n such that π is regular at each point in π -1 (α, s) close to the origin. If we denote by g the function defined by g(x) = ḡ(x, s), this means that g|f -1 (α) admits only Morse critical points in a neighborhood of the origin.

Proof of Theorem 2.1. Let ω : R n → R be the distance function to the origin. Let ε > 0 be sufficiently small so that

g |f -1 (0)\{0} has no critical point in f -1 (0) \ {0} ∩ {ω < ε}. Let δ be such that 0 < |δ| ≪ ε ≪ 1. We want to express χ(W ε (f,g-δ) ) in terms of deg 0 k(f, g).
Let α be a regular value of f such that 0 < |α| ≪ |δ| and the following properties are satisfied :

(1) W ε (f -α,g-δ) is diffeomorphic to W ε (f,g-δ) , (2) the critical points of g |f -1 (α)∩{ω<ε} lie in {|g| < δ} ∩ {ω < ε 2 }.
Hence the critical points of g |∂W ε f -α are correct. Furthermore by the previous lemmas, we can assume that g |f -1 (α)∩{ω<ε} has only Morse critical points, that at the correct critical points of g |W ε f -α lying in {g > 0} (resp. {g < 0}), ∇g |W ε f -α points outwards (resp. inwards) and that there are no correct critical points of g |W ε f -α in g -1 (0). We assume that δ > 0 and we apply Morse theory for manifolds with boundary to obtain :

χ(W ε f -α ∩ {g ≥ -δ}, W ε (f -α,g+δ) ) = i (-1) λ(p i ) ,
where {p i } is the set of critical points of g |f -1 (α)∩{ω<ε} , and :

χ(W ε f -α ∩ {g ≤ -δ}, W ε (f -α,g+δ
) ) = 0. Summing these equalities and using the Mayer-Vietoris sequence gives :

χ(W ε f -α ) -χ(W ε (f -α,g+δ) ) = i (-1) λ(p i ) . By Lemma 2.5, i (-1) λ(p i ) is equal to deg 0 k(f, g). By Khimshiashvili's formula, χ(W ε f -α ) = 1 -sign(-α) n • deg 0 ∇f . Now by Proposition 1.1 in [FK], we know that deg 0 ∇f = 0 if n is odd. This gives the result for the fibre W ε (f,g-δ)
with δ < 0. The formula for the fibre W ε (f,g-δ) with δ > 0 is obtained replacing g with -g. It remains to prove the third formula. Let δ be such that (α, δ) is a regular value of (f, g) and 0 ≤ |δ| ≪ |α| ≪ ε. Since n is even, we have :

χ(W ε f -α ∩ {g ≥ δ}) -χ(W ε (f -α,g-δ) ) = i|g(p i )>δ (-1) λ(p i ) , χ(W ε f -α ∩ {g ≤ δ}) -χ(W ε (f -α,g-δ) ) = - i|g(p i )<δ (-1) λ(p i ) .
Making the difference and using Lemma 2.5 leads to the result.

Corollary 2.8. If 0 is an isolated critical point of f and is isolated in k(f, g) -1 (0), then one has :

if n is odd : χ(∂W ε (f,g) ) = 2 -2deg 0 k(f, g), if n is even : χ(∂W ε f ∩ {g ≥ 0}) -χ(∂W ε f ∩ {g ≤ 0}) = 2deg 0 k(f, g). Proof.
The first point is easy. For the second point, see [START_REF] Dutertre | Degree formulas for a topological invariant of bifurcations of function germs[END_REF], Theorem 5.2. Now let us suppose that g also has an isolated critical point at the origin and consider the mapping l(f, g) : (R n , 0) → (R n , 0) defined by : l(f, g) = (g, v 2 g, . . . , v n g).

In [FK], Theorem 4.1, Fukui and Khovanskii prove that if 0 is isolated in l(f, g) -1 (0) and if the set of critical points of f does not intersect W ε g-δ then:

deg 0 l(f, g) = -sign(-δ) n • χ(W ε g-δ ∩ {f ≥ 0}) -χ(W ε g-δ ∩ {f ≤ 0}) .
In our situation the second condition is fullfilled because f has an isolated critical point. We want to give an other interpretation to this degree. We will prove the following theorem.

Theorem 2.9. If f and g have an isolated critical point at the origin and 0 is isolated in l(f, g) -1 (0) then :

if n is even : χ(W ε (f -δ,g) ) = 1 -deg 0 ∇g -sign(δ) • deg 0 l(f, g), if n is odd : χ(W ε f -δ ∩ {g ≥ 0}) -χ(W ε f -δ ∩ {g ≤ 0}) = deg 0 ∇g + sign(δ) • deg 0 l(f, g), where 0 < |δ| ≪ ε ≪ 1.
The proof of this theorem goes nearly like the proof of Theorem 2.1. We need some lemmas.

Lemma 2.10. For δ = 0 sufficiently small, (δ, 0) is a regular value of (f, g).

Lemma 2.11. Let p be a regular point of g. The function f |g -1 (g(p)) has a critical point at p if and only if v i g(p) = 0 for all i ∈ {2, . . . , n}.

Proof. The function f |g -1 (g(p)) has a critical point at p if and only if ∇f (p) and ∇g(p) are colinear. Since these two vectors are non zero, this is equivalent to the fact that g |f -1 (f (p)) has a critical point at p. It is enough to use Lemma 2.3.

Lemma 2.12. The origin is an isolated singularity of f -1 (0) ∩ g -1 (0) if and only if 0 is isolated in l(f, g) -1 (0). Lemma 2.13. Let α = 0 be a small regular value of g. Let p be a point in g -1 (α). The function f |g -1 (α) has a non-degenerate critical point at p if and only if l(f, g)(p) = (α, 0, . . . , 0) and detDl(f, g)(p) = 0. Furthermore if λ(p) is the Morse index of f |g -1 (α) at p and if µ(p) is the real number such that ∇f (p) = µ(p) • ∇g(p) then we have :

(-1) λ(p) = (-1) n-1 • sign [µ(p) n • detDl(f, g)(p)] .
Proof. Since α is a regular value of g, there exists j such that g x j (p) = 0. Assume that j = 1. From [START_REF] Szafraniec | A formula for the Euler characteristic of a real algebraic manifold[END_REF], p is a non-degenerate critical point of f |g -1 (α) if and only if :

det ∇g(p) -∇m 1i (p) 2≤i≤n = 0.
Furthermore, we have :

(-1) λ(p) = (-1) n-1 • sign g x 1 (p) n • det ∇g(p) -∇m 1i (p)
.

Since g x 1 (p) = 0, f x 1 (p) does not vanish for otherwise µ(p) and ∇f (p) would vanish as well. Then the computations done in Lemma 2.5 show that :

sign det ∇g(p) -∇v i g(p) = sign f x 1 (p) n-2 • det ∇g(p) -∇m 1i (p) ,
and it is easy to conclude.

The following lemma deals with the critical points of f |∂W ε g-α . Lemma 2.14. For all α and ε such that 0 < |α| ≪ ε ≪ 1, one has :

• at all correct critical points of

f |∂W ε g-α with f > 0, ∇f |g -1 (α) points outwards, • at all correct critical points of f |∂W ε g-α with f < 0, ∇f |g -1 (α) points inwards,
• there are no correct critical points of f |∂W ε g-α in f -1 (0). Similarly, we have : Lemma 2.15. For ε sufficiently small, one has :

• at all correct critical points of

f |S n-1 ε with f > 0, ∇f points outwards, • at all correct critical points of f |S n-1 ε with f < 0, ∇f points inwards, • there are no correct critical points of f |S n-1 ε in f -1 (0).
Lemma 2.16. We can choose α small enough and we can pertube g into g in such a way that f |W ε g-α has only Morse critical points. Proof. With the method of Lemma 2.7, we can prove that there exists a small perturbation g of g such that f |W ε g-α has only Morse critical points outside {f = 0}. But Lemma 2.2 states that (0, α) is a regular value of (f, g) for α small enough.

Proof of Theorem 2.9. The case n even is proved as in Theorem 2.1. So let us assume that n is odd. Let ω : R n → R be the distance function to the origin. Let ε > 0 be sufficiently small so that f |g -1 (0)\{0} has no critical point in g -1 (0) \ {0} ∩ {ω < ε}. Let (δ, α) be a regular value of (f, g) such that :

(1) 0

< |α| ≪ |δ| ≪ ε, (2) the critical points of f |g -1 (α) lie in {|f | < δ} ∩ {ω < ǫ 2 }, (3) {g?0}∩W ε f -δ is diffeomorphic to {g?α}∩W ε f -δ
, where ? ∈ {≤, =, ≥}. Thanks to the three previous lemmas, we can assume as in Theorem 2.1 that we are in a good situation to apply Morse theory for manifolds with corners. Let us assume that δ > 0. By Morse Theory, we obtain :

χ({g ≥ α} ∩ {f ≥ δ} ∩ B n ε ) -χ({g ≥ α} ∩ W ε f -δ ) = 0, (1)
χ({g ≥ α} ∩ {f ≤ δ} ∩ B n ε ) -χ({g ≥ α} ∩ W ε f -δ ) = i|µ(p i )<0 (-1) λ(p i ) , (2)
χ({g ≤ α} ∩ {f ≥ δ} ∩ B n ε ) -χ({g ≤ α} ∩ W ε f -δ ) = 0, (3)
χ({g ≤ α} ∩ {f ≤ δ} ∩ B n ε ) -χ({g ≤ α} ∩ W ε f -δ ) = -deg 0 ∇f + i|µ(p i )>0 (-1) λ(p i ) . (4)
In the equality (4), the terms -deg 0 ∇f appears because we can pertube f in such a way that its critical points lie in {|g| ≤ α} ∩ {|f | ≤ δ}. The combination (1) + ( 2) -(3) -(4) together with the Mayer-Vietoris sequence gives :

χ({g ≥ α} ∩ B n ε ) -χ({g ≤ α} ∩ B n ε ) -χ({g ≥ α} ∩ W ε f -δ ) +χ({g ≤ α} ∩ W ε f -δ ) = - i signµ(p i ) • (-1) λ(p i ) + deg 0 ∇f.
We have already seen that deg 0 ∇f = 0. Moreover, by the remark after Theorem 3.2 in [START_REF] Dutertre | Courbures et singularités réelles[END_REF], we have :

χ({g ≥ α} ∩ B n ε ) -χ({g ≤ α} ∩ B n ε ) = deg 0 ∇g.
Using Lemma 2.13, we find that :

χ({g ≥ 0} ∩ W ε f -δ ) -χ({g ≤ 0} ∩ W ε f -δ ) = deg 0 ∇g + deg 0 l(f, g).
The proof for δ negative is obtained replacing f with -f .

A generalization of Fukui's formula

In this section, we present a generalization of Fukui's formula mentionned in the introduction. As in the previous section, we need to put a condition. More precisely, let (x 0 , x 1 , . . . , x n ) be a coordinate system in R 1+n and let F : (R 1+n , 0) → (R, 0) be an analytic function-germ with an isolated critical point at the origin. We assume that F satisfies the following condition (Q) : there exists C ∞ vector fields V 2 , . . . , V n on R 1+n such that V 2 (p), . . . , V n (p) span the tangent space at p to F -1 (F (p)) ∩ x -1 0 (x 0 (p)) whenever p is a regular point of (F, x 0 ) and such that (e 0 , ∇F (p), V 2 (p), . . . , V n (p)) agrees with the orientation of R 1+n . Here e 0 is the vector (1, 0, . . . , 0).

Let G : (R 1+n , 0) → (R, 0) be an other analytic function-germ. We define three mappings H(F, G), I(F, G) and J(F, G) : (R 1+n , 0) → (R 1+n , 0) by :

H(F, G) = (F, G, V 2 G, . . . , V n G), I(F, G) = (x 0 , G, V 2 G, . . . , V n G), J(F, G) = (x 0 F, G, V 2 G, . . . , V n G).
Our first aim is to prove the following theorem : Theorem 3.1. If F has an isolated critical point at the origin, 0 is isolated in J(F, G) -1 (0) and (0, δ, 0) is a regular value of (F, G, x 0 ), then 0 is isolated in H(F, G) -1 (0) and we have :

deg 0 H(F, G) = sign(-δ) n • χ W ε (F,G-δ) ∩ {x 0 ≥ 0} -χ W ε (F,G-δ) ∩ {x 0 ≤ 0} , deg 0 J(F, G) = sign(-δ) n • χ W ε (F,G-δ) -χ W ε (G-δ,x 0 )
, where 0 < |δ| ≪ ε ≪ 1.

To establish this theorem, we need a series of lemmas. From now on, we will assume that the three assumptions of Theorem 3.1 are fullfilled. For all (i, j) ∈ {1, . . . , n} 2 , we will set

M ij = ∂(F,G) ∂(x i ,x j ) .
Lemma 3.2. For δ = 0 sufficiently small, (0, δ) is a regular value of (F, G).

Lemma 3.3. The origin is an isolated singularity of F -1 (0) ∩ G -1 (0).

Lemma 3.4. Let δ = 0 be sufficiently small so that F -1 (0) ∩ G -1 (δ) is smooth submanifold (or empty) of codimension 2 near the origin. Let p be a point in

F -1 (0) ∩ G -1 (δ). The function x 0|F -1 (0)∩G -1 (δ) has a critical point at p if and only if H(F, G)(p) = (0, δ, 0, . . . , 0).
Proof. The function x 0|F -1 (0)∩G -1 (δ) has a critical point at p if and only if

F (p) = 0, G(p) = δ and rank   1 0 . . . 0 F x 0 (p) F x 1 (p) . . . F xn (p) G x 0 (p) G x 1 (p) . . . G xn (p)   < 3.
First let us suppose that p is a critical point of x 0|F -1 (0)∩G -1 (δ) and remark that necessarly x 0 (p) = 0 because (0, δ, 0) is a regular value of (F, G, x 0 ). This implies that p is a regular point of (F, x 0 ) for the critical points of x 0|F -1 (0)\{0} lie in {x 0 = 0} by the Curve Selection Lemma and, so,

(V 2 (p), . . . , V n (p)) is a basis of T p [F -1 (0) ∩ x -1 0 (x 0 (p))].
Since ∇G(p) belongs to the normal space at p to F -1 (0) ∩ x -1 0 (x 0 (p)), we find that for each i ∈ {2, . . . , n}, V i (p), ∇G(p) = 0.

Let us show the inverse implication. Let p be such that H(F, G)(p) = (0, δ, 0, . . . , 0). If (F, x 0 ) is not regular at p then x 0 (p) = 0 and (0, δ, 0) is not a regular value of (F, G, x 0 ), which is impossible. Hence (V 2 (p), . . . , V n (p)) is a basis of T p [F -1 (0) ∩ x -1 0 (x 0 (p))] and ∇G(p) is normal to this last tangent space.

Lemma 3.5. Under the assumptions of Lemma 3.4, x 0|F -1 (0)∩G -1 (δ) has a non-degenerate critical point at p if and only if H(F, G)(p) = (0, δ, 0, . . . , 0) and detDH(F, G)(p) = 0. Furthermore if λ(p) is the Morse index of this function at p then :

(-1) λ(p) = (-1) n • sign G(p) x 0 (p) n • detDH(F, G)(p) .
Proof. First observe that, since (0, δ) is a regular value of (F, G) and the M ij 's, i, j ∈ {1, . . . , n}, vanish at p, there exists k ∈ {1, . . . , n} such that ∂(F,G) ∂(x 0 ,x k ) (p) = 0. Assume that k = 1. This implies that F x 1 (p) = 0 for otherwise G x 1 (p) = 0 and F x j (p) = 0 for j ∈ {2, . . . , n}, which means that p is not a regular point of (F, x 0 ) and x 0 (p) = 0.

From [START_REF] Szafraniec | A formula for the Euler characteristic of a real algebraic manifold[END_REF], p is a Morse critical point of x 0|F -1 (0)∩G -1 (δ) if and only if det

  ∇F (p) ∇G(p) ∇N i (p)   2≤i≤n = 0,
where

N i = ∂(x 0 ,F,G) ∂(x 0 ,x 1 ,x i ) = M 1i .
Moreover, one has :

(-1) λ(p) = sign   det   ∇F (p) ∇G(p) ∇M 1i (p)   • ∂(F, G) ∂(x 0 , x 1 ) (p) n   .
Let us relate det(∇F (p), ∇G(p), ∇M 1i (p)) to det(∇F (p), ∇G(p), ∇V i G(p)).

For i ∈ {2, . . . , n}, let U i (p) be the vector : (0, F x i (p), 0, . . . , 0, -F x 1 (p), 0, . . . , 0),

where -F x 1 (p) is the (i + 1)-th coordinate. Then (U 2 (p), . . . , U n (p)) is a basis T p [F -1 (0) ∩ x -1 0 (x 0 (p))] and det(e 0 , ∇F (p), U 2 (p), . . . , U n (p)) = (-1) n-1 • F x 1 (p) n-2 • n i=1 F x i (p) 2 .
Hence there exists a (n -1) × (n -1)-matrix B(p) such that :

  e 0 ∇F (p) U i (p)   =   1 0 0 0 1 0 0 0 B(p)   •   e 0 ∇F (p) V i (p)   , with sign[detB(p)] = (-1) n-1 • sign[F x 1 (p) n-2
]. As we proceed in Lemma 2.5, we have :

sign [det (e 0 , ∇F (p), ∇U i G(p))] = (-1) n-1 • sign F x 1 (p) n-2 • det (e 0 , ∇F (p), ∇V i G(p)) .
Since e 0 is a linear combination of ∇F (p) and ∇G(p), it is easy to see that :

sign [det (∇F (p), ∇G(p), ∇U i G(p))] = (-1) n-1 • sign F x 1 (p) n-2 • det (∇F (p), ∇G(p), ∇V i G(p)) .
Using the fact that U i G(p) = -M 1i (p), we find that :

(-1) λ(p) = sign   det   ∇F (p) ∇G(p) ∇V i G(p)   • ∂(F, G) ∂(x 0 , x 1 ) (p) n • F x 1 (p) n-2   .
It remains to study the sign of ∂(F,G) ∂(x 0 ,x 1 ) (p). By the Curve Selection Lemma, we can assume that p is on the image of an analytic arc γ :]0, ε[→ F -1 (0) such that M ij (γ(t)) = 0 for t ∈]0, ε[ and (i, j) ∈ {1, . . . , n} 2 . One has

n i=1 F x i (γ) • γ ′ i = 0 since F • γ = 0 and (G • γ) ′ = n i=1 G x i (γ) • γ ′ i .
Multiplying the first equality by G x 1 , the second by F x 1 and making the difference leads to :

F x 0 G x 1 -G x 0 F x 1 = - (G • γ) ′ γ ′ 0 • F x 1 . Hence if δ = 0 is small enough, sign ∂(F,G) ∂(x 0 ,x 1 ) n • F n-2 x 1
= -sign G x 0 at p. The following lemma deals with the critical points of x 0|∂W ε (F,G-δ) . Lemma 3.6. Assume that (0, δ, 0) is a regular value of (F, G, x 0 ) for δ sufficiently small. Then, for ε and δ such that 0 < |δ| ≪ ε ≪ 1 :

• the vector ∇x 0|F -1 (0)∩G -1 (δ) points outwards at all correct critical points of x 0|∂W ε (F,G-δ) with x 0 > 0, • the vector ∇x 0|F -1 (0)∩G -1 (δ) points inwards at all correct critical points of x 0|∂W ε (F,G-δ) with x 0 < 0, • there are no correct critical points of x 0|∂W ε (F,G-δ) in {x 0 = 0}. Proof. The proof is the same as [START_REF] Dutertre | Degree formulas for a topological invariant of bifurcations of function germs[END_REF], Lemma 4.1.

Lemma 3.7. If for δ small enough (0, δ, 0) is a regular value of (F, G, x 0 ), then we can perturbe G into G in such a way that

x 0|W ε (F, G-δ) has only Morse critical points in W ε (F, G-δ) \ {x 0 = 0}. Proof.
The proof is similar to the proofs of Lemma 2.7 and Lemma 4.2 in [START_REF] Dutertre | Degree formulas for a topological invariant of bifurcations of function germs[END_REF]. Let us describe it briefly. Let (x 0 , . . . , x n ; t 1 , . . . , t n ) = (x; t) be a coordinate system of R 2n+1 and let

Ḡ(x, t) = G(x) + t 1 x 1 + • • • + t n x n .
For (i, j) ∈ {1, . . . , n} 2 , we define

M ij (x, t) by M ij (x, t) = ∂(F, Ḡ) ∂(x i ,x j ) . Note that M ij (x, t) = m ij (x) + F x i t j -F x j t i . Let Γ be defined by : Γ = (x, t) ∈ R 2n+1 | F (x) = 0 and M ij (x, t) = 0 for (i, j) ∈ {1, . . . , n} 2 .
In the same way as in Lemma 2.7 and Lemma 4.2, we can prove that Γ\{x 0 = 0} is a smooth manifold (or empty) of dimension n + 1. Then we conclude with the following mapping :

π : Γ \ {x 0 = 0} → R 1+n (x, t) → ( Ḡ(x, t), t).
Lemma 3.8. The function G |{x 0 =0} has an isolated critical point at the origin.

Proof. Since J(F, G) has an isolated zero at 0, the point (0, 0, 0) is isolated in I(F, G) -1 (0). This would not be the case if 0 in R n was not an isolated critical point of G |{x 0 =0} . Lemma 3.9. Let δ = 0 be sufficiently small so that {x 0 = 0} ∩ G -1 (δ) is a smooth submanifold of codimension 2 (or empty) near the origin. Let s be a point in {x 0 = 0} ∩ G -1 (δ). The function F |{x 0 =0}∩G -1 (δ) has a critical point at s if and only if I(F, G)(s) = (0, δ, 0, . . . , 0).

Proof. Since (0, δ, 0) is a regular value of (F, G, x 0 ), we can apply the proof of Lemma 2.11. Lemma 3.10. Under the assumptions of Lemma 3.9, F |G -1 (δ)∩x -1 0 (0) has a non-degenerate critical point at s if and only if I(F, G)(s) = (0, δ, 0, . . . , 0) and detDI(F, G)(s) = 0. Furthermore if µ(s) is the Morse index of this function at s then :

(-1) µ(s) = (-1) n-1 • sign G(s) F (s) n • detDI(F, G)(s) .
Proof. The proof is the same as Lemmas 2.5, 2.13 and 3.5. We leave it to the reader.

The following lemma deals with the critical points of F |∂W ε (G-δ,x 0 ) . Lemma 3.11. Assume that (0, δ, 0) is a regular value of (F, G, x 0 ) for δ sufficiently small. Then, for ε such that 0 < |δ| ≪ ε ≪ 1 :

• the vector ∇F |x -1 0 (0)∩G -1 (δ) points outwards at all correct critical points of

F |W ε (G-δ,x 0 ) with F > 0, • the vector ∇F |x -1 0 (0)∩G -1 (δ)
points inwards at all correct critical points of F |W ε (G-δ,x 0 ) with F < 0, • there are no correct critical points of F |W ε (G-δ,x 0 ) in F -1 (0). Lemma 3.12. We can perturbe G into G in such a way that F |W ε ( G-δ,x 0 ) has only Morse critical point.

Proof. The same method as Lemma 2.16 can be applied, because we have assumed that (0, δ, 0) is a regular value of (F, G, x 0 ).

Proof of Theorem 3.1. It is easy to see that 0 is isolated in H(F, G) -1 (0) and I(F, G) -1 (0). Let ω : R 1+n → R be the distance function to the origin. Because 0 is isolated in H(F, G) -1 (0), x 0|F -1 (0)∩G -1 (0)\{0} has no critical point and then, choosing δ sufficiently small, we can assume that x 0|F -1 (0)∩G -1 (δ)∩{ω<ε} admits its critical points in W ε/4 (F,G-δ) . Thus the critical points of x 0|∂W ε (F,G-δ) are correct. By Lemmas 3.6 and 3.7, we can suppose that x 0|W ε (F,G-δ) is a correct Morse function, that its critical points lie in B ε/2 , that at the correct critical points of x 0|W ε (F,G-δ) lying in {x 0 > 0} (resp. in {x 0 < 0}), ∇x 0|F -1 (0)∩G -1 (δ) points outwards (resp. inwards) and that there are no correct critical points of x 0|F δ in {x 0 = 0}. Applying Morse Theory for manifolds with boundary, we find :

χ(W ε (F,G-δ) ∩ {x 0 ≥ 0}, W ε (F,G-δ,x 0 ) ) = i|x 0 (p i )>0 (-1) λ(p i ) ,
where {p i } is the set of Morse critical points of x 0|W ε (F,G-δ) . Similarly, we have :

χ(W ε (F,G-δ) {x 0 ≤ 0}, W ε (F,G-δ,x 0 ) ) = (-1) n-1 i|x 0 (p i )<0
(-1) λ(p i ) .

By Lemma 3.4, p is a critical point of x 0|W ε (F,G-δ) if and only if H(F, G)(p) = (0, δ, 0, . . . , 0). Hence H(F, G) -1 (0, δ, 0, . . . , 0) is the set of critical points of x 0|W ε (F,G-δ) . Since x 0|W ε (F,G-δ) is a Morse function, detDH(F, G)(p) = 0
for each p in H -1 (0, δ, 0, . . . , 0) by Lemma 3.5. Hence (0, δ, 0, . . . , 0) is a regular value of H(F, G) and

deg 0 H = p∈H -1 (0,δ,0,...,0) sign [detDH(F, G)(p)] .
Combining this with the above equalities and Lemma 3.5, we obtain the first equality. Let us study the critical points of F |W ε (G-δ,x 0 ) . Thanks to Lemmas 3.11 and 3.12, we can assume that we are in a good situation to apply Morse theory. We have :

χ(W ε (G-δ,x 0 ) ∩ {F ≥ 0}) -χ(W ε (F,G-δ,x 0 ) ) = j|F (s j )>0 (-1) µ(s j ) ,
where {s j } is the set of Morse critical points of F |W ε (G-δ,x 0 ) . Similarly, we have :

χ(W ε (G-δ,x 0 ) ∩ {F ≤ 0}) -χ(W ε (F,G-δ,x 0 ) ) = (-1) n-1 j|F (s j )>0
(-1) µ(s j ) .

Hence, we get :

χ(W ε (G-δ,x 0 ) ) -χ(W ε (F,G-δ,x 0 ) ) = j|F (s j )>0 (-1) µ(s j ) + (-1) n-1 • j|F (s j )<0
(-1) µ(s j ) .

Applying Lemma 3.10, this gives :

χ(W ε (G-δ,x 0 ) ) -χ(W ε (F,G-δ,x 0 ) ) = -sign(-δ) n • j sign[F (s j )] • detDI(F, G)(s j ).
Similarly, we have :

χ(W ε (F,G-δ) ) -χ(W ε (F,G-δ,x 0 ) ) = sign(-δ) n • i sign[x 0 (p i )] • detDH(F, G)(p i ).
But the sets {p i } and {s j } are exactly the preimages of (0, δ, 0, . . . , 0) by J(F, G). Furthermore, each p i is a regular point of J(F, G) and

sign [detDJ(F, G)(p i )] = sign [x 0 (p i ) • detDH(F, G)(p i )] .
Each s j is a regular value of J(F, G) as well and

sign [detDJ(F, G)(s j )] = sign [F (s j ) • detDI(F, G)(s j )] .
With all these informations, it is easy to conclude. We want to apply these formulas when F and G are one-parameter deformations of two germs f and g. First let us define the function-germ

F 0 : (R n , 0) → (R, 0) by F 0 (x 1 , . . . , x n ) = F (0, x 1 , . . . , x n ).
Lemma 3.13. Assume that the function-germ F 0 has an isolated critical point at the origin.. Then for δ sufficiently small, (0, δ, 0) is a regular of (F, G, x 0 ). Let us suppose that δ > 0, then for 0 < δ ≪ ε ≪ 1, one has :

W ε (F,G-δ) ∩ {x 0 ≥ 0} ≃ ∂W ε F ∩ {G ≥ 0} ∩ {x 0 ≥ 0} ≃ W ε (F,x 0 -δ) ∩ {G ≥ 0}, W ε (F,G-δ) ∩ {x 0 ≤ 0} ≃ ∂W ε F ∩ {G ≥ 0} ∩ {x 0 ≤ 0} ≃ W ε (F,x 0 +δ) ∩ {G ≥ 0}, W ε (F,G+δ) ∩ {x 0 ≥ 0} ≃ ∂W ε F ∩ {G ≤ 0} ∩ {x 0 ≥ 0} ≃ W ε (F,x 0 -δ) ∩ {G ≤ 0}, W ε (F,G+δ) ∩ {x 0 ≤ 0} ≃ ∂W ε F ∩ {G ≤ 0} ∩ {x 0 ≤ 0} ≃ W ε (F,x 0 +δ) ∩ {G ≤ 0}, where ≃ means diffeomorphic to.
Proof. Let us prove the first line. It is just an adaptation to our case of the deformation argument given by Milnor [START_REF] Milnor | Singular points of complex hypersurfaces[END_REF]Lemma 11.3]. We can construct a vector field v 1 on W ε F \{G = 0} such that v 1 (x), ∇G(x) and v 1 (x), x are both positive. Similarly there exists a vector field v 2 on W ε (F,x 0 ) \ {G = 0} such that v 2 (x), ∇G(x) and v 2 (x), x are both positive. Using a collar, we can extend v 2 to a vector field ṽ2 defined in a neighborhood of W ε (F,x 0 ) \{G = 0} in W ε F ∩ {x 0 ≥ 0} \ {G = 0} such that ṽ2 (x), ∇G(x) and ṽ2 (x), x are positive. Gluing v 1 and ṽ2 gives a new vector field w on

W ε F ∩ {x 0 ≥ 0} \ {G = 0}. The diffeomorphism between W ε (F,G-δ) ∩ {x 0 ≥ 0} and ∂W ε F ∩ {G ≥ 0} ∩ {x 0 ≥ 0} is obtained integrating the trajectories of w. Similarly W ε (F,x 0 -δ) ∩ {G ≥ 0} is diffeomorphic to ∂W ε F ∩ {G ≥ 0} ∩ {x 0 ≥ 0} because, by Lemma 3.3, F -1 (0) ∩ G -1 (0) is smooth outside the origin.
We want to compute χ(W ε (F,G,x 0 -δ) ). By the Mayer-Vietoris sequence, we know that :

χ(W ε (F,x 0 -δ) ) = χ(W ε (F,x 0 -δ) ∩ {G ≥ 0})+ χ(W ε (F,x 0 -δ) ∩ {G ≤ 0}) -χ(W ε (F,G,x 0 -δ)
). Hence, by Lemma 3.13, we find that if δ > 0, then :

χ(W ε (F,G,x 0 -δ) ) = χ(W ε (F,G-δ) ∩ {x 0 ≥ 0})+ χ(W ε (F,G+δ) ∩ {x 0 ≥ 0}) -χ(W ε (F,x 0 -δ) ), χ(W ε (F,G,x 0 +δ) ) = χ(W ε (F,G-δ) ∩ {x 0 ≤ 0})+ χ(W ε (F,G+δ) ∩ {x 0 ≤ 0}) -χ(W ε (F,x 0 +δ)
). The Euler-Poincaré characteristic of W ε (F,x 0 ±δ) can be computed thanks to formulas established in [Fu], as explained in [START_REF] Dutertre | Courbures et singularités réelles[END_REF]Theorem 3.2]. More precisely, let L(F ) : (R 1+n , 0) → (R 1+n , 0) be the mapping defined by L(F ) = (F, F x 1 , . . . , F xn ). If L(F ) and ∇F 0 have an isolated zero at the origin, then ∇F has an isolated zero at the origin and the following theorem explains how to compute W ε (F,x 0 ±δ) .

Theorem 3.14. Let δ and ε be such that 0 [Fu] and [START_REF] Dutertre | Courbures et singularités réelles[END_REF]. At this point, we have assumed that :

< |δ| ≪ ε ≪ 1. If n is even then : χ(W ε (F,x 0 -δ) ) = 1 -deg 0 ∇F 0 . If n is odd then : χ(W ε (F,x 0 -δ) ) = 1 -deg 0 ∇F -sign(δ) • deg 0 L(F ). Proof. See
(1) F has an isolated critical point at the origin, (2) J(F, G) has an isolated zero at the origin, (3) (0, δ, 0) is a regular value of (F, G, x 0 ), (4) F 0 has an isolated critical point at the origin. By the Curve Selection Lemma, the assumption (4) implies the assumption (3). Moreover, it means that 0 is isolated in

{F = F x 1 = • • • = F xn = x 0 = 0}. Since this last set is equal to {F = F x 1 = • • • = F xn = 0}
near the origin thanks to the Curve Selection Lemma and the fact that F -1 (0) has an isolated singularity, we have that (4) implies that 0 is isolated in L(F ) -1 (0). So, under the assumption (4), we can apply the above theorem.

It remains to compute χ(W ε (F,G±δ) ∩ {x 0 ?0}), ? ∈ {≤, ≥}. By the Mayer-Vietoris sequence, we have :

χ(W ε (F,G-δ) ) = χ(W ε (F,G-δ) ∩ {x 0 ≥ 0})+ χ(W ε (F,G-δ) ∩ {x 0 ≤ 0}) -χ(W ε (F,G-δ,x 0 )
). But Theorem 2.1 enables us to compute χ(W ε (F,G-δ) ) and χ(W ε (F,G-δ,x 0 ) ). Let G 0 : (R n+1 , 0) → (R, 0) be defined by G 0 (x 1 , . . . , x n ) = G(0, x 1 , . . . , x n ) and let us assume that it has an isolated critical point at the origin. Then using Theorem 3.1 and Khimshiashvili's formula, we find that :

χ(W ε (F,G-δ) ) = 1 + sign(-δ) n • [deg 0 J(F, G) -deg 0 ∇G 0 ]
. Now observe that, since F satisfies the condition (Q), F 0 satisfies the condition (P ) of Section 2 with the vector fields V 0 2 , . . . , V 0 n given by :

V 0 i (x 1 , . . . , x n ) = V i (0, x 1 , . . . , x n ). Let k(F 0 , G 0 ) : (R n , 0) → (R n , 0) be defined by : k(F 0 , G 0 ) = (F 0 , V 0 2 G 0 , . . . , V 0 n G 0 ).
By Theorem 2.1, we have :

if n is even : χ(W ε (F,G-δ,x 0 ) ) = 1 -deg 0 ∇F 0 + sign(δ) • deg 0 k(F 0 , G 0 ), if n is odd : χ(W ε (F,G-δ,x 0 ) ) = 1 -deg 0 k(F 0 , G 0 )
. Let us focus first on the case n even. We have :

χ(W ε (F,G-δ) ∩ {x 0 ≥ 0}) -χ(W ε (F,G-δ) ∩ {x 0 ≤ 0}) = deg 0 H(F, G), χ(W ε (F,G-δ) ∩ {x 0 ≥ 0}) + χ(W ε (F,G-δ) ∩ {x 0 ≤ 0}) = 2 + deg 0 J(F, G) -deg 0 ∇G 0 -deg 0 ∇F 0 + sign(δ) • deg 0 k(F 0 , G 0 ).
This gives :

χ(W ε (F,G-δ) ∩ {x 0 ≥ 0}) = 1 + 1 2 [deg 0 J(F, G) -deg 0 ∇G 0 -deg 0 ∇F 0 + sign(δ) • deg 0 k(F 0 , G 0 ) + deg 0 H(F, G)] , χ(W ε (F,G-δ) ∩ {x 0 ≤ 0}) = 1 + 1 2 [deg 0 J(F, G) -deg 0 ∇G 0 -deg 0 ∇F 0 + sign(δ) • deg 0 k(F 0 , G 0 ) -deg 0 H(F, G)] .
Collecting all these informations, we obtain :

χ(W ε (F,G,x 0 -δ) ) = 1 + deg 0 J(F, G) -deg 0 ∇G 0 + sign(δ) • deg 0 H(F, G), χ(W ε (F,x 0 -δ) ∩ {G ≥ 0}) -χ(W ε (F,x 0 -δ) ∩ {G ≤ 0}) = deg 0 k(F 0 , G 0 ). If n is odd, we have : χ(W ε (F,G-δ) ∩{x 0 ≥ 0})-χ(W ε (F,G-δ) ∩{x 0 ≤ 0}) = -sign(δ)•deg 0 H(F, G), χ(W ε (F,G-δ) ∩ {x 0 ≥ 0}) + χ(W ε (F,G-δ) ∩ {x 0 ≤ 0}) = 2 -sign(δ) • [deg 0 J(F, G) -deg 0 ∇G 0 ] -deg 0 k(F 0 , G 0 ).
This gives :

χ(W ε (F,G-δ) ∩ {x 0 ≥ 0}) = 1 - 1 2 [sign(δ) • (deg 0 J(F, G) -deg 0 ∇G 0 +deg 0 H(F, G)) + deg 0 k(F 0 , G 0 )] , χ(W ε (F,G-δ) ∩ {x 0 ≤ 0}) = 1 - 1 2 [sign(δ) • (deg 0 J(F, G) -deg 0 ∇G 0 -deg 0 H(F, G)) + deg 0 k(F 0 , G 0 )] .
Finally we find :

χ(W ε (F,G,x 0 -δ) ) = 1 -deg 0 k(F 0 , G 0 ) -deg 0 ∇F -sign(δ) • deg 0 L(F ), χ(W ε (F,x 0 -δ) ∩ {G ≥ 0}) -χ(W ε (F,x 0 -δ) ∩ {G ≤ 0}) = -deg 0 J(F, G) + deg 0 ∇G 0 -sign(δ) • deg 0 H(F, G).
Here, we have to remark that :

χ(W ε (F,G,x 0 -δ) ) = 1 2 χ(∂W ε (F,G,x 0 -δ) ) = 1 2 χ(∂W ε (F,G,x 0 ) ) = 1 -deg 0 k(F 0 , G 0 ),
by Corollary 2.8. Hence, we get that deg 0 ∇F = deg 0 L(F ) = 0.

We can reformulate these results in terms of one-parameter deformations of function-germs. Let (x 1 , . . . , x n ) be a coordinate system of R n . Let f : (R n , 0) → (R, 0) be a function-germ with an isolated critical point at the origin. Let g : (R n , 0) → (R, 0) be a function-germ with an isolated critical point at the origin such that the mapping k(f, g) : (R n , 0) → (R n , 0) has an isolated zero where k(f, g) is defined as in Section 2. Let (λ, x 1 , . . . , x n ) be a coordinate system in R 1+n and let F : (R 1+n , 0) → (R, 0) (resp. G : (R 1+n , 0) → (R, 0)) be a one-parameter deformation of f (resp g), i.e F (0, x) = f (x) (resp. G(0, x) = g(x)). We will use the notations f t (x) = F (t, x) and g t (x) = G(t, x). We assume that :

(1) F has an isolated critical point at the origin, (2) the mapping J(F, G) has an isolated zero at the origin, (3) F satisfies the condition (Q) (which implies that f satisfies the condition (P )). We note that F 0 and G 0 have an isolated critical point because F 0 = f and G 0 = g. So we are in situation to apply the above process.

Corollary 3.15. For t and ε with 0 < |t| ≪ ε ≪ 1, we have : -if n is odd :

χ(W ε (ft,gt) ) = 1 -deg 0 k(f, g), χ(W ε ft ∩ {g t ≥ 0}) -χ(W ε ft) ∩ {g t ≤ 0}) = -deg 0 J(F, G) + deg 0 ∇g -sign(t) • deg 0 H(F, G), -if n is even : χ(W ε (ft,gt) ) = 1 + deg 0 J(F, G) -deg 0 ∇g + sign(t) • deg 0 H(F, G), χ(W ε ft ∩ {g t ≥ 0}) -χ(W ε ft ∩ {g t ≤ 0}) = deg 0 k(f, g).
Let us consider the following deformations of f and g :

F (λ, x) = f (x) -γ 1 (λ) and G(λ, x) = g(x) -γ 2 (λ),
where γ = (γ 1 , γ 2 ) : (R, 0) → (R 2 , 0) is an analytic arc such that γ(t) = 0 if t = 0 and γ ′ 1 (t) = 0 if t = 0. With this last condition, the function F has an isolated critical point at the origin. Furthermore, we assume that f satisfies the condition (P ). This implies that F satisfies the condition (Q) with V i (λ, x) = v i (x) for i = 2, . . . , n. Let us denote by Disc(f, g) the discriminant of the mapping (f, g). The following lemma tells us when the points in the image of γ are regular value of (f, g) near the origin.

Lemma 3.16. The origin (0, 0) is isolated in H(F, G) -1 (0) if and only if 0 is isolated in Disc(f, g) ∩ γ(I), where I is a small open interval in R containing 0.

Proof. The point (0, 0) is isolated in H(F, G) -1 (0) if and only if for all (t, x) = (0, 0) such that F (t, x) = G(t, x) = 0, there exists i ∈ {2, . . . , n} such that v i G(t, x) = 0. Let us remark that if x = 0 is such that F (0, x) = G(0, x) = 0 then v i G(0, x) = 0 for some i in {2, . . . , n} because f -1 (0) ∩ g -1 (0) has an isolated singularity. Therefore the point (0, 0) is isolated in H(F, G) -1 (0) if and only if for all (t, x) with t = 0 such that F (t, x) = G(t, x) = 0 there exists i ∈ {2, . . . , n} such that v i G(t, x) = 0. This is equivalent to the fact that for all t = 0 and for all x such that f (x) = γ 1 (t) and g(x) = γ 2 (t), ∇f (x) and ∇g(x) are not colinear.

Corollary 3.15 can be restated in this situation.

Corollary 3.17. Assume that f and g have an isolated singularity and that γ ′ 1 (t) = 0 if t = 0. Assume that J(F, G) and k(f, g) have an isolated zero at the origin then for t and ε with 0 < |t| ≪ ε ≪ 1, we have : -if n is odd :

χ(W ε (f -γ 1 (t),g-γ 2 (t)) ) = 1 -deg 0 k(f, g), χ(W ε f -γ 1 (t) ∩ {g ≥ γ 2 (t)}) -χ(W ε f -γ 1 (t) ∩ {g ≤ γ 2 (t)}) = -deg 0 J(F, G) + deg 0 ∇g -sign(t) • deg 0 H(F, G), -if n is even : χ(W ε (f -γ 1 (t),g-γ 2 (t)) ) = 1 + deg 0 J(F, G) -deg 0 ∇g + sign(t) • deg 0 H(F, G), χ(W ε f -γ 1 (t) ∩ {g t ≥ γ 2 (t)}) -χ(W ε f -γ 1 (t) ∩ {g ≤ γ 2 (t)}) = deg 0 k(f, g).
Let us examine the situation when λ 1 (t) = t and λ 2 (t) = 0. In this case, we can check that deg 0 J(F, G) = 0 and that deg 0 H = -deg 0 l(f, g), where l(f, g) is defined in Section 2. Hence, we recover the results of Theorem 2.9.

Explicit formulas

In this section, we present some situations where the conditions (P ) and (Q) are satisfied. 4.1. Case n = 2, 4 or 8. As explained in [FK], when n = 2, 4 or 8, the condition (P ) is satisfied for any function-germ f : (R n , 0) → (R, 0). If ∂ x i denotes the vector e i = (0, . . . , 0, 1, 0 . . . , 0) where 1 is the i-th coordinate, then the vectors v 2 , . . . , v n are given by, if n = 2 :

v 2 = -f x 2 ∂ x 1 + f x 1 ∂ x 2 , if n = 4 : v 2 = -f x 2 ∂ x 1 + f x 1 ∂ x 2 -f x 4 ∂ x 3 + f x 3 ∂ x 4 , v 3 = -f x 3 ∂ x 1 + f x 4 ∂ x 2 + f x 1 ∂ x 3 -f x 2 ∂ x 4 , v 4 = -f x 4 ∂ x 1 -f x 3 ∂ x 2 + f x 2 ∂ x 3 + f x 1 ∂ x 4 , if n = 8 : v 2 = -f x 2 ∂ x 1 + f x 1 ∂ x 2 -f x 4 ∂ x 3 + f x 3 ∂ x 4 -f x 6 ∂ x 5 + f x 5 ∂ x 6 + f x 8 ∂ x 7 -f x 7 ∂ x 8 , v 3 = -f x 3 ∂ x 1 + f x 4 ∂ x 2 + f x 1 ∂ x 3 -f x 2 ∂ x 4 -f x 7 ∂ x 5 -f x 8 ∂ x 6 + f x 5 ∂ x 7 + f x 6 ∂ x 8 , v 4 = -f x 4 ∂ x 1 -f x 3 ∂ x 2 + f x 2 ∂ x 3 + f x 1 ∂ x 4 -f x 8 ∂ x 5 + f x 7 ∂ x 6 -f x 6 ∂ x 7 -f x 5 ∂ x 8 , v 5 = -f x 5 ∂ x 1 + f x 6 ∂ x 2 + f x 7 ∂ x 3 + f x 8 ∂ x 4 +f x 1 ∂ x 5 -f x 2 ∂ x 6 -f x 3 ∂ x 7 -f x 4 ∂ x 8 , v 6 = -f x 6 ∂ x 1 -f x 5 ∂ x 2 + f x 8 ∂ x 3 -f x 7 ∂ x 4 +f x 2 ∂ x 5 + f x 1 ∂ x 6 + f x 4 ∂ x 7 -f x 3 ∂ x 8 , v 7 = -f x 7 ∂ x 1 -f x 8 ∂ x 2 -f x 5 ∂ x 3 + f x 6 ∂ x 4 +f x 3 ∂ x 5 -f x 4 ∂ x 6 + f x 1 ∂ x 7 + f x 2 ∂ x 8 , v 8 = -f x 8 ∂ x 1 + f x 7 ∂ x 2 -f x 6 ∂ x 3 -f x 5 ∂ x 4 +f x 4 ∂ x 5 + f x 3 ∂ x 6 -f x 2 ∂ x 7 + f x 1 ∂ x 8 .
The condition (Q) is also fullfilled, the vectors V i being given by, if n = 2 :

V 2 = -F x 2 ∂ x 1 + F x 1 ∂ x 2 ,
if n = 4 :

V 2 = -F x 2 ∂ x 1 + F x 1 ∂ x 2 -F x 4 ∂ x 3 + F x 3 ∂ x 4 , V 3 = -F x 3 ∂ x 1 + F x 4 ∂ x 2 + F x 1 ∂ x 3 -F x 2 ∂ x 4 , V 4 = -F x 4 ∂ x 1 -F x 3 ∂ x 2 + F x 2 ∂ x 3 + F x 1 ∂ x 4 ,
if n = 8 :

V 2 = -F x 2 ∂ x 1 + F x 1 ∂ x 2 -F x 4 ∂ x 3 + F x 3 ∂ x 4 -F x 6 ∂ x 5 + F x 5 ∂ x 6 + F x 8 ∂ x 7 -F x 7 ∂ x 8 , V 3 = -F x 3 ∂ x 1 + F x 4 ∂ x 2 + F x 1 ∂ x 3 -F x 2 ∂ x 4 -F x 7 ∂ x 5 -F x 8 ∂ x 6 + F x 5 ∂ x 7 + F x 6 ∂ x 8 , V 4 = -F x 4 ∂ x 1 -F x 3 ∂ x 2 + F x 2 ∂ x 3 + F x 1 ∂ x 4 -F x 8 ∂ x 5 + F x 7 ∂ x 6 -F x 6 ∂ x 7 -F x 5 ∂ x 8 , V 5 = -F x 5 ∂ x 1 + F x 6 ∂ x 2 + F x 7 ∂ x 3 + F x 8 ∂ x 4 +F x 1 ∂ x 5 -F x 2 ∂ x 6 -F x 3 ∂ x 7 -F x 4 ∂ x 8 , V 6 = -F x 6 ∂ x 1 -F x 5 ∂ x 2 + F x 8 ∂ x 3 -F x 7 ∂ x 4 +F x 2 ∂ x 5 + F x 1 ∂ x 6 + F x 4 ∂ x 7 -F x 3 ∂ x 8 , V 7 = -F x 7 ∂ x 1 -F x 8 ∂ x 2 -F x 5 ∂ x 3 + F x 6 ∂ x 4 +F x 3 ∂ x 5 -F x 4 ∂ x 6 + F x 1 ∂ x 7 + F x 2 ∂ x 8 , V 8 = -F x 8 ∂ x 1 + F x 7 ∂ x 2 -F x 6 ∂ x 3 -F x 5 ∂ x 4 +F x 4 ∂ x 5 + F x 3 ∂ x 6 -F x 2 ∂ x 7 + F x 1 ∂ x 8 .
So all the results of Section 2 and Section 3 can be applied. Note also that the vector fields v i and V i are analytic.

4.2. Case f x 1 ≥ 0 and F x 1 ≥ 0. The condition (P ) is satisfied for a function-germ f : (R n , 0) → (R, 0) such that f x 1 ≥ 0 (see [START_REF] Fukui | Mapping degree and Euler characteristic[END_REF]p151]). The vectors v 2 , . . . , v n are defined by :

v i = -f x i ∂ x 1 - n j=2 (f x i f x j -δ i,j T )∂ x j ,
where T = f x 1 + n j=2 f 2 x j and δ i,j is the Kronecker symbol. Here we notice that there is a mistake in the computation of the determinant of the matrix M defined p.151 in [FK]. This determinant is (-1) n T n-1 n i=0 g 2 x i . That is why our v i 's are the opposite of the v i 's defined by Fukui and Khovanskii.

If F x 1 ≥ 0, the condition (Q) is satisfied with the vectors V i 's defined by :

V i = -F x 1 ∂ x 1 - n j=2 (F x i F x j -δ i,j T ′ )∂ x j ,
where T ′ = F x 1 + n j=2 F 2 x j . Let us remark that in this situation the computation of χ(W ε (F,G-δ) ) can be simplified thanks to Theorem 2.1. Actually, the function F satisfies the condition (P ) with the following vectors :

Z 0 = F x 0 ∂ x 1 + n j=0 | j =1 (F x i F x j -δ i,j S)∂ x j , Z i = -F x i ∂ x 1 - n j=0 | j =1
(F x i F x jδ i,j S)∂ x j , i = 2, . . . , n, where S = F x 1 + F 2

x 0 + n j=2 F 2 x j . Let K(F, G) : (R n+1 , 0) → (R n+1 , 0) be defined by : K(F, G) = (F, Z 0 G, Z 2 G, . . . , Z n G). Since F -1 (0) ∩ G -1 (0) has an isolated singularity at the origin (Lemma 3.3) then K(F, G) has an isolated zero at the origin (Lemma 2.4). Hence, by Theorem 2.1 and since deg 0 ∇F = 0 for F x 1 ≥ 0, we have : if n is odd : χ(W ε (F,G-δ) ) = 1 + sign(δ) • deg 0 K(F, G), if n is even : χ(W ε (F,G-δ) ) = 1deg 0 K(F, G). So Corollary 3.15 can be rewritten without the assumption that g has an isolated critical point at the origin. Namely, with the obvious assumptions, we obtain : Corollary 4.1. For t and ε with 0 < |t| ≪ ε ≪ 1, we have : -if n is odd :

χ(W ε (ft,gt) ) = 1 -deg 0 k(f, g), χ(W ε ft ∩ {g t ≥ 0}) -χ(W ε ft ∩ {g t ≤ 0}) = +deg 0 K(F, G) -sign(t) • deg 0 H(F, G),
-if n is even :

χ(W ε (ft,gt) ) = 1 -deg 0 K(F, G) + sign(t) • deg 0 H(F, G),

  0 /I, Mathematics Subject Classification (2000) : 14P15, 58K15.

If the deformation (F, G) of (f, g) is of the form F (λ, x) = f (x)γ 1 (λ), G(λ, x) = f (x)-γ 2 (λ), then we just need to suppose that f x 1 ≥ 0. Therefore Corollary 3.17 becomes : Corollary 4.2. Assume that f and g have an isolated singularity and that γ ′ 1 (t) = 0 if t = 0. Assume that J(F, G) and k(f, g) have an isolated zero at the origin then for t and ε with 0 < |t| ≪ ε ≪ 1, we have : -if n is odd :

-if n is even :

Let us end with an example. Let

4 and g(x 1 , x 2 , x 3 , x 4 ) = x 1 x 2 + x 3 x 4 . These functions have an isolated critical point at the origin and deg 0 ∇f = -1 and deg 0 ∇g = 1. The mappings k(f, g) and l(f, g) of Section 2 are :

) is equal to 2. By Theorem 2.1 and Theorem 2.9, we get that

By Corollary 3.17, we have :

Let us compute χ(W ε (f -t,g-t) ) using Corollary 3.17. The mappings H and J of Section 3 are given by :

Let us search the points (t, x) such that H(t, x) = 0. If x 2 = 0 then clearly x 1 = x 3 = x 4 = t = 0. If x 2 = 0 then x 3 = x 4 = 0 and :

, which is a contradiction. Hence H admits an isolated zero at the origin. Furthermore deg 0 H = 0. To see this, let (t, x) be such that H(t, x) = (0, 0, β, 0, 0) where β < 0. Necessarly x 2 = 0 and x 3 = x 4 = 0. Hence x 1 , x 2 and t satisfy the system :

. This last equality is equivalent to 3t 2 = -γ 2 , which is impossible.

Let us search the points (t, x) such that J(t, x) = 0. As above, if

which is a contradiction. The case x 2 1 + x 2 2t = 0 is also impossible as we have already explained. Hence J admits an isolated zero at the origin. Let β < 0 and let us search the points (t, x) such that J(t, x) = ( β 2 8 , 0, β, 0, 0). Necessarly x 2 = 0 and x 3 = x 4 = 0. Hence x 1 , x 2 and t satisfy the system :

8 and x 1 and x 2 have the same sign. Putting γ = β 2 and λ = t + β 2 8t , we find that

. Hence, we get that 3t 4 = β 4 64 . Thus ( β 2 8 , 0, β, 0, 0) has two preimages q 1 = (t 0 , a 1 , b 1 , 0, 0) and q 2 = (t 0 , a 2 , b 2 , 0, 0), where t 0 > 0, a 1 , b 1 > 0 and a 2 , b 2 < 0. An easy computation shows that DJ(q i ) = -128b 2 i t 0 (a ib i ) 2 . Finally we find that deg 0 J = -2. Corollary 3.17 gives that χ(W ε (f -t,g-t) ) = -2. Let us now compute χ(W ε (f -t,g-1 4 t) ). The mappings H and J are :

). We use the same technics as in the previous example. We find that H and J have an isolated root at the origin. If β < 0 then (0, 0, β, 0, 0) has two preimages by H : p 1 = (t 0 , a 1 , b 1 , 0, 0) and p 2 = (t 0 , a 2 , b 2 , 0, 0) where t 0 > 0, a 1 , b 1 > 0 and a 2 , b 2 < 0. A computation gives that DH(p i ) = -48b 2 i t 0 , which implies that deg 0 H = -2. Let us search the preimages of ( β 2 8 , 0, β, 0, 0), β < 0, by J. If (t, x) is such a preimage then necessarly x 2 = 0, x 3 = x 4 = 0 and t > 0. Moreover x 1 , x 2 and t satisfy the system :

This gives that -3 4 t 2 = β 4 63t 2 , a contradiction. We have proved that deg 0 J = 0. Applying Corollary 3.17, we obtain that χ(W ε (f -t,g - 1 4 t) ) = -2 if t > 0 and χ(W ε (f -t,g-1 4 t) ) = 2 if t < 0.


 Introduction History and geographical location of the unit

The "Laboratoire des Neurosciences de Brest, LNB", EA 4685, was created in 2012 and grouped clinicians and scientists from skin, pathology, neurosurgery, neurology, biochemistry. The scientific themes focused on pruritus mainly but also on cancer, glioblastoma, and autism. The previous AERES committee suggested to the group to focus on nerves and skin. For the next contract, a new acronym, LIEN, for Laboratory for Epithelial-Neural Interaction, is proposed. The laboratory is located at the faculty of medicine of Brest in close proximity to clinical departments of the clinicians of the unit.

Management team

The laboratory of Neuroscience de Brest is a single-team unit headed by Mr Laurent MISERY, assisted by a deputy-director, Mr Jean-Luc CARRÉ, elected for 5 years by the general assembly of all lab members. The Laboratoire des Neurociences de Brest (LNB) is focused on pathological perceptions (the way our body senses its environment during the course of a disease) by a combination of approaches including cell biology, pharmacology, physiology, epidemiology or ethology in the field of skin biology and neuroscience. During these last 5 years, the research activities of LNB were divided into several complementary axes including:

HCERES

Studies of interaction between human skin and the sensory nervous system by co-cultures of sensory neurons with keratinocyte or intact human skin (in skin organ culture).

Epidemiological and histopathological studies of conditions with sensitive skin and/or pruritus.

Study of the mechanisms of pruritus and pain perception, geared toward the development of novel pharmacological treatment strategies.

Study of the psychological aspects of skin disorders and autism, including the role of chloride channels and their inhibition by bumetadine.

Studies on the pathophysiology of cancers, including Merkel Cell carcinoma and glioma, had been part of the initial objectives of the LNB but were progressively dropped as had been recommended, even though some members of the lab remained involved in some collaborative studies on this topic during the last years.

Global assessment of the unit

The present application is a renewal of a single-team unit entitled Laboratoire des Neurosciences de Brest (LNB) since its creation in 2012. The unit requests a new name, Laboratoire des Interactions Epithélium Neurone (LIEN), for the next contract. This new name describes nicely the original and unique scientific position of the unit at a biologically, clinically, and commercially very important but underserviced research frontier that intimately links dermatology and neuroscience.

The unit exemplarily brings together researchers with multiple and complementary expertise, which are united by a joint focus on pruritus. This model interdisciplinarity is one of the major strengths of the unit, which is on a strong upwards trajectory towards impacting the course of research in its chosen field at the international level. With its focus on pruritus, the unit is well prepared to become one of the very few internationally recognized center of excellence on this topic. However, this will require committed institutional support, namely by the recruitment of additional full-time researchers.

The research projects of the LNB have been very successful during the last years, with important, novel findings leading to a better understanding of the interaction between neural and epithelial cells in human skin and a better management of irritant/sensitive skin. The members of the unit have published very satisfactorily and frequently, with emphasis on publications in leading skin research journals, and have thus increased the international visibility of the unit steadily. They have even been able to obtain recently very innovative, as yet unpublished results, concerning the novel ways how neurons interact with human keratinocytes. The unit members also have raised very substantial financial support both from foundations and from industry, including major global players in the industry that targets skin.

Strengths and opportunities in the context

The unit represents a unique gathering of expertise in multiple and complementary skills from basic to clinical research, from dermatology to neuroscience, from bench to social sciences that is not found elsewhere in pruritus research worldwide. This genuine interdisciplinarity permits a level of global comprehension of pruritus and the addressing of complex, fundamentally and translationally important biological questions in this area that the few existing competing major pruritus groups (USA, Ireland, Germany) are challenged to rival.

It deserves special emphasis that, within Europe, the LNB arguably now holds the leading role among those dermatologically focused pruritus research units, which combine excellent clinical research (incl. trials) with first-rate lab research, not the least due to its landmark human skin reinnervation and neuron-keratinocyte interaction assays. Worldwide, the LNB is one of the very few centers devoted to the underserviced and under-researched, but both clinically and commercially highly relevant, areas of pruritus, neurodermatology, and psychodermatology -all the field of dermatology awaiting the development of more satisfactory therapies. Given that chronic pruritus is one of the most debilitating symptoms in all of clinical medicine, the LNB fills a major gap in translational biomedical research and addresses an unmet, major health care challenge. That the unit does this with a strictly interdisciplinary strategy and at a high level of creativity and innovation puts the LNB in an excellent position to influence where these fields are moving on an international level. The LNB is very well positioned to realize translational research, better than other competing European institutions which are focused on basic research in neurophysiology and neuroimmunology of itch.

The LNB has demonstrated a very good to excellent scientific publication output, especially in clinical research. Namely, the LNB's review published in Nat Rev Neurol documents the rapidly growing international reputation it has acquired in neuropathic pruritus research over the last years.

Given that major studies in the field typically take 3-5 years from initiation to publication, it is to be expected that this relatively young unit does not yet have many first and/or senior author publications in top journals. Moreover, the expert committee was impressed by very exciting, as yet unpublished, data generated within the unit (e.g., demonstration of a keratinocyte-neuronal synapsis), which promises to lead to a landmark publication within the near future and to change currently dominant research paradigms in pruritus research.

The unit has built impressive national and international collaboration networks and has managed to achieve recognition as a key research center, center of reference, and resource center in the field of human skin neuroscience and pruritus research.

Given that, over the past decade, funding support for translational skin research increasingly came from industry, the LNB's excellent funding record from industry sources must be considered an important strength of the unit, since this will greatly facilitate the unit's capacity to become a magnet for industry studies in human skin neurosciences and pruritus research.

The unit has documented a strong involvement in teaching and in the management of research education and is committed to the public diffusion of scientific culture. The LNB's national and international reputation is attested by the organization of several large-audience congresses and many invitations of the unit's leader to lectures both nationally and internationally (European Society for Dermatological Research, American Academy of Dermatology).

Weaknesses and threats in the context

The unit has an insufficient number of full time-researchers and harbours several members with precarious employment status.

The unit has a low number of original research paper with a first-and/or senior author in the first skin research journal.

In contrast with its impressive list of industry contracts, the number of patents that have been filed is still relatively low.

Currently, the laboratory is mainly based on, and supported by, the leadership and scientific recognition of its director. The emergence of other leaders from within its ranks needs to be systematically encouraged and fostered.

The relative geographic isolation from other research centers in the west of France renders scientific interactions and collaborations somewhat more difficult than elsewhere. This requires greater efforts to connect the unit to its field at both the national and international level, and to make it an attractive work place also for students and scientists beyond the region.

The unit has only few international students, scientists and visitors or invited speakers and conducts its daily operation exclusively in French, thus limiting accessibility and attractiveness beyond France.

Recommendations

The expert committee views the unit as having a strong potential for becoming a credible candidate for application for INSERM status on the next evaluation. However, this requires constant support on its successful upwards trajectory from the university and the hospital. The unit should propose clear strategies for recruiting researchers in the next years: this should also parallel a higher ratio of scientists/MD in the group.

These new full-time researchers should then fully focus on the two areas where the unit the committee feels the unit has its greatest strengths: in-depth study of the keratinocyte-neurons synapse(s), and further development and characterization of ex vivo-reinnervated, organ-cultured human skin.

As industry funding is notoriously unpredictable and does not always favour the generation and timely publication of top-level science and/or ambitious long-term projects, the unit would be well-advised to broaden its funding base at least somewhat, namely so as to include French and European public or charitable organisms (ANR, FRM,…).

The expert committee recommends that the clinicians in the unit are granted a higher percentage of their work time that is dedicated exclusively to research, and pertains that only those clinicians can realistically be expected to contribute significantly to the unit's research output who can devote 50% of their time to pruritus-related research.

The unit should work to systematically promote the emergence of at least one additional senior scientist and one clinician who become internationally recognized in the fields of human skin neuroscience and pruritus, to assist the director in publicizing and promoting the unit's mission, achievement, and scientific appeal, and to gradually build an independent career in this specific field of research, also for facilitating an eventual handover in due course. The internal scientific life of the unit should be further vitalized and internationalized, e.g. by extending journal club activities held in English and organizing more seminars with external invited speakers.

The unit should place greater emphasis on publishing every year a first-and/or senior author original research paper in the #1 skin research journal (J Invest Dermatol), in favour of reducing its quantitative output in minor journals. This will further strengthen its international visibility & leadership within the skin research community, namely in human skin neuroscience and pruritus research. The LNB's range of funding sources should be broadened so as to go beyond the private sector and should also target government, E.U., and foundation sources. In the E.U. funding context, it should be possible to forge strategic research alliances with other key pruritus research centers (e.g. Dublin, Münster, Mannheim), e.g. by offering the unit's skin reinnervation and keratinocyte-neuron interaction models to collaborating external investigators.


president Michel Cosnard

Guus Van Dongen

Evaluation report

This report is the sole result of evaluation by the expert committee, the composition of which is specified below.

The assessments contained herein are the expression of an independent and collegial reviewing by the committee.

Unit name:

Animal The AMaROC research unit (Animal cancers as Models for Research in comparative OnCology) officially emerged on the 1 st of January 2012 after the last national evaluation procedure. The unit is located in Oniris -Nantes Atlantic National College of Veterinary Medicine, Food Science and Engineering, site Chantrerie (where the veterinary school, platforms and hospital are located). AMaROC's domain of research is comparative oncology, i.e. naturally occurring cancers in dogs and cats as models for research and advancements in human medicine. At the time of emergence, the AMaROC unit brought together several competences from several complementary fields. This includes veterinarians and physicians associated to multi-domain scientific competences, such as biology, pathology, clinical practice and physics in order to favor translational researchers in oncology. The team participates in DGER ("Direction Générale de l'Enseignement et de la Recherche" from the Ministry of Agriculture).

Management team

This is a small team where the direction team is composed by the director assisted by the senior scientists responsible for the 3 research axes. The research of the unit has a unique position and expertise, and tight connections with local partners for 3 research areas (axes): (1) studies to look for pathophysiological homology between spontaneous canine and feline cancers and human cancers, with focus on mammary cancer, glioma and diffuse large B-cell lymphomas; (2) studies related to dosimetry in nuclear oncology, aiming at the improvement of dosimetric methods and the assessment of dose-effect relationships in internal radiation therapy; (3) development of a cutting edge translational infrastructure for preclinical trials in domestic animals, according to guidelines and standards as used in human medicine.

HCERES

Global assessment of the unit

AMaROC is providing a unique environment for preclinical research in vivo and has a strong unique national position in the veterinary field. Because of this, important partnerships have been established, while three INCa projects were obtained. AMaROC is a small team with a light organization. Each of the 3 axes is well defined, with at least 6 team members involved, and with well-chosen strategic collaborations. Also the interaction between the different axes is sound. A plan is available for strategic, operational and scientific meetings. The scientific agenda is evolving in time. There is a good interaction between AMaROC and CRIP ("Centre de Recherche et d'Investigation Préclinique"), which is important for the availability of infrastructure/equipment/platforms. There is no formal contract between these two structures.

Within the team everything is well organized, everybody knows what to do and expect. Despite the fact that they are located in different buildings, the communication is fluid between the members. Weaknesses and threats in the context  diversity of projects. For each project the question should be asked: do canine and feline tumor models provide added value?

 publication output needs improvement in numbers and quality;

 staffing is mainly based on young researchers with a heavy teaching load.

Recommendations

 implement a more "publication directed policy": get more, faster and better publications;

 try to obtain an IBISA label for the biobank;

 obtain several additional habilitations (HDR), enabling the supervision of more PhD projects;

 try to establish and/or participate in European/international networks (e.g. COSTS);

 increase the number of persons in research.

  Strengths and opportunities in the context strong network and biobanking; capacity to manage large cohorts; attempt for in-depth characterization of models; setting up very attractive imaging infrastructure; strong connection with the CRCNA ("Centre de Recherches sur le Cancer de Nantes-Angers");  synergy between various competences: excellent expertise in clinical management of cancer in animals; very good fund raising.

  cancers as Models for Research in comparative OnCology

	1  Introduction
	History and geographical location of the unit
	Unit acronym:	AMaROC
	Label requested:	UPR
	Current number:
	Name of Director
	(2015-2016):	Mr Jérôme ABADIE
	Name of Project Leader
	(2017-2021):	Mr Jérôme ABADIE
	Expert committee members
	Mr Jean ROSENBAUM
	Representative of supervising institutions and bodies:
		Ms Dominique BUZONI-GATEL, École Nationale Vétérinaire, Agroalimentaire et
		l'Alimentation Nantes-Atlantique, ONIRIS
	Head of Doctoral School:
		Mr Bruno LE BIZEC, École Doctorale Biologie Santé Nantes Angers, ED n°502

Chair:

Mr Guus

VAN

DONGEN, VU University Medical Center, Amsterdam, The Netherlands Experts: Mr Pascal BOIREAU, ANSES Maisons Alfort (representative of the CNECA) Mr Jean-Ehrland RICCI, University of Nice Scientific delegate representing the HCERES:


M Jérôme

M Charles

M Jean Rosenbaum

M Pierre Sai

Observers AERES scientific advisor

The visit was organized on January 20, 2011 and befgan at 8:30 am. Following a presentation by J. Abadie, the committee met separately the students and post-docs, technicians, and scientists. The committee then met privately for drafting the report. The visit ended at 12h15.

 History and geographical localization of the research unit, and brief presentation of its field and scientific activities

This team is a recent creation within the premise of the Veterinary School of Nantes. It results from the fusion of subgroups originally working in different departments of the Veterinary Hospital (Pathology, imaging and surgery). It aims at developing clinical studies on cancer in domestic animals (dogs and cats), be the corner stone of the future Center of clinical investigation and research the Vet school envision to create. The leading idea being that cancers in cats and dogs could be excellent models of the disease in humans and fill the gap between mouse models and the clinical reality in humans.

 Management team

This is a small team where the direction team is limited to the Director himself.

 Staff members (on the basis of the application file submitted to the AERES) Malignancies principally investigated will be mammary cancer in cats and dogs, non-Hodgkin B cell lymphoma, CNS cancer and soft tissue sarcomas in dogs. This is a timely project since there is a growing consensus in favor of building bridges between spontaneous cancer in domestic animals and human malignancies, based on indications of pathophysiological similarities.

 Strenghts and opportunities

 Excellent expertise in clinical management of cancer in animals;

 Strong network of veterinarians ;

 Unique opportunity to have access to novel and original cancer models in dogs and cats ;

 Strong links built between the veterinay clinic and the Cancer Center ;

 Unique opportunity to make imaging studies and radioimmunotherapy studies in animal models that strongly mimic real clinical situations in humans ;

 The greater genetic homogeneity of dogs make them powerfull models for genetic studies on cancer predisposition, drug senstivity or study genome/environment interactions.

 Weaknesses and threats

 Staffing is mainly based on young researchers with an important teaching load.

 Pre-clinical studies are based on spontaneous cancer in pets. This involves identical difficulties in recruiting patients in trials as for trials involving patients. The recruitment capacity will be determining.

 Recommendations

 Care should be taken to increase the number of regular meetings of the whole staff, which at present works in 2 to 3 different buildings. Furthermore, practical measures should be envisaged to facilitate interactions with teams at the CRCNA as part of regular participation to meetings of these teams.

 A significant effort should be envisaged to increase the number (or the Full time involvement) of researchers in charge of these ambitious projects. PhD level engineers should be highered. 3  Specific comments



 Appreciation on the results

There is a growing consensus on the strong relevance of cancer in dogs, particularly for what concerns breast cancer, gliomas, non Hodgkin B cell lymphomas and sarcomas. The Vet School in Nantes has for almost 10 years proven its willingness to develop interface research with human health aspects and has made important efforts to structure its capacities in this direction. This project aiming at creating de novo this preclinical facility has been developed in strong interaction with the CRCNA (Centre de Recherche en Cancérologie de Nantes Angers) and IRCNA (Institut Régional du Cancer de Nantes Angers). Personnel from both institution participate to this project. This team has no previous activity. Team members have had a constant publication activity in veterinary medicine and now aim to open their scope to more general cancer journals.

 Appreciation on the impact, the attractiveness of the research unit and of the quality of its links with international, national and local partners Scientists are already active in collaborative projects funded by INCa or ANR. A large ANR grant on environmental risk is currently under review.

 Appreciation on the management and life of the research unit

The leader of the team is bright and energetic and seems well respected by the other members. Internal animation has been judged somewhat insufficient (see recommendations) and should increase.

 Appreciation on the scientific strategy and the project Two principal aims are pursued:

1develop a clinical trial platform within the premise of the animal clinic at the veterinary school. The increased development of the disease in animals will allow to obtain rapid return on treatment response to novel therapeutics. Furthermore, the pathophysiological setting being very similar to that in humans, valuable information will be produced in terms of recurrence and metastatic spread.

2domestic animals are exposed to identical environments than their owners (apart from food). They thus constitute highly valuable witnesses of the relationship between environmental exposure to chemical or physical polution and cancer. Furthermore, their genetic homogeneity make them particularly suited to linkage or SNP studies.

Scientific projects were overall judged promising. However, for what concerns the project on environmental risk, there were some questions on the adequation of manpower needed to accomplish it.



  Date and execution of the visit

  Production results A1: Number of permanent researchers with teaching duties (recorded in N1) who are active in research 3 A2: Number of permanent researchers without teaching duties (recorded in N2) who are active in research 2 A3: Ratio of members who are active in research among staff members [(A1 + A2)/(N1 + N2)] 60% A4: Number of HDR granted during the past 4 years 0 A5: Number of PhD granted during the past 4 years 1

  It is based on the unique expertise the veterinary school of Nantes has acquired in this field.

	2  Overall appreciation on the research unit
	 Summary
	This is a new project aiming at creating a novel and original research unit based on cancer in domestic animals
	(cats and dogs).
	Past	Future
	N1: Number of researchers with teaching duties (Form 2.1 of the application file)	5
	N2: Number of full time researchers from research organizations (Form 2.3 of the application file)	0
	N3: Number of other researchers including postdoctoral fellows (Form 2.2 and 2.4 of the application file)	2
	N4: Number of engineers, technicians and administrative staff with a tenured position (Form 2.5 of the application file)	3
	N5: Number engineers, technicians and administrative staff
	without a tenured position (Form 2.6 of the application file)
	N6: Number of Ph.D. students (Form 2.7 of the application file)
	N7: Number of staff members with a HDR or a similar grade	1

Sciences du Vivant et Environnement

			Statistiques de notes globales par domaines scientifiques
					(État au 06/05/2011)
	Note globale	SVE1_LS1_LS2	SVE1_LS3	SVE1_LS4	SVE1_LS5	SVE1_LS6	SVE1_LS7	SVE2 _LS3 *	SVE2_LS8 *	SVE2_LS9 *	Total
	A+	7	3	1	4	7	6		2		30
	A	27	1	13	20	21	26	2	12	23	145
	B	6	1	6	2	8	23	3	3	6	58
	C	1					4				5
	Non noté	1									1
	Total	42	5	20	26	36	59	5	17	29	239
	A+	16,7%	60,0%	5,0%	15,4%	19,4%	10,2%		11,8%		12,6%
	A	64,3%	20,0%	65,0%	76,9%	58,3%	44,1%	40,0%	70,6%	79,3%	60,7%
	B	14,3%	20,0%	30,0%	7,7%	22,2%	39,0%	60,0%	17,6%	20,7%	24,3%
	C	2,4%					6,8%				2,1%
	Non noté	2,4%									0,4%
	Total	100,0%	100,0%	100,0%	100,0%	100,0%	100,0%	100,0%	100,0%	100,0%	100,0%
	*
	Intitulé UR / équipe	C1	C2	C3	C4	Note globale
	CANCERS ANIMAUX: MODÈLES POUR LA RECHERCHE EN ONCOLOGIE COMPARÉE	Non noté	B	A	B	B
	C1 Qualité scientifique et production
	C2 Rayonnement et attractivité, intégration dans
	l'environnement
	C3 Gouvernance et vie du laboratoire
	C4 Stratégie et projet scientifique

les résultats SVE2 ne sont pas définitifs au 06/05/2011. Intitulés des domaines scientifiques Sciences du Vivant et Environnement • SVE1 Biologie, santé SVE1_LS1 Biologie moléculaire, Biologie structurale, Biochimie SVE1_LS2 Génétique, Génomique, Bioinformatique, Biologie des systèmes SVE1_LS3 Biologie cellulaire, Biologie du développement animal SVE1_LS4 Physiologie, Physiopathologie, Endocrinologie SVE1_LS5 Neurosciences SVE1_LS6 Immunologie, Infectiologie SVE1_LS7 Recherche clinique, Santé publique • SVE2 Ecologie, environnement SVE2_LS8 Evolution, Ecologie, Biologie de l'environnement SVE2_LS9 Sciences et technologies du vivant, Biotechnologie SVE2_LS3 Biologie cellulaire, Biologie du développement végétal


president Michel Cosnard

Joao Rocha

Evaluation report

This report is the sole result of evaluation by the expert committee, the composition of which is specified below.

The assessments contained herein are the expression of an independent and collegial reviewing by the committee.

Unit name:

Laboratoire

HCERES nomenclature

Principal -ST4, Secondary -ST2; keywords -ceramics, structural materials, materials for electronics CRISMAT is an internationally-leading multidisciplinary unit in the fields of materials chemistry and physics, gathering considerable know-how in the synthesis and elaboration, physical characterisation and theory across a broad range of materials areas. Its main asset is the remarkable multidisciplinary integration of scientists with different expertise and interests to carry out fundamental or applied research. Studies encompass areas such as: novel multiferroics, thermoelectrics, materials for lithium and sodium batteries, magnetic materials, complex oxide and hybrid hetero-structures and layered oxides, ferrites, functional ceramics, composites and structural materials, materials (and systems) for electronics. The unit stands out for its talented Transmission Electron Microscopy (TEM) and X-Ray Diffraction (XRD) characterisation skills and advanced crystallography methods, and for its experience in establishing relationships between crystal structure and a plethora of properties including dielectricity, ferroelectricity, thermoelectricity, photoluminescence, magneto-transport, superconductivity, energy storage, electro/magneto/thermomechanics. The unit performs first-class work exploiting engineered interfaces and heterostructures. CRISMAT is also highly skilled in advanced elaboration techniques, such as pulsed laser ablation, spark plasma sintering and hydrothermal methods, and in ceramics shaping and texturing. Since 2011, the unit's international status and visibility have been raised, as witnessed, e.g., by the increasing numbers of papers published in high-impact journals (including in Science and Nature series), which are receiving more citations, or the organisation of prestigious events, such as the Gordon Research Conference in Solid State Chemistry.

Scientific domains

Global assessment of the unit

CRISMAT is a first-class unit, internationally leading in materials chemistry and physics, having a strong leadership, clear scientific objectives and the drive to achieve them. The unit gathers considerable know-how in the synthesis and elaboration, physical characterisation and theory across a broad range of materials areas. The scientific output of the lab is high, with many papers published in prominent journals and receiving a large number of citations. CRISMAT's international reputation is also witnessed from, among others, the number of invited conferences delivered and scientific events organised, participation in international projects and networks, membership of journal boards and learning societies. The highly skilled CRISMAT knowledge base has created opportunities for a strong and sustained involvement with local and national companies, in particular through LaMIPS. There is, nevertheless, further room for strengthening such collaboration and better securing the generated IP. Concerning high-level training and mentoring of Ph.D. students and post-docs, CRISMAT provides a strong contribution with an excellent scientific culture and equipment infrastructure, frequent exchanges with reputed international institutions and with companies, and regional, national and European networking. A renewed leadership and an ambitious scientific programme for the future ensure that CRISMAT's reputation will endure in the coming five-year period.

Strengths and opportunities in the context

reputation as internationally leading in materials science, with true multidisciplinarity in chemistry and physics;

strong leadership and clear scientific objectives;

outstanding expertise in materials synthesis and elaboration, and in TEM and XRD;

very good regional, national and European networking;

high scientific output in journals of very good to excellent impact;

strong and sustained involvement with local and national companies;

excellent conditions for training and mentoring of Ph.D. students and post-docs.

Weaknesses and threats in the context

despite an excellent productivity, uneven contribution of teams and individual researchers;

number of papers in very high impact factor (>6-7) journals still not at the level it could be;

  location of the unit CRISMAT was created in 1986 as a unité mixte de recherche, headed by Mr Bernard RAVEAU until 2003 and since then by Mr Antoine MAIGNAN. In January 2015, Mr Wilfrid PRELLIER became deputy director and will become the director in the new contract. CRISMAT is affiliated to ENSICAEN and Université de Caen Basse-Normandie and associated to the CNRS, primarily to the Institut National de Chimie (INC), and secondarily to the Institut National de Physique (INP). CRISMAT is housed in two ENSICAEN buildings covering a total area of 6000 m 2 (one of the buildings is shared with two other labs, LCMT and LCS). Personnel in the joint research lab LaMIPS are housed at the Caen NXP and PRESTO sites at, respectively, Colombelles and Giraf. CRISMAT hosts first-class equipment facilities, particularly for transmission electron microscopy and X-ray diffraction. Management team Direction: Dr. Antoine MAIGNAN Directeur-Adjoint: Dr. Wilfrid PRELLIER

  de Cristallographie et Science des Matériaux

	Representatives of supervising institutions and bodies:
		Mr Marco DATURI, ENSICAEN
		Mr Daniel DELAHAYE, Université de Caen
		Mr Niels KELLER, CNRS INP
		Ms Aurélie MENARD, CNRS DR19
		Mr Jean-François TASSIN, CNRS INC
	Unit acronym: Head of Doctoral School:	CRISMAT
	Label requested:	UMR Mr Frédéric JURIE, Doctoral school n°181 "SIMEM"
	Current number:	6508
	Name of Director
	(2015-2016):	Mr Antoine MAIGNAN
	Name of Project Leader (2017-2021):	Mr Wilfrid PRELLIER
	Expert committee members
	Chair:	Mr Joao ROCHA, University of Aveiro, Portugal
	Experts:	Mr Éric BEAUREPAIRE, Université de Strasbourg
		Mr Olivier FRUCHART, Université de Grenoble
		Mr Christian MASQUELIER, Université de Picardie (representative of the CNU)
		Mr Matthew ROSSEINSKY, Royal Society, University of Liverpool, UK
		Mr Philippe THOMAS, SPCTS, Université de Limoges (representative of the CNRS)
		Ms Corinne ULHAQ, Université de Strasbourg
	Scientific delegate representing the HCERES:
		Mr Marc DRILLON

Number of Research Supervisor Qualifications (HDR) obtained during the period

less than optimal participation in and leadership of European (and ANR) projects; -few patents filed;small number of post-docs.

Recommendations

The panel notes the excellent productivity of the unit but considers there is room for achieving a more even contribution of teams and individual researchers; the exciting research being carried out deserves even more visibility and, thus, the trend to publish in high impact factor journals should be sustained. There is opportunity for further strengthening the links between theory and experiment, and to develop the former.

There is opportunity to raise CRISMAT's international profile by capitalising on the excellent science performed and participating in and leading more H2020 projects, receiving more invitations to talks at high-profile conferences, attracting more post-docs (e.g., through European programmes, such as Marie Curie).

Increase the number of patents filed and profit from the inception of Normandy Valorisation; raise the awareness of all unit members, including students, for the opportunity to commercially exploit the knowledge created at CRISMAT, including software, databases, materials, techniques, processes, methods, etc.

More may be done to encourage the emergence of new leadership across the teams in the future; the panel strongly supports the idea of organising all-lab and group-seminars open to all unit members, including the technical staff; spread good team animation practices; intensify scientific exchange within and between teams, profiting even more from scientific complementarities and exploiting new opportunities; increase visibility of technical staffs in the unit's life. Number of Ph.D. students and post-docs mentored may be further increased.