Alain Yger
email: alain.yger@math.u-bordeaux1.fr

Averaging residue currents and the Stückrad-Vogel algorithm

Averaging residue currents and the Stückrad-Vogel algorithm

sheaf O V = O X /I V , where I V,x = {h x ∈ O X ,x ; h x = 0 on V x }. Let [V ] be the integration current on the complex subspace (V, O V ). Let ∆ 1 , ..., ∆ m be m ≤ n-M Cartier divisors on X , with respective holomorphic global sections s 1 , ..., s m , such that for all x ∈ V ∩ m j=1 s -1 j (0), the germs (s 1,x , ..., s m,x ) define a regular sequence in O V,x . We denote as O V [s] the ideal sheaf of O V obtained as the image of m j=1 O V (-∆ j ) by the interior product with (s 1 , ..., s m ). Note that, if the closed hypersurfaces s -1 j (0), j = 1, ..., m, intersect properly on V (that is dim(V ∩ m j=1 s -1 j (0)) ≤ n -M -m) or, which is equivalent, define a complete intersection on V , the required condition holds provided that the ambient manifold X is replaced by some convenient neighborhood U of V ∩ s -1 (0) := V ∩ m j=1 s -1 j (0). What seems to be today the most robust approach towards the so-called Coleff-Herrera current m j=1 ∂(1/s j )∧[V ] (originally introduced by N. Coleff and M. Herrera in their pioneer work [START_REF] Coleff | Les courants résiduels associés à une forme méromorphe[END_REF]) is the following result (initially obtained in [START_REF] Berenstein | Residue currents and Bézout identities[END_REF] for m = 2, then finally extended by H. Samuelsson [START_REF] Samuelsson | Analytic continuation of residue currents[END_REF] for arbitrary m) :

Theorem 1 (robust approach to the Coleff-Herrera residue current attached to a complete intersection). In the above context, for any choice of C ∞ metrics | | j on the line bundles O X (∆ j ), the holomorphic

m r=0 1≤j1<•••<jr≤m D (M,M +r) (X , r l=1 O X (-∆ j l ))-valued map (λ 1 , ..., λ m ) ∈ {λ ; Re λ j > 1, j = 1, ..., m} -→ 1 j=m 1 -|s j | 2λj j + 1 2iπ ∂ |s j | 2λj j s j ∧ [V ] (1)
can be analytically continued as an holomorphic map to a product of halfspaces {Re λ j > -η} for some η > 0. Its value at λ = 0 coincides with its D M,M +r (X , m j=1 O X (-∆ j ))-component and defines a ∂-closed bundlevalued current which is independent of the metrics | | j , j = 1, ..., m. This current, supported by V ∩s -1 (0), is denoted as m j=1 ∂(1/s j )∧ [V ]. Considered as a ∂-closed current on the complex space (V, O V ), it is locally annihilated by any local section of the ideal sheaf O V [s]. Moreover, when V = X , it realizes the local duality with respect to the ideal sheaf O X [s], namely

h x • m j=1 ∂(1/s j ) = 0 ⇐⇒ h x ∈ (O X [s]) x = (s 1,x , ..., s m,x )O X ,x . (2)
The intimate relationship between the sheaf of differential operators with meromorphic coefficients and the local description of such a current arises from the following remark, which refers to the sheaves of Coleff-Herrera currents (here bundle-valued, holomorphic or meromorphic, see [START_REF] Vidras | Coleff-Herrera currents revisited[END_REF]) introduced by J.E. Björk in [START_REF] Björk | Residues and D-modules[END_REF], together with their companion local structure theorems (see [START_REF] Björk | Residues and D-modules[END_REF], or also [START_REF] Vidras | Coleff-Herrera currents revisited[END_REF] for a survey in this volume). The current m j=1 ∂(1/s j ) ∧ [V ] splits locally about each point of its sup-port V ∩ s -1 (0) as T x ∧ ω x , where T x is a local section of the Coleff-Herrera sheaf CH X ,V ∩s -1 (0) (• ; S x , m j=1 O X (-∆ j )), and ω x is a local section of Ω M +m X (•, m 1 O X (∆ j )). Here S x denotes a germ of hypersurface at x polar respect to V x ∩ s -1 x (0), i.e. (V x ∩ s -1 x (0)) \ S x = V x ∩ s -1 x (0). Moreover, when V is Cohen-Macaulay about x 0 ∈ V ∩ s -1 (0) (that is O V,x0 is Cohen-Macaulay), the integration current [V ] factorizes about x 0 as τ x0 ∧ x0 , where x0 is a local section of the Coleff-Herrera sheaf CH X ,V , and x0 is a section of the sheaf Ω M X (see [START_REF] Andersson | Coleff-Herrera currents, duality, and Noetherian operators[END_REF]), which implies in this particular case that one can take about x 0 the current section T x0 to be a local section T x0 of CH X ,V ∩s -1 (0) (•,

m 1 O X (-∆ j )) instead of CH X ,V ∩s -1 (0) (• ; S x0 , m 1 O X (-∆ j )
). This occurs in particular when V is defined as a reduced complete intersection about the point x 0 .

Remark 1. Theorem 1 provides a robust approach (via analytic continuation) towards the Coleff-Herrera residue current for complete intersections on a purely dimensional analytic set (V, O V ), as developed in [START_REF] Coleff | Les courants résiduels associés à une forme méromorphe[END_REF]. It is important here to point out that one may replace in (1) the integration current [V ] by any global section of the Coleff-Herrera sheaf CH X ,V (•, E), where E denotes a finite rank holomorphic bundle over X . When m ≤ M , the first assertion in Theorem 1 remains valid under the complete intersection hypothesis about the s j 's on V , which provides some kind of robustness for multiplicative residue calculus involving Coleff-Herrera currents (see for example Proposition 3 below). The proof can be carried in a similar way than to that in [START_REF] Samuelsson | Analytic continuation of residue currents[END_REF], taking advantage of the wedge product operation introduced in [START_REF] Björk | Regularizations of residue currents[END_REF] (see also [START_REF] Vidras | Coleff-Herrera currents revisited[END_REF]). Be careful however that the definition of the sheaf CH X ,V (•, E), whose global sections are (0, M ) E-valued currents in the ambient manifold X with support lying in V , depends on the embedding V ⊂ X . Nevertheless, Remark 1 will be important with respect to the role of such Coleff-Herrera residue currents in the construction of analytic tools for division theory, in accordance with that played by integration currents in intersection theory (see subsection 2.2 below).

We considered so far the holomorphic sections s 1 , ..., s m of the hermitian line bundles (O X (∆ j ), | | j ) independently, then introduced the Coleff-Herrera current m j=1 ∂(1/s j ) ∧ [V ] through a multiplicative procedure which is reminiscent of the multiplicative operational formalism carried by the Cauchy kernel

dz 1 /z 1 ∧ • • • ∧ dz m /z m . Instead of that, one could interpret s = s 1 ⊕ • • • ⊕ s m as an holomorphic section of the m-holomorphic bun- dle m j=1 O X (∆ j ), equipped with the metric 2 = | | 2 1 ⊕ • • • ⊕ | | 2 m .
The robustness of the approach towards the current m j=1 ∂(1/s j ) ∧ [V ] (in the complete intersection setting, see Theorem 1) motivates this alternative one. We still use analytic continuation, but this time respect to a single auxiliary complex parameter λ instead of several. Such an approach was introduced in [START_REF] Berenstein | Residue currents and Bézout identities[END_REF][START_REF] Passare | Residue currents of the Bochner-Martinelli type[END_REF], then extensively developed (after being interpreted within the frame of holomorphic hermitian bundles) in [START_REF] Andersson | Residue currents and ideals of holomorphic functions[END_REF][START_REF] Andersson | Coleff-Herrera currents, duality, and Noetherian operators[END_REF][START_REF] Andersson | Residue currents with prescribed annihilator ideals[END_REF][START_REF] Andersson | Decomposition of residue currents[END_REF]. It is based on an averaging procedure.

Theorem 2 (Bochner-Martinelli approach). Let X , V , the ∆ j 's and the s j 's be as in the preamble of this section. Let be an arbitrary hermitian metric on the m-dimensional bundle

m j=1 O X (∆ j ) = E ∆ , s = s 1 ⊕ • • • ⊕ s m , and s * be the conjugate section of s, that is s * (x)(ξ) = ξ, s(x) x for x ∈ X , ξ ∈ E ∆,x . The holomorphic m r=0 D M,M +r (X , r E * ∆ )-valued map λ -→ (1 -s 2λ ) + ∂ s 2λ ∧ m r=1 1 (2iπ) r s * ∧ (∂s * ) r-1 s 2r ∧ [V ] (3)
(defined for Re λ >> 1) extends as an holomorphic map in a half-plane Re λ > -η for some η > 0. It coincides at λ = 0 coincides with its

D M,M +m (X , m E * ∆ ) component, that is, independently of the choice of the metric on E ∆ , the Coleff-Herrera current m j=1 ∂(1/s j ) ∧ [V ].
Remark 2. In the particular case where all line bundles O X (∆ j ), j = 1, ..., m, are trivial over X , and the holomorphic sections s j are holomorphic functions in X defining a complete intersection on V , one can interpret the current

(1 -s 2λ ) + ∂ s 2λ ∧ m r=1 1 (2iπ) r s * ∧ (∂s * ) r-1 s 2r ∧ [V ] λ=0 (now is the euclidean norm on C m , s * (x)(ξ) = m 1 s j (x) ξ j for ξ ∈ C m ) as an averaged value of the currents 1 j=m 1 -| u j , s | 2λj + 1 2iπ ∂ | u j , s | 2λj u j , s ∧ [V ] λ1=•••=λm=0
(where (u 1 , ..., u m ) ∈ (P n-1 (C)) m , and s, u j = m k=0 u j k s k for k = 1, ..., m) defined as in in [START_REF] Achilles | Intersection numbers, Segre numbers and generalized Samuel multiplicities[END_REF], for (u 1 , ..., u m ) generic in (P m-1 (C)) m , respect to the normalized tensorized Fubini-Study metric on (P n-1 (C)) m . Averaging Coleff-Herrera currents (realized in a multiplicative form as in Theorem 1) prevents usually from keeping track of the algebraic or arithmetic structure of the data (when there is one). Keeping track of such a structure is indeed better possible through a multiplicative approach such as in Theorem 1. Nevertheless, within the complete intersection frame, the fact that the current m j=1 ∂(1/s j ) ∧ [V ] remains preserved under such averaging (Theorem 2) allows to reinterpret in geometric terms (and prove) the key (algebraic) operational property of multivariate residue calculus (see for example [START_REF] Griffiths | Principles of Algebraic Geometry[END_REF][START_REF] Berenstein | Residue Calculus and effective Nullstellensatz[END_REF][START_REF] Elkadi | Residue calculus and applications[END_REF]), the so called Transformation Law. In order to formulate such a computational rule (which appears to be the geometric counterpart of Wiebe's theorem [START_REF] Wiebe | Über homologische Invarianten lokaler Ringe[END_REF]) in a rather general form, one needs to introduce two sequences of Cartier divisors on the ambient complex manifold X , namely (∆ 1 , ..., ∆ m , T 0 , ..., T k-1 ) and ( ∆ 1 , ..., ∆ m ) (m + k ≤ n -M ), together with respective holomorphic sections (s 1 , ..., s m , t 0 , ..., t k-1 ) = (s, t), ( s 1 , ..., s m ) = s ; when k = 0, one takes {T 0 , ..., T k-1 } = ∅. The geometric hypothesis are :

codim V (V ∩ s -1 (0)) = codim V (V ∩ s -1 (0)) = m ; (V ∩ s -1 (0)) ∩ t -1 l (0) = V ∩ s -1 (0) ∀ l = 0, ..., k -1 ; codim V (V ∩ s -1 (0) ∩ t -1 (0)) = codim V (V ∩ s -1 (0) ∩ t -1 (0)) = m + k. (4) Let E ∆ = m j=1 O X (∆ j ) and E ∆ = m j=1 O X ( ∆ j ).
As a pendant algebraic hypothesis, one assumes that there is a meromorphic section H of Hom C (E ∆ , E ∆ ), together with positive integers ν 0 , ..., ν k , such that s = H • s and

D = det H ⊗ k-1 κ=0 t ⊗νκ κ ∈ O X X , Hom C (E ∆ , E ∆ ) ⊗ k-1 κ=0 O X (ν κ T κ ) . (5)
Theorem 3 (Transformation Law for Coleff-Herrera residue currents). Under the above geometric and algebraic hypothesis (4) and (5), one has the following identity between bundle valued currents :

m j=1 ∂ 1 s j ∧ k-1 κ=0 ∂ 1 t κ ∧ [V ] = m j=1 ∂ 1 s j ∧ k-1 κ=0 ∂ 1 t νκ+1 κ ∧ [V ] ⊗ D Proof.
In order to get this result, we introduce metrics | | κ on the divisors T κ , κ = 0, ..., k -1, and compare the values at µ 0 = • • • = µ k-1 (following the analytic continuation) of the two holomorphic maps (for Re µ κ >> 1, κ = 0, ..., k -1):

µ -→ 0 κ=k-1 ∂ |t κ | 2µκ κ t κ ∧ m j=1 ∂ 1 s j ∧ [V ] µ -→ 0 κ=k-1 ∂ |t κ | 2µκ κ t κ ∧ m j=1 ∂ 1 s j ∧ [V ] ⊗ D t ν0 0 • • • t ν k-1 k-1 . ( 6
)
These two current valued maps coincide for Re µ κ >> 1, κ = 0, ..., k -1, because of the robustness assumption in Theorem 2, together with the holonomy and the Standard Extension Property of sections of the Coleff-Herrera sheaves CH X ,V ∩s -1 (0 [START_REF] Vidras | Coleff-Herrera currents revisited[END_REF], condition 1 in Definition 2). The fact that they share the same value at µ = 0 about any x ∈ V ∩ s -1 (0) ∩ t -1 (0) follows from the robustness of the approach towards the Coleff-Herrera residue via analytic continuation (Theorem 1). On the other hand, both maps (6) vanish at µ = 0 about any x ∈ V ∩ ( s -1 (0) \ s -1 (0)) ∩ t -1 (0) because of Cramer's rule, combined with the local duality property (2) in Theorem 1 (here is the crucial algebraic point on which relies Wiebe's theorem in the algebraic context).

) (• ; S) or CH X ,V ∩ s -1 (0) (• ; S), where S = k-1 κ=0 t -1 κ (0) (see
In order to focus on the importance of the Transformation Law, let us conclude this section with few comments inspired by arithmetic considerations. Let {p 1 , ..., p m } be a collection of polynomials in Z[X 1 , ..., X n ] (here m ≤ n), with respective degrees d 1 , ..., d m , whose homogenizations define a complete intersection in C n = P n \ {z 0 = 0}. Denote as T [p 1 , ..., p m ] the

T [p 1 , ..., p l+1 ] = 1 2iπ ∂ |p l+1 | 2λ l+1 p l ∧ T [p 1 , ..., p l ] λ=0 , l = 1, ..., m -1. ( 7
)
The metric | | l one takes here on

O P n (C) (d l+1 ) is the Fubini-Study metric, but
this is in fact irrelevant, since the result after taking λ = 0 in ( 7) is unaffected by such a choice. For u 1 , ..., u n-m generic in Z m+1 \ {0}, the linear subspace

Π u := {[z 0 : • • • : z n ] ∈ P n (C) ; u j , z = 0 , j = 1, ..., m} is such that dim C n (Π u ∩ {p 1 = • • • = p m = 0}) = 0. Consider the m 1 O P n (C) (-d j )-valued (n -m, n) ∂-closed current T [p 1 , ..., p m ] ∧ [Π u ]. For any global smooth (m, 0) form b in P n (C) with values in m 1 O P n (C) (d j ), one has, T [p 1 , ..., p m ] ∧ [Π u ], b = (1 -|z 0 | 2λ ) T [p 1 , ..., p m ] ∧ [Π u ] λ=0 , b + |z 0 | 2λ T [p 1 , ..., p m ] ∧ [Π u ] λ=0 , b = (T [p 1 , ..., p m ] ∧ [Π u ]) |{z0=0} , b + (T [p 1 , ..., p m ] ∧ [Π u ]) |C n , b .
An explicit computation of

(T [p 1 , ..., p m ] ∧ [Π u ]) |C n , b (8)
can be carried thanks to the Transformation Law (such as formulated in Theorem 3 with k = 0), following the procedure which is described in [START_REF] Elkadi | Residue calculus and applications[END_REF]. In particular

, if b[q, J] is the form expressed in C n , in affine coordinates ζ 1 , ..., ζ n , as b[q, J] = q(ζ) (1 + |ζ| 2 ) (d+deg q)/2 m l=1 d ζ j l 1 + |ζ| 2 , d = d 1 + • • • + d m for some q ∈ Z[X 1 , ..., X n ] and 1 ≤ j 1 < • • • < j m ≤ n, one gets for (8) a somehow explicit rational expression R q,J [u] ∈ Q.
A key point in such a procedure is that it provides a final estimate on the logarithmic height h of the rational function R q,J [u] of the form

h(R q,J [u]) ≤ h(q) + κ 1 (n)D(D + deg q) max(h(p j ), log u j ), (9)
where D = m 1 (d j +1), in accordance with the geometric Bézout theorem and its arithmetic counterpart. If q (T [p 1 , ..., p m ] ∧ [Π u ]) {z0=0} , one gets a rational expression with logarithmic height control [START_REF] Berenstein | Residue currents and Bézout identities[END_REF] for

T [p 1 , ..., p m ]∧[Π u ], b[q, J] .

Coleff-Herrera residue currents and ordered sequences of sections of Cartier divisors

Let X be an n-dimensional complex manifold. Given a purely dimensional cycle [V ] ∈ Z n-M (X ) (identified here with the (M, M ) associated integration current on it and with support denoted as V ) and a closed submanifold Z ⊂ X , the splitting operation

[V ] = [V ] Z + [V ] X \Z , (10)
which consists in separating (as components of [V ] Z ) the components of [V ] with support lying in Z, from the others, is one of the major operational tools in geometric intersection theory. Transposed in algebraic terms, it leads to the notion of gap sheaf (see for an introduction to the subject Part I in [START_REF] Massey | Numerical control over Complex Analytic Singularities[END_REF]).

Example 1. [START_REF] Stückrad | An algebraic approach to the intersection theory. Queen's Papers in Pure[END_REF][START_REF] Tworzewski | Intersection theory in complex analytic geometry[END_REF][START_REF] Kollár | Effective Nullstellensatz for arbitrary ideals[END_REF]

Let [V 1 ], ..., [V m ] be m purely dimensional algebraic cycles in P n (C), [V ] be the (n -M )-cycle in X = P n (C) ( n = m(n + 1) -1, M = m j=1 codim P n (C) V j ) corresponding to the ruled join J(V 1 , ..., V m ) = [Z 1 : • • • : Z m ] ∈ P m(n+1)-1 (C) ; z j ∈ V j ∀ j = 1, ..., m
(multiplicities been taken into account), and Z be the diagonal subspace of P n (C) defined as a complete intersection as the set of points [Z 1 :

• • • : Z m ] ∈ P n (C) such that Z k = Z 1 for any k ∈ {2, .
.., m} (see [START_REF] Flenner | Joins and intersections[END_REF]). Note that, in this particular case, the coherent O P n (C) -ideal sheaf I Z is globally generated by (m -1)(n + 1) holomorphic sections σ l of the same line bundle O P n (C) [START_REF] Achilles | Intersection numbers, Segre numbers and generalized Samuel multiplicities[END_REF]. Moreover the zero sets σ -1 l (0), l = 1, ..., (m -1)(n + 1), define here Z as a complete intersection in the ambient n-dimensional manifold X = P n (C).

What we propose to do now is to transpose within the frame of global sections of the Coleff-Herrera sheaves CH X ,V (or CH X ,V (• ; S)) (instead of the geometric frame of integration currents) the splitting operation [START_REF] Berenstein | Residue Calculus and effective Nullstellensatz[END_REF], together with the construction such an operation generates when applied inductively. We should point out here that such a construction is directly inspired from the ideas introduced in [START_REF] Coleff | Les courants résiduels associés à une forme méromorphe[END_REF] by N. Coleff and M. Herrera (see for example [START_REF] Solomin | Le résidu logarithmique dans les intersections non complètes[END_REF]).

The Coleff-Herrera product

Let X and V be as in Section 1. Consider also an ordered sequence ∆ 1 , ..., ∆ m (one drops here the assumption m ≤ n -M ) of Cartier divisors on X , equipped respectively with hermitian metrics | | j , j = 1, ..., m, and with holomorphic global sections s 1 , ..., s m . The major difference with Section 1 is that we forget here the geometric assumption that the closed hypersurfaces s -1 j (0) ⊂ X , j = 1, ..., m, intersect as a complete intersection on the closed analytic subset V . Let T be a global section over X of the Coleff-Herrera sheaf CH X ,V (•, E), where E denotes some finite rank holomorphic bundle over X . Let

m j=1 ∂(1/s j ) ∧ T be the current in m r=0 1≤j1<•••<jr≤m D (0,M +r) (X , r l=1 O X (-∆ j l ) ⊗ E) defined inductively from ∂ 1 s 1 ∧ T = 1 -|s 1 | 2λ 1 + 1 2iπ ∂ |s 1 | 2λ 1 s 1 ∧ T λ=0 by the iterative process l+1 j=1 ∂ 1 s j ∧T = 1-|s l+1 | 2λ l+1 + 1 2iπ ∂ |s l+1 | 2λ l+1 s l+1 ∧ l j=1 ∂ 1 s j ∧T λ=0 (11)
for l = 1, ..., m -1. Remark that in spite of the notation m j=1 ∂(1/s j ) ∧ T (that one uses here for the sake of simplicity), this current is not in general a (0, m + M ) current. The above inductive construction is independent of the choice of the metrics | | j on the line bundles O X (∆ j ), but of course depends on the ordering of the sequence of divisors {∆ 1 , ..., ∆ m }. When the hypersurfaces s -1 j (0), j = 1, ..., m, define a complete intersection on V , this current coincides with its D (0,M +m) (X , m j=1 O X (-∆ j ) ⊗ E) component, and can be recovered in a robust way in a neighborhood of V ∩ s -1 (0) as the value at λ 1 = • • • = λ m = 0 of an holomorphic function in m variables in a product of half planes Re λ j > -η for some η > 0 (see Remark 1 above). The iterative construction is justified by the local structure theorems for sections of the Coleff-Herrera sheaves CH X ,W (• ; S, E), where W denotes a purely dimensional closed analytic subset of V and S a closed hypersurface in a neighborhood of W in X such that

W \ S = W . When j = (j 1 , ..., j r ) is a r-uplet (0 ≤ r ≤ m, j = ∅ if r = 0) of strictly increas- ing integers 1 ≤ j 1 < • • • < j r ≤ m, the component ( m j=1 ∂(1/s j ) ∧ T ) j in D (0,M +r) (X , r l=1 O X (-∆ j l ) ⊗ E) is a global section of the Coleff-Herrera sheaf CH X ,Vj [s] (• ; S j [s], E j ), where E j = E j [∆, E] = r l=1 O X (-∆ j l ) ⊗ E, V j [s] is a purely (n -M -r)-dimensional closed subset of V and S j [s] some closed hypersurface in a neighborhood of V j [s] such that V j [s] \ S j [s] = V j [s].
The construction of V j [s] is carried through the procedure that leads to the construction of the multi-cycle of contact (see [START_REF] Tworzewski | Intersection theory in complex analytic geometry[END_REF]) between a given cycle [V ] and a smooth closed submanifold Z ⊂ X contained in the intersection of an ordered sequence H 1 , ..., H m (m ≤ dim V ) of closed hypersurfaces. More precisely, V j [s] can be reached through the following iterated splitting operation : when W is a closed purely dimensional subset in X and S denotes a closed hypersurface in a neighborhood of W , then W is (geometrically) decomposed as

W = W S ∪ W X \S , (12)
where W S denotes the union of irreducible components of W lying entirely in S and W X \S the union of the remaining ones. The closed analytic set V j [s] appears then as the end term in the inductive sequence :

V j,1 [s] = V s -1 1 (0) if 1 / ∈ {j 1 , ..., j r } ; else V j,1 [s] = V X \s -1 1 (0) ∩ s -1 1 (0); . . . . . . . . . V j,k+1 [s] = (V j,k [s]) s -1 k+1 (0) if k + 1 / ∈ {j 1 , ..., j r } ; else V j,k+1 [s] = (V j,k [s]) X \s -1 k+1 (0) ∩ s -1 k+1 (0); . . . . . . . . . V j [s] = V j,m [s] = (V j,m-1 [s]) s -1 m (0) if m / ∈ {j 1 , ..., j r } ; else V j [s] = V j,m [s] = (V j,m-1 [s]) X \s -1 m (0) ∩ s -1 m (0). ( 13
)
If m ≤ n -M and r = m, V {1,...,m} [s] is the so-called essential intersection

(V ∩s -1 1 (0)∩• • •∩s -1 m (0)) ess while, if r = 0, V ∅ [s]
is the union of the irreducible components of V which lie entirely in the intersection s -1

1 (0) ∩ • • • ∩ s -1 m (
0). The same iterative procedure allows also to define the Coleff-Herrera current m j=1 ∂(1/s j ) ∧ T , when T is a global section of the Coleff-Herrera sheaf CH X ,V (• ; S, E) for some closed hypersurface S in a neighborhood of V such that V \ S = V . As before, each component (

m j=1 ∂(1/s j ) ∧ T ) j is then a global section of the Coleff-Herrera sheaf CH X ,Vj [s] (• ; S j [s, S], E j ) for some convenient hypersurface S j [s, S] in a neighborhood of V j [s] such that V j [s] \ S j [s, S] = V j [s]
. One can point out (see [START_REF] Andersson | Nonproper intersection theory and positive currents I, local aspects[END_REF]) that analytic continuation with respect to a single complex parameter (as in Theorem 2) provides a direct approach to such a current. Proposition 1. Let X , V , the ∆ j 's, the metrics | | j , the holomorphic sections s j 's, and the Coleff-Herrera current T (with eventual poles) be as above. Let We intend from now on to focus on the advantages (or disadvantages) that carries the idea of averaging (as done in Section 1 when stating Theorem 1, then Theorem 2) Coleff-Herrera residues of the form

γ 1 > γ 2 > • • • > γ m ≥ 1 be m positive integers and > 0. Then the current- valued map λ ∈ λ ∈ C ; |arg ]π,π[ (λ)| < π 2(γ 1 + ) ; |λ| >> 1 -→ 1 j=m 1 -|s j | 2λ γ j j + 1 2iπ ∂ |s j | 2λ γ j j s j ∧ T (14
m j=1 ∂(1/s j ) ∧ T or m j=1 ∂(1/s j ) ∧ T ,
when V is a purely dimension (n -M )-closed analytic subset in a complex manifold X , s 1 , ..., s m , being holomorphic sections of arbitrary Cartier divisors on X .

Vogel sequences and Vogel residue currents

The first positive point with respect to the averaging idea arises from the theory of improper intersection on X , as developed in the algebraic context in [START_REF] Stückrad | An algebraic approach to the intersection theory. Queen's Papers in Pure[END_REF] (see also [START_REF] Kollár | Effective Nullstellensatz for arbitrary ideals[END_REF]), and in the analytic context in [START_REF] Tworzewski | Intersection theory in complex analytic geometry[END_REF]. Let X and V be as in Section 1, and I be a coherent ideal sheaf in O X , with Z(I) being the support of the quotient sheaf O X /I. We recall that, at the local level, a local Vogel sequence (in the geometric sense) at x ∈ V (for I x , on the germ of complex analytic space (V x , (O V ) x ) is a sequence (s 1,x , ..., s n-M,x ) in the ideal I x such that there is a neighborhood U x of x in X , together with representatives s 1,x , ..., s n-M,x of the germs in this neighborhood, with

codim V (U x ∩ V ) \ Z(I) ∩ l j=1 s -1 j,x (0) = l or + ∞, l = 1, ..., n -M. ( 15
)
Such geometric conditions (15) (considered in the semi global setting) are already sufficient in order to imply the following Proposition.

Proposition 2. Let U be an open set in C n , V be a purely M -codimensional closed analytic subset of U , I be an ideal in O C n (U ) (with zero set Z(I) in U and generators σ 0 , ..., σ L ). Let (s 1 , ..., s n ) be a sequence of elements in I such that

codim V V \ Z(I) ∩ l j=1 s -1 j (0) = l or + ∞, l = 1, ..., n -M. ( 16
)
If T is an element in CH U,V (U, C), then n-M j=1 ∂ 1 s j ∧ T = T |Z(I) + n-M l=1 T [s 1 , ..., s l ] |Z(I) , (17)
where T [s 1 , ..., s l ] is defined inductively after l -1 operations via

T [s 1 ] = 1 (2iπ) ∂ |s 1 | 2λ s 1 ∧ T |U \s -1 1 (0) λ=0 T [s 1 , ..., s l +1 ] = 1 (2iπ) ∂ |s l | 2λ s l ∧ T [s 1 , ..., s l ] |U \s -1 l (0) λ=0
. , by the closed analytic set V ∩Z(I). As noticed in [START_REF] Andersson | Decomposition of residue currents[END_REF], such a (0, l +M ) current (T [s 1 , ..., s l ]) |Z(I) vanishes as soon as l + M < codim(V ∩ Z(I)), which means that only the terms with index l between codim(V ∩ Z(I)) -M and n -M remain in the development [START_REF] Elkadi | Bornes pour les degrés et les hauteurs dans le problème de division[END_REF]. The first term

Moreover
T |Z(I) is of course only present if codim (V ∩ Z(I)) = codim V = M .
Proof. Decompose T as

T = (1 -|s 1 | 2λ ) T λ=0 + |s 1 | 2λ T λ=0 = T |s -1 1 (0) + T |U \s -1 1 (0) . (18)
The fact that T |s -1 1 (0) has its support included in V ∩ s -1 1 (0) implies

∂ 1 s 1 ∧ T |s -1 1 (0) = 1 -|s 1 | 2λ + 1 2iπ ∂ |s 1 | 2λ s 1 ∧ T s -1 1 (0) λ=0 = T |s -1 1 (0) . Since T U \s -1 1 (0) is a Coleff-Herrera current with respect to V U \s -1 1 (0) , one has ∂ 1 s 1 ∧ T |U \s -1 1 (0) = 1 2iπ ∂ |s 1 | 2λ s 1 ∧ T U \s -1 1 (0) λ=0 = T [s 1 ].
We now observe that, since all components in the support

V s -1 1 (0) of T |s -1 1 (0)
lie in V ∩ s -1 1 (0) and have codimension M as V , the geometric condition ( 16)

for l = 2 implies n-M j=2 ∂ 1 s j ∧ T |s -1 1 (0) = T |s -1 1 (0) = T |Z(I) .
It remains to continue the process to compute

1 -|s 2 | 2λ + 1 2iπ ∂ |s 2 | 2λ s 2 ∧ T [s 1 ] λ=0 .
In order to do this, one decomposes T [s 1 ] as in ( 18):

T [s 1 ] = (1-|s 2 | 2λ )T [s 1 ] λ=0 + |s 2 | 2λ T [s 2 ] λ=0 = T [s 1 ] |s -1 2 (0) +T [s 1 ] |U \s -1 2 (0) .
The geometric condition ( 16) for l = 3 implies now that

1 -|s 2 | 2λ + 1 2iπ ∂ |s 2 | 2λ s 2 ∧ T [s 1 ] |s -1 2 (0) λ=0 = T [s 1 ] |Z(I) .
The contribution

1 -|s 2 | 2λ + 1 2iπ ∂ |s 2 | 2λ s 2 ∧ T [s 1 ] |U \s -1 2 (0) λ=0
, equals, as for the first step,

1 (2iπ ∂ |s 2 | 2λ s 2 ∧ T [s 1 ] |U \s -1 2 (0) λ=0 = T [s 1 , s 2 ].
The procedure can be thus repeated, which leads to [START_REF] Elkadi | Bornes pour les degrés et les hauteurs dans le problème de division[END_REF]. In order to prove that all T [s 1 , ..., s l ] are ∂ closed, it is enough to prove it for T [s 1 ] (then the proof goes inductively). The result for T [s 1 ] follows immediately from the fact that

T [s 1 ] = 1 (2iπ) ∂ |s 1 | 2µ s 1 ∧ |s 1 | 2λ T λ=0 µ=0 , (19)
by computing ∂ of both sides in [START_REF] Flenner | Joins and intersections[END_REF].

Proposition 3. Let U be an open set in C n , M ≤ n, f 1 , ..., f M be M holo- morphic functions in U , and T = T [f ] the (0, M )-Coleff-Herrera current T [f ] = M j=1 ∂ 1 f j (0,M ) (with respect to (V [f ]) ess = (f -1 1 (0) ∩ • • • ∩ f -1 M (0)) ess ). Let I be an ideal in O X (U ), such that (V [f ]) ess \ Z(I) = (V [f ]
) ess , with generators (σ 0 , ..., σ L ). If u 1 , ..., u n-M , one after each other, in this order, are generic in P L (C), such that in particular (s 1 , ...., s n-M ) := ( u 1 , σ , ..., u n-M , σ ) fulfills conditions [START_REF] Elkadi | Une version effective du théorème de Briançon-Skoda dans le cas algébrique discret[END_REF]

with V = (V [f ]) ess , then the current n-M j=1 ∂ 1 u j , σ ∧ T [f ] (20)
is annihilated as a current by I n-M .

Proof. Before presenting a sketch of the proof, let us focus on a simple situation (to which the general case will in fact be reduced). Suppose for the moment that T ∈ CH U,V (U, C), and that σ and h are two holomorphic functions in a neighborhood of V , such that (σ, h) defines a complete intersection on V . Then, we claim that the current

∂ |σh| 2λ σh ∧ T λ=0 |σ -1 (0) = (1 -|σ| 2µ ) ∂ |σh| 2λ σh ∧ T λ=0 µ=0
is annihilated by σ. This follows from the fact that the map

(λ 1 , λ 2 ) ∈ {Re λ 1 >> 1, Re λ 2 >> 1} -→ |h| 2λ1 h |σ| λ2 T
extends as an holomorphic function of two variables to a product of half spaces {Re

λ 1 > -η} × {Re λ 2 > -η} for some η > 0. Since V \ σ -1 (0) = V \ h -1 (0) = V , it follows that ∂ |h| 2λ1 h |σ| 2λ2 T λ1=λ2=0 = ∂ |h| 2λ h ∧ T λ=0 = ∂ T h ,
which is a Coleff-Herrera current on U with respect to V ∩ {h = 0}. Since σ does not vanish identically on any component the set V ∩ {h = 0}, one has

(1 -|σ| 2µ ) ∂ T h µ=0 = 0,
which proves the claim.

Consider now the Coleff Herrera current T [f ]. Let U π → U be the normalized blowup of U along I and E(I) be its exceptional divisor (multiplicities being taken into account). Though U is not smooth, one can (locally) consider an embedding U ⊂ Ω ⊂ C N and treat the current (

M 1 ∂(1/π * [f j ])) (0,M ) as the ∂-closed (N -n, M + N -n) current ( M 1 ∂(1/π * [f j ])) (0,M ) ∧ [ U ] in Ω. It factorizes in Ω as the product of a (0, M + N -n) Coleff-Herrera current with eventual poles (with respect to some closed M -codimensional analytic subset V [f ] of U ) times an holomorphic (N -n, 0) form. More- over, V [f ] \ π -1 (Z(I)) = V [f ]. Actually, one can approach the current ( M 1 ∂(1/π * [f j ])) (0,M ) by χ π * (σ) ≥ M 1 ∂(1/π * [f j ]
) when > 0 tends to 0, and χ denotes a smooth cutoff function as in [START_REF] Björk | Regularizations of residue currents[END_REF]. The exceptional divisor E(I) defines then a divisor

E(I) | V [f ] on V [f ]. Choose u 1 (generically) such that π * [s 1 ] | V [f ] vanishes on each irreducible component of the sup- port E(I) | V [f ]
exactly at the multiplicity on this component of the divisor

E(I) | V [f ] . This means that π * [s 1 ] | V [f ] can be expressed locally about a point x ∈ V ∩ π -1 (Z(I)) as π * [s 1 ] = σ | V [f ] h 1 ,
where σ is the generator at x for the stalk of the principal sheaf ideal (π * [σ 0 ], ..., π * [σ L ]), and {h 1 = 0} defines with σ on V [f ] (about x) a complete intersection. Then it follows from the case studied at the beginning that the current

T [f ][s 1 ] |Z(I) = ∂ |s 1 | 2λ s 1 ∧ T [f ] λ=0 |Z(I)
is annihilated by I. Suppose that l ≥ 2 and that we know that T [s 1 , ..., s l-1 ] is annihilated by I l-1 , once s 1 , ..., s l-1 have been conveniently chosen. Then one has

I l-1 T [f ][s 1 , s 2 , ..., s l-1 ] |Z(I) = 0, which implies that if h ∈ I l-1 , the current h T [f ][s 1 , s 2 , ..., s l-1 ] |U \Z(I) = h T [f ][s 1 , s 2 , ..., s l-1 ] is a Coleff-Herrera current. Repeating the argument used for l = 1, with T [f ] replaced by h T [f ][s 1 , ..., s l-1 ] |U \Z(I)
, one can choose s l in a generic way (genericity depends here on the previous choices of s 1 , ..., s l-1 , but is independent of h), so that, for any

h ∈ I l-1 , h T [f ][s 1 , ..., s l ] |Z(I) = ∂ |s l | 2λ s l ∧ h T [f, s 1 , ..., s l-1 ] |U \Z(I) λ=0 |Z(I)
is annihilated by I. It follows then that, with this convenient choice of s l , the current (T [f ][s 1 , ..., s l ]) |Z(I) is annihilated by I l . That concludes the proof.

It is natural to call the sequence ( u 1 , σ , ..., u n-M , σ ) a Vogel sequence for I with respect to the Coleff-Herrera current T [f ], if such a sequence is constructed (in a generic way in terms of the u j ) through the inductive procedure described above step-by-step. The corresponding current

n-M 1 ∂(1/ u j , σ ) ∧ T [f ] is then called a Vogel residue current (for I with respect to T [f ]).
Remark 5. When [V ] is a purely dimensional cycle in X (here assimilated to its associated integration current), and Z ⊂ X is a complex submanifold, it is proved in [START_REF] Cygan | Intersection theory and separation exponent in complex analytic geometry[END_REF], and reinterpreted in algebraic terms in [START_REF] Kollár | Effective Nullstellensatz for arbitrary ideals[END_REF], that the (multi) cycle of intersection [V ] • Z, defined via the Vogel procedure from a prescribed Vogel sequence, is such that its Chow ideal [START_REF] Kollár | Effective Nullstellensatz for arbitrary ideals[END_REF] for the definition of this notion) lies in the integral closure of the sheaf of ideals (I([V ]), I Z ). Briançon-Skoda's theorem then implies that

I Chow ([V ] • [Z]) (see
(I Chow ([V ] • [Z])) n ⊂ (I([V ]
), I Z ). Proposition 3 can thus be understood as an analog result, when [V ] is replaced by the Coleff-Herrera T [f ], and intersection theory transposed at the (algebraic) level of residue currents instead of the (geometric) one of integration currents.

Averaging Vogel residue currents

Let X be a n-dimensional complex manifold, U be an open set in X , M ≤ n, T be a (0, M ) E-valued Coleff-Herrera current in U (or T a (0, M ) E-valued Coleff-Herrera current in U with poles), with respect to a closed analytic subset V ⊂ U of pure codimension M . Let I be an ideal sheaf in O X (U ). Assume that σ 0 , ..., σ L generate I(U ) globally in U . One can think of (σ 0 , ..., σ L ) as an holomorphic section of the trivial bundle E σ = U × C L+1 , equipped with its standard hermitian metric. It is natural to propose as an averaged current

(P L (C)) m m j=1 ∂ 1 u j , σ ∧ (T or T ) dω L (u 1 ) ⊗ • • • ⊗ dω L (u m ) (21)
(where m = min(L + 1, n -M + 1), ω L denotes the Fubini metric on P L (C) and u, σ

= L 0 u l σ l for u = [u 0 : • • • : u L ] ∈ P L (C))
, the current obtained as the value at 0 of the

n-M r=0 D (0,M +r) (X , r E * σ ⊗ E) current-valued map λ -→ 1 -σ 2λ + ∂ σ 2λ ∧ dim V r=1 1 (2iπ) r σ * ∧ (∂σ * ) r-1 σ 2r ∧ (T or T ). (22)
Here σ * = σ = L 0 σ l ⊗ e * l and dσ = L 0 dσ l ⊗ e l (if (e 0 , ..., e L ) denotes a standard base of sections for E σ = U × C L+1 ). Because of the local structure theorems for sections of the Coleff-Herrera sheaves (see [START_REF] Vidras | Coleff-Herrera currents revisited[END_REF]), this map is holomorphic in some half-plane Re λ > -η for some η > 0. It was indeed proved in [START_REF] Andersson | Nonproper intersection theory and positive currents I, local aspects[END_REF], as a consequence of Crofton's formula, that, if m = min(L + 1, dim V + 1), the averaging

(P L (C)) m m j=1 ∂ 1 u j , σ ∧ [V ] ∧ m j=1 d u j , σ dω L (u 1 ) ⊗ • • • ⊗ dω L (u m )
leads to the current

M σ V := (1 -σ 2λ + ∂ σ 2λ ∧ dim V r=1 1 (2iπ) r σ * ∧ (∂σ * ) r-1 σ 2r ∧ [V ] λ=0 (dσ, ..., dσ). (23)
The action on (dσ, ..., dσ) corresponds here to the contraction operation between the r E * σ and E σ as follows :

(Φ * 1 ∧ • • • ∧ Φ * r ) (dσ, ..., dσ) = r j=1 Φ * j (dσ) r! , r = 1, ..., dim V,
for any Φ * j of E * σ -valued currents. The current ( 23) is d-closed and positive in this case. Even though it is not an integration current, the vector of the Lelong numbers (e 0 (x), ..., e n-M (x)) of its various components (from type (0, 0) till type (n-M, n-M )), at any point x ∈ V ∩Z(I), coincides with the vector of Segre numbers (see [START_REF] Gaffney | Segre numbers and hypersurface singularities[END_REF]) at x for I x with respect to V , that is the minimum (respect to lexicographic order) of all u-plets (deg x γ x,0 , ..., deg x γ x,n-M ), where γ is a Vogel cycle at x and γ x,k , k = 0, ..., n-M , denotes its component at x with codimension k in V (as a subcycle of V x ). Moreover, the relevant part in the Siu decomposition of the averaged current M σ is expressed as

M σ V,relevant = dim V k=0 ι β k ι [Z k ι ],
where the Z k ι are the distinguished varieties of I with codimension k (distinguished varieties being defined as the images of the irreducible components of the exceptional divisor E in the normalized blowup of U along the coherent ideal sheaf I). Here the β k ι are positive integer coefficients. This generalized version of H. King's formula is proved in [START_REF] Andersson | Nonproper intersection theory and positive currents I, local aspects[END_REF]. This shows that, at the semi global level, averaging Coleff-Herrera currents, once they are conveniently multiplied in order to become intersection currents, is an operation that fits well with improper intersection theory, such as implemented in [START_REF] Stückrad | An algebraic approach to the intersection theory. Queen's Papers in Pure[END_REF][START_REF] Tworzewski | Intersection theory in complex analytic geometry[END_REF]. Note that the Segre numbers remain unchanged if one replaces s = (σ 0 , ..., σ L ) by some u-plet (σ 0 , ..., σL ) which generate in O X (U ) an ideal with, locally about each point, the same integral closure than I. This will emphasizes the role of a current M σ (or, to be more precise, residue currents involved in its factorization) with respect to Briançon-Skoda's type theorems.

In order to extend these ideas to the global setting, one assumes that the coherent sheaf I is globally generated in X by holomorphic sections σ 0 , ..., σ L of the same line bundle O X (∆). Referring to Example 1, the holomorphic sections of O P m(n+1)-1 (C) (1) defining the diagonal subspace Z in the join as a complete intersection, provide a useful illustration of such a situation, the coherent sheaf I being in this case the radical sheaf

I Z . Let E σ = L+1 O X (∆). Choose a metric | | on O X (∆), that induces the metric = | |⊕• • •⊕| | on E σ .
Let E be a finite rank holomorphic bundle over X . When T (resp. T ) is an element in CH X ,V (X , E) (resp. in CH X ,V (X ; S, E)), it is natural to propose, as the averaged current [START_REF] Jelonek | On the effective Nullstellensatz[END_REF], the current obtained as the value at 0 of the n-M r=0 D (0,M +r) (X , r E * σ ⊗ E) current-valued map [START_REF] Gaffney | Segre numbers and hypersurface singularities[END_REF], where σ * denotes the conjugate section of σ respect to the metric on E σ . If π : V -→ V de-notes the normalized blowup of the complex space (V, (O X ) |V ) along the O V ideal sheaf I |V , generated locally by the holomorphic sections x → σ l (ι(x)) (ι being the embedding V ⊂ X ), l = 0, ..., L, and E denotes the exceptional divisor in this normalized blowup, then one can check that this averaging leads, from the point of view of intersection theory: that is, when T or T are replaced by the (M, M )-current [V ], to the construction of the Vogel current

Vog V,σ,| | = 1 -σ 2λ + ∂ σ 2λ ∧ dim V r=1 1 (2iπ) r σ * ∧ (∂σ * ) r-1 σ 2r ∧ [V ] λ=0 (dσ, ..., dσ) = [V Z(I) ] + dimV r=1 π * [E] ∧ c 1 O V (-E) ⊗ π * [∆ |V ] r-1 = dim V r=0
Vog (...); r .

(

) 24
We denote as ∆ |V the line bundle O V (∆). Here π * [σ] = e [0] × τ , where τ is an holomorphic non vanishing u-plet of sections of

O V (-E) ⊗ π * [∆ |V ]. The metric on O V (-E) is defined by |e [0] | E = σ • π . It induces a metric on the divisor O V (-E) ⊗ π * ((O X (∆)) |V ), so that dd c log |e [0] | 2 E = [E] + c 1 (O V (-E)) = [E] + dd c log τ 2 -π * (c 1 (∆ |V )),
that is

dd c log τ 2 = c 1 (π * [∆ |V ]) + c 1 (O V (-E)) = c 1 O V (-E) ⊗ π * [∆ |V ] .
The current Vog (...) = dim V r=0 Vog (...); r (that can also be considered as a current on (V, O V )) is related to the Segre current

Seg V,σ,| | = [V Z(I)∩V ] + dimV r=1 π * [E] ∧ c 1 O V (-E) r-1 = dimV r=0 Seg V,σ,| | ;r (25) thanks to the algebraic relations Vog V,σ,| | ; r = r-1 l=0 r -1 l Seg V,σ,| | ; r-l ∧ c 1 (∆ |V ) l , r = 1, ..., dim V.
An important particular case occurs where T is a Coleff-Herrera residue of the form M j=1 ∂(1/s j ), s 1 , ..., s M being respective holomorphic sections of Cartier divisors ∆ j , j = 1, ..., M , on X , such that the s -1 j (0) define a complete intersection. In this particular case, one can consider the (M +L+1)holomorphic bundle

E ∆,σ = M j=1 O X (∆ j ) ⊕ E σ ,
m j=1 ∂ 1 u j , σ ∧ M j=1 ∂ 1 s j , u j ∈ P L (C), 1 ≤ j ≤ m = min(L + 1, dim V + 1)
the current obtained (still through the analytic continuation process) as the value at λ = 0 of the

n-M r=0 D (0,M +r) (X , r E * ∆,σ ) current-valued map λ -→ 1 -s ⊕ σ 2λ + ∂ s ⊕ σ 2λ ∧ dim V r=1 1 (2iπ) r (s ⊕ σ) * ∧ (∂(s ⊕ σ) * ) r-1 s ⊕ σ 2r , (26)
where s ⊕ σ = s 1 ⊕ • • • ⊕ s M ⊕ σ and (s ⊕ σ) * denotes its conjugate section respect to the metric which has been chosen on E ∆,σ . The two currents, which belong to

n-M r=0 D (0,M +r) (X , r E * ∆,σ ), 1 -σ 2λ + ∂ σ 2λ ∧ dim V r=1 1 (2iπ) r σ * ∧ (∂σ * ) r-1 σ 2r ∧ M j=1 ∂(1/s j ) λ=0 and 1 -s ⊕ σ 2λ + ∂ s ⊕ σ 2λ ∧ dim V r=1 1 (2iπ) r (s ⊕ σ) * ∧ (∂(s ⊕ σ) * ) r-1 s ⊕ σ 2r λ=0 (27)
coincide when codim (s -1 (0) ∩ σ -1 (0)) = M + L + 1, see [START_REF] Wulcan | Products of residue currents of the Cauchy-Fantappié-Leray type[END_REF], but differ in general. They can both be used (see [START_REF] Andersson | Residue currents and ideals of holomorphic functions[END_REF][START_REF] Andersson | Decomposition of residue currents[END_REF]) to materialize the residual obstruction for the exactness of the generically exact Koszul complex

0 → M +L+1 E * ∆,σ s⊕σ -→ M +L E * ∆,σ -→ • • • s⊕σ -→ l+1 E * ∆,σ s⊕σ -→ l E * ∆,σ -→ • • • s⊕σ -→ E * ∆,σ s⊕σ -→ X × C, (28)
since they are both annihilated (as currents) by the operator 2iπ s⊕σ -∂, where s⊕σ denotes the interior multiplication by the holomorphic section s ⊕ σ of E ∆,σ .

3 About a result by M. Hickel, M. Andersson and E. Götmark

In order to emphasize what averaged Vogel residue currents introduced in Section 2 could be useful for, and, in parallel, to illustrate how far using their use reveals to be successful, let us focus on the effective geometric global formulation of Briançon-Skoda's theorem in C[X 1 , ..., X n ], as obtained first by M. Hickel in [START_REF] Hickel | Solution d'une conjecture de C. Berenstein et A. Yger et invariants de contact à l'infini[END_REF], and then reformulated (and thus reproved), using the frame developed above, by M. Andersson and E. Götmark in [START_REF] Andersson | Explicit representation of membership in polynomial ideals[END_REF]. Before stating the result, one needs to recall basic facts about Lojasiewicz exponents at infinity in

C n ⊂ P n (C) = C n ∪ {z 0 = 0}. Let p 1 , ..., p m be m polynomials in C[X 1 , ..., X n ],
for some strictly positive constant κ. In particular, for p to be in this case a proper map from C n into C n , it is enough that ν ∞ (P ) < deg p j , j = 1, ..., m.

Notice that, when m = n and dim C n p -1 (0) ≤ 0, the current

n 1 ∂ 1 P j |{z0=0}
is annihilated by z nν∞(P ) 0

. This can be seen using (as in the proof of Proposition 3) the normalized blowup of P n (C) along the coherent ideal sheaf I {z0=0} . On the other hand, the current

n 1 ∂ 1 P j |C n
is annihilated by any homogeneous polynomial of the form Q such that the map ζ → Q(1, ζ)/ p(ζ) n is locally bounded in C n (see [START_REF] Andersson | Explicit representation of membership in polynomial ideals[END_REF]).

M. Hickel's result ( [START_REF] Hickel | Solution d'une conjecture de C. Berenstein et A. Yger et invariants de contact à l'infini[END_REF], revisited with the analytic methods presented above in [START_REF] Andersson | Explicit representation of membership in polynomial ideals[END_REF]) can be stated as follows (we just mention here the result when m ≤ n).

Theorem 4 ( [START_REF] Andersson | Explicit representation of membership in polynomial ideals[END_REF][START_REF] Hickel | Solution d'une conjecture de C. Berenstein et A. Yger et invariants de contact à l'infini[END_REF]). Assume m ≤ n and let p 1 , ..., p m be m polynomials in C[X 1 , ..., X n ]. Let q be a polynomial in C[X 1 , ..., X n ] such that the function |q|/|p| m is locally bounded in C n . Then, there exist polynomials a 1 , ..., a m ∈ C[X 1 , ..., X n ] such that q ≡ m j=1 a j p j with deg a j p j ≤ deg q + [mν ∞ (p)] + 1, where [γ] denotes the integer part of the rational number γ.

As a consequence of this theorem, it appears that the effective realization of the global Briançon-Skoda theorem (q being in the ideal (p 1 , ..., p m ) in As pointed out in [START_REF] Kollár | Effective Nullstellensatz for arbitrary ideals[END_REF], when dealing with intersection problems involving arithmetic aspects, it is more adequate to use a specific Vogel cycle of intersection instead of what could be understood as an averaged version, for example the intersection cycle constructed in [START_REF] Tworzewski | Intersection theory in complex analytic geometry[END_REF] and reinterpreted in [START_REF] Andersson | Nonproper intersection theory and positive currents I, local aspects[END_REF] in the currential setting. The methods presented here (end of Section 1 and subsection 2.2) aim precisely to give some support to the following conjecture :

C[X 1 , ..., X n ] if the germ q ζ , for each ζ ∈ p -1 (0),
Conjecture 1 (Global arithmetic Briançon-Skoda) Let p 1 , ..., p m be m polynomials in Z[X 1 , ..., X n ] with degrees bounded by d and logarithmic sizes bounded by h. Let q ∈ Z[X 1 , ..., X n ] be such the function q/|p| min(m,n) is locally bounded in C n . Then, one can find δ ∈ N * , a 1 , ..., a m ∈ Z[X 1 , ..., X n ] such that δ q κ(n) = m j=1 a j p j , max j deg(a j p j ) ≤ κ(n) deg q + κ 0 (n) d γ0 min(n,m) max j h(δ), h(a j p j ) ≤ κ(n) h(q) + κ 1 (n) (h + log m) d γ1 min(n+1,m) ,

where κ(n), κ 0 (n), κ 1 (n) are numerical constants depending only on the number n of variables, γ 0 , γ 1 being universal constants (idealistically κ(n) = 1, κ 0 (n) = n, γ 0 = γ 1 = 1).

This result was obtained by M. Elkadi in [START_REF] Elkadi | Une version effective du théorème de Briançon-Skoda dans le cas algébrique discret[END_REF] when dim(p -1 (0)) = 0. On the other hand, the arithmetic membership problem can be solved with such bounds [START_REF] Stückrad | An algebraic approach to the intersection theory. Queen's Papers in Pure[END_REF] when m ≤ n and (p 1 , ..., p m ) defines a complete intersection in C n (see [START_REF] Elkadi | Bornes pour les degrés et les hauteurs dans le problème de division[END_REF]), which includes in particular the case of the arithmetic nullstellensatz ( [START_REF] Berenstein | Residue Calculus and effective Nullstellensatz[END_REF], [START_REF] Krick | Sharp estimates for the arithmetic Nullstellensatz[END_REF], [START_REF] D'andrea | Heights of varieties in multiprojective spaces and arithmetic nullstellensatz[END_REF]). This fits well with the ideas that govern the construction of Vogel sequences. Nevertheless, it is known now that methods based on multidimensional residue calculus (relying essentially on Cauchy-Weil integral formula and associated Bergman-Weil developments) do not provide the sharpest bounds for the arithmetic nullstellensatz, which is truly a problem related to arithmetic intersection theory (see the recent approach in [START_REF] D'andrea | Heights of varieties in multiprojective spaces and arithmetic nullstellensatz[END_REF], based on a arithmetic version of O. Perron's theorem used for the algebraic nullstellensatz in [START_REF] Jelonek | On the effective Nullstellensatz[END_REF]). As a consequence, this makes more clear that such methods were in fact more in the spirit of Conjecture 1. Using the Stückrad-Vogel approach [START_REF] Stückrad | An algebraic approach to the intersection theory. Queen's Papers in Pure[END_REF], combined with a precise analysis of the Vogel residue currents involved (description of their annihilators, explicit computations of the restrictions to C n of the auxiliary Coleff-Herrera currents involved in their expansion, see Propositions 2 and 3 above, together with the concluding comments in Section 1), seems to be a natural way to tackle such a conjecture. It is indeed necessary to overcome the difficulty which is inherent to the fact that averaging such Vogel residue currents in order to get suitable Bochner-Martinelli currents (for control of the degree in effectivity questions) does not preserve the arithmetic structure of the data (which would be necessary in order to get in parallel control on the heights). It seems also opportune to mention that the initial approach to Theorem 4 by F. Amoroso in [START_REF] Amoroso | On a conjecture of C. Berenstein and A. Yger[END_REF] relies on the Northcott-Rees notion of superficial elements in ideals, that is also present in the construction of Vogel sequences (more specifically of filtered sequences, see [START_REF] Achilles | Intersection numbers, Segre numbers and generalized Samuel multiplicities[END_REF]).

This presentation of the Coleff-Herrera machinery, combined in subsection 2.2 with that of the Stückrad-Vogel approach [START_REF] Stückrad | An algebraic approach to the intersection theory. Queen's Papers in Pure[END_REF][START_REF] Tworzewski | Intersection theory in complex analytic geometry[END_REF][START_REF] Andersson | Nonproper intersection theory and positive currents I, local aspects[END_REF], transposed to the context of residue currents instead of integration currents on cycles, intends to be a modest invitation towards such an approach to effectivity questions in arithmetic polynomial geometry, when they require operational tools related to duality, such as Briançon-Skoda's theorem or multidimensional operational residue calculus.

m 1 O

 1 P n (C) (-d j )-valued Coleff-Herrera global current in P n (C) obtained from the current T [p 1 ] = ∂(1/p 1 ) through the inductive process

)

  extends as a holomorphic map in an open neighborhood of the closed sector {|arg ]π,π[ (λ)| ≤ π/(2γ 1 )}. Its value at λ = 0 equals the current m j=1 ∂(1/s j )∧ T .

equipped with the metric M 1 |

 1 | j ⊕ (where | | j denotes an hermitian metric on O X (∆ j ), j = 1, ..., M ), and propose as an alternative averaged version for all Vogel residue currents

  together with their homogenizations P 1 , ..., P m , considered as respective holomorphic sections of the Cartier divisors O P n (C) (deg p j ). Let E = ι E ι be the (reduced) exceptional divisor of the normalized blowupP n (C) π -→ P n (C) of P n (C)along the coherent ideal sheaf I(P ), generated locally by the holomorphic sections P 1 , ..., P m . Let µ ι (P ) be the multiplicity of I(P ) • O P n (C) , and µ ι (z 0 ) be the multiplicity ofI {z0=0} • O P n (C) along the same component. The Lojaciewicz exponent ν ∞ (P ) is defined as ν ∞ (P ) = sup ι µ ι (P ) µ ι (z 0 ) .In the special case where m = n and dim C n p -1 (0) ≤ 0, one has the Lojasiewicz inequality at infinity in C n ⊂ P n (C), namely n j=1 |p j (ζ)| |ζ| deg pj ≥ κ |ζ| ν∞(P ) , |ζ| >> 1,

1 O 1 ∂ 1 u

 111 lies in the m-power of the integral closure of the ideal (p 1,ζ , ..., p ,ζ ) in the local ring O C n ,ζ ) can be achieved with degree estimates for the quotients a j controlled by the geometric Bézout theorem, which means that the effectivity of the problem is governed by geometric intersection theory. The proof of Theorem 4 relies on the use of the averaged Bochner-Martinelli version (22) (E σ = m P n (C) (d j ) with the Fubini-Study metric on each component, σ = P , V = P n (C) = X ), of all currents m j , P j , u 1 , ..., u m ∈ P m-1 (C), m = min(dim({P = 0}) + 1, m), with respect to the tensorized Fubini Study metric on P m-1 (C) ⊗ • • • ⊗ P m-1 (C) (m times).

  all currents T [s 1 , ..., s l ], l = 1, ..., n -M , are ∂-closed, and thus are Coleff-Herrera currents. Here T |Z(I) means [(1 -|σ| 2λ )T ] λ=0 , while T |U \Z(I) means [|σ| 2λ T ] λ=0 , when T is a Coleff-Herrera current.

Remark 3. Formula (17) remains valid if T is replaced by a meromorphic Coleff-Herrera current T ∈ CH U,V (U ; S, C) for some closed hypersurface S in U such that V \ S = V . But the second assertion does not remain true. Remark 4. All currents (T [s 1 , ..., s l ]) |Z(I) are supported, as the current T |Z(I)


Matthieur Perreira

Da Silva

Vincent Courboulay

Pascal Estraillier

IMAGE COMPLEXITY MEASURE BASED ON VISUAL ATTENTION

Digital images can be analyzed at wide range of levels going from pixel arrangement to semantics. As a consequence, finding a visual complexity estimator is a difficult task. In this article we propose a definition of attention based perceptual complexity. We study the performance of human eye movements and of different models of computational attention against a ground-truth of image complexity based on the observation time of an image description task. The results obtained show that besides its lack of semantic processing, attentional behavior is a good estimator of image complexity.

INTRODUCTION 1.Context

The study of visual complexity is particularly relevant to both cognitive and computer science studies. Yet, it is still illdefined.

The pioneers in this field are the psychologists Snodgrass and Vanderwart [START_REF] Snodgrass | A standardized set of 260 pictures: norms for name agreement, image agreement, familiarity, and visual complexity[END_REF] which, in the early 1980's established a classification of the complexity of a set of black and white line art drawings. This classification was based on subjective observers ranking. Later, in order to obtain a more objective measure of complexity, more "algorithmic" measures were introduced (number of line segments, lines crossings, etc.). These, however, were not calculated computationally.

In the 90's, the image processing community tackled this question in order to solve different kinds of problems. Firstly, evaluating image complexity allows measuring the performance of algorithms relative to the kind of images treated [START_REF] Peters | Image complexity metrics for automatic target recognizers[END_REF]. Another domain which exploits complexity is content based image retrieval (CBIR) [START_REF] Perkiö | Modelling image complexity by independent component analysis , with application to content-based image retrieval[END_REF]: instead of searching for images with the same attributes (shape, color, etc.) it can be interesting to request images with the same complexity as the query image. Image complexity can also be used in watermarking [START_REF] Yaghmaee | Estimating Watermarking Capacity in Gray scale Images based on Image Complexity[END_REF]: the more complex the image is, the easier it is to insert information without altering its quality. The list of possible applications is obviously much longer (image recognition, compression, etc.), but these few examples highlight the diversity of possibilities.

To sum up, estimating the complexity of an image is interesting for a wide range of applications (from psychology to computer science). However, its estimation is facing a major problem: the definition of complexity. Webster's dictionary (1986) defines a complex object as "an arrangement of parts, so intricate as to be hard to understand or deal with". According to W. Li, the definition of complexity should be very close to certain measures of difficulty (of construction, description, etc.) concerning the object or the system studied [START_REF] Li | On the Relationship Between Complexity and entropy for markov chains and regular languages[END_REF].

The measures of complexity arising from such a definition are numerous. Lloyd [START_REF] Lloyd | Measures of complexity: a nonexhaustive list[END_REF], proposes a non exhaustive list, classified into three categories: how hard is it to describe, how hard is it to create, and what is its degree of organization?

In the field of visual complexity, proposals are more scarce but equally varied. To name few: fractal theory [START_REF] Siu-Ngan | An Evaluation of Fractal Methods for Characterizing Image Complexity[END_REF], fuzzy theory [START_REF] Cardaci | A fuzzy approach to the evaluation of image complexity[END_REF] or, more frequently information theory [START_REF] Rigau | An informationtheoretic framework for image complexity[END_REF]. Regarding this last point, and as previously mentioned, recent work [START_REF] Mulhern | Confounds in pictorial sets: The role of complexity and familiarity in basiclevel picture processing[END_REF][START_REF] Forsythe | Visual Complexity : Is That All There Is ?[END_REF] presents the strong correlation between human evaluation of complexity and Kolmogorov complexity (JPEG and GIF compression ratios).

Hypothesis

The methods mentioned above evaluate the complexity of images, regardless of their perception by our visual and attentional systems. Intuitively, the heatmaps generated by a computational model of attention seem to vary (spreading, patterns) according image complexity (figure 1).

In order to verify this hypothesis, we propose to construct a measure of complexity based, no longer on the "direct" analysis of their content, but from this perceptual complexity through the attentional filter. This measure does not aim to compete or to replace the traditional measures, but rather to provide an additional tool for estimating the complexity of images.

In the following section, we describe a few computational models of attention as well as our contributions to the modeling of visual attention. In section 3, we describe how we have evaluated the potential of these models. Section 4 provides the results of this evaluation.

COMPUTATIONAL ATTENTION

In order to validate the hypothesis that attention is correlated to image complexity, we have chosen to evaluate three well known bottom-up attention models. They were chosen for their proven performances in the field of attention modeling, their public availability [START_REF] Itti | A model of saliency-based visual attention for rapid scene analysis[END_REF] [13] and / or their link with publicly available eye-tracking experiments databases [13] [14].

We have also evaluated different versions of our own computational model of attention [START_REF] Perreira | Evaluation of preys / predators systems for visual attention simulation[END_REF], partially derived from Laurent Itti's work. The main characteristics of these models are summarized in the following subsections.

Some existing models

Itti's model [START_REF] Itti | A model of saliency-based visual attention for rapid scene analysis[END_REF] is one of the most popular attention models.

It is based on a hierarchical approach using multi-resolutions decomposition and center-surround filtering. It provides a centralized representation of attention through the generation of a global saliency map. Le Meur's model [START_REF] Meur | A coherent computational approach to model bottom-up visual attention[END_REF] is an improvement of [START_REF] Itti | A model of saliency-based visual attention for rapid scene analysis[END_REF], built on a more realistic (yet more complex) generation of features and conspicuity maps as well as an improved conspicuity maps fusion scheme.

Bruce [START_REF] Bruce | Saliency, attention, and visual search: An information theoretic approach[END_REF] proposes an alternative approach based in information theory. It combines independent component analysis (ICA) with a measure of self-information in order to estimate the saliency of each pixel of an image.

Our contributions

All of the previously cited models are based on a central representation of saliency. The model proposed in [START_REF] Perreira | Evaluation of preys / predators systems for visual attention simulation[END_REF] provides an alternative processing of visual attention, based on competition, leading to a distributed representation of attention. This competition is achieved by using a prey / predators dynamical system.

The advantages of such a system are numerous:

it handles naturally the competition between the different kinds of visual information (intensity, color, orientation, etc.): thus, there is no need for additional normalization steps;

it is inherently dynamical: the evolution of the focus of attention is generated naturally by the evolution of the prey / predators system;

it is adaptable: thanks to its different parameters it can be tuned to many different applications. In particular, it provides a noise parameter allowing to balance between focusing exclusively on a few salient points and exploring the whole image;

In addition to this original competitive maps fusion process, we have improved the computation of conspicuity and singularity maps so that they are faster to generate. As a consequence, images can be analyzed using more scales, typically 6 against 3 for the default implementation of [START_REF] Itti | A model of saliency-based visual attention for rapid scene analysis[END_REF].

An extension of our basic system introduces a feedback mechanism which allows adapting the dynamics and scene exploration behavior of our system to a specific context. It modulates the activity of the dynamical system with a map built from previous attentional focuses (with a weight inversely proportional to their "oldness"). This mechanism is controlled by a feedback parameter f ∈ [-1, 1] which allows to:

maximize scene exploration and accelerate the system dynamics if f ∈ [-1; 0];

stabilize the focus points on previously visited locations and slow down system dynamics if f ∈ [0; 1] .

Experiments ran on the test databases described in section 3 shows that our system is closer to human attentional behavior if f = 0.2. This feedback value is used in the feedback based version of our system.

The aim of the experiments described in the next section is twofold. Firstly, it should validate that saliency based models of attention can help predicting the complexity of an image. Secondly, it should determine if the dynamical behavior of our attentional model can provide additional information leading to a better evaluation of visual complexity.

EXPERIMENTS

Building a ground-truth of image complexity

Considering that image complexity is linked to its difficulty of description, we have measured the mean observation time of a set of M = 148 images belonging to two databases (containing images and eye-tracking data) made publicly available by Bruce [START_REF] Bruce | Saliency, attention, and visual search: An information theoretic approach[END_REF] and Le Meur [START_REF] Meur | A coherent computational approach to model bottom-up visual attention[END_REF]. We asked N = 12 participants to watch each image of the database enough time to be able to describe it with a medium level of details. Observation times were normalized, averaged, and sorted in order to obtain a mean observer ranking M OR.

In order to evaluate the intra-observers variation, we built each observer's own ranking OR i with i ∈ [1...N ]. For every rank k ∈ [1...M ] we can define a distance from the mean ranking MOR such as:

dOR[k] =| OR i [k] -M OR[k] |.
We can then build a ROC curve by considering a binary classifier which correctly estimates image complexity if dOR

[k] < S with S ∈ [1...M ].
The ROC curve represents the number of good predictions relative to threshold S. The area under ROC curve provides a single estimator of the complexity estimation quality. The mean area under ROC curve of all observers is linked to the intra-observer variation: it represents the highest reference score to which computational models can be compared.

Models and measures

We evaluated the following methods in order to obtain a complexity based ranking of the database images:

-Random ranking of the images. This represents our lower bound of complexity evaluation;

-JPEG compression ratio of the original images. Provides a default Kolmogorov complexity estimator, di-rectly based on image pixels;

-JPEG compression ratio of the eye-tracking heatmaps which are part of the databases provided by [13] [14]. This gives an estimate of how human eye movements can predict complexity;

-JPEG compression ratio of the saliency maps generated by several computation models of attention [START_REF] Itti | A model of saliency-based visual attention for rapid scene analysis[END_REF] [13] [START_REF] Meur | A coherent computational approach to model bottom-up visual attention[END_REF];

-JPEG compression ratio of the heatmaps built from simulated focus points obtained using [START_REF] Perreira | Evaluation of preys / predators systems for visual attention simulation[END_REF] after 300 iterations; Since our system is dynamical, we also built two specific measures in order to directly exploit the focus points coordinates:

-DCR: the deflate compression ratio [START_REF] Lloyd | Measures of complexity: a nonexhaustive list[END_REF] of the focus points coordinates measures if there are spatial redundancy or patterns in the evolution of the focus point.

-SLFE: the saccade length Fourier entropy [START_REF] Toh | Spectral entropy as speech features for speech recognition[END_REF] measures if there are temporal redundancies in the evolution of the distance between two consecutive focus points.

RESULTS AND DISCUSSION

Independently from our own attention model, we can see in table 1 that JPEG compression ratio of attentional maps seems to be a better complexity estimator than simple JPEG compression ratio of the original images [START_REF] Mulhern | Confounds in pictorial sets: The role of complexity and familiarity in basiclevel picture processing[END_REF]. This is true for computationally generated maps, but also for maps built from eye-tracking experiments. Interestingly, complexity estimation obtained using computational models of attention outperforms estimation obtained using heatmaps generated from eye-tracking experiments (ground truth attention maps).

JPEG compression of the heatmaps generated by our dynamical attention model (table 2) provides slightly lower results than the other attention models but compares well to eye tracking, This is probably due to the use of randomness in its dynamical equations (which injects additional complexity to the heatmaps generated).

However, if we measure the complexity of the trajectories generated by the model and its feedback based refined version (SLFE estimator), the performances increase and even exceed those obtained by the best saliency map based models.

CONCLUSION

This article proposes a new method for estimating the complexity of natural images. It is a difficult task that has been addressed by only few previous works.

We show that attentional maps, and in particular saliency maps and heatmaps generated by computational models of attention, can serve as a good estimator of image complexity. We also show that the dynamics of the focus points generated by our non centralized attention model can provide a better estimator of visual complexity. Since our computational model of attention is capable of handling video inputs, future experiments will focus on studying the validity of our approach for video complexity estimation with promising applications in, for example, automatic bit-rate adjustment of video streams during their broadcast.

Fig. 1 :

 1 Fig. 1: The subjective evaluation of visual complexity is a difficult task. Is (a) more complex than (b)? This question may be partially answered by observing their corresponding attention maps (c) and (d).

Fig. 2 :

 2 Fig. 2: ROC curves as an image complexity estimator.

Table 1 :

 1 Evaluation of attention based complexity estimators.

	Complexity estimator	-	JPEG	JPEG		-		JPEG	JPEG	JPEG
	Area under ROC curve	0.666	0.691	0.713		0.782		0.729	0.734	0.730
			Heatmap		Dynamical system
		Data	Perreira	Perreira	Perreira	Perreira	Perreira	Perreira
				feedback		feedback		feedback
	Complexity estimator	JPEG	JPEG	DCR	DCR	SLFE	SLFE
	Area under ROC curve	0.709	0.708	0.711	0.710	0.735	0.745

Table 2 :

 2 Evaluation of complexity estimation using our visual attention model.


Matthieu Perreira

Da Silva

Vincent Courboulay

Pascal Estraillier

OBJECTIVE VALIDATION OF A DYNAMICAL AND PLAUSIBLE COMPUTATIONAL MODEL OF VISUAL ATTENTION

Keywords: Visual attention, evaluation, objective measures, plausibility

Building a generic and highly capable vision system is still an open research problem. Actually, real-world vision systems need to face the challenge of dimensionality and ambiguity of data. To tackle this problem we introduced, in [1], a dynamic computational model of visual attention. This latter selects the most salient scene information while being able to adapt its behavior to the needs of a generic vision system. In this article, we focus on the objective validation of the plausibility of this original model. To check this property we compare (through three classical measures) the results obtained by several algorithms to an eye-tracking ground truth. Additionally, we study the influence of the model parameters on plausibility.

INTRODUCTION

Despite 40 years of research on computer vision, strong increase of computer processing power and huge improvements of image sensors quality (now 3D), building smart and generic computer vision system is still unreachable. «There are today numerous sophisticated methods for extracting visual information, but they seldom work consistently and robustly in the real, dynamically changing world» Eklundh et Christensen [START_REF] Eklundh | Computer vision: Past and future[END_REF] Yet, this general statement can be put into perspective when considering more specific application fields like face detection and tracking, industrial defect detection, scene reconstruction, motion analysis, content based information retrieval and so on. Actually, when some hypothesis related to the context can be stated, fine tuned algorithms, outperforming humans, can be built. However, as soon as a more general problem is considered (e.g. complex or changing environment) numerous problems arise : ambiguous data interpretation, combinatorial explosion due to data exploration, etc. Finding efficient mechanisms for selecting relevant features is one possible way that can be explored in order to tackle this problem.

The attentional system is one example of such a mechanism. It has been deeply studied by psychologists and neuro-psychologists since the 1950's. More recently, thanks to neurosciences progresses, computational attention models have emerged. These models help the study and understanding of specific aspects of human attention (see [START_REF] Itti | Neurobiology of attention[END_REF] for more details).

Real time execution constraints must also be taken into account when trying to adapt a computational attention model to computer vision. These facts lead to some questions:

• How can we make the attention model fast enough so that it remains a pre-processing ?

• To which extent should we imitate biological mechanism ?

• How can we integrate contextual information to attentional mechanisms ?

Concerning the last question, neuropsychologists have identified two complementary attentional pathways. The bottom-up pathway modulates attention according to the saliency of perceived stimuli, whereas the top-down pathway modulates attention according to the current context and objective. In [START_REF] Da Silva | Evaluation of preys / predators systems for visual attention simulation[END_REF] we introduced a dynamic and real-time computational bottom-up attention model which does not include a topdown pathway but is nevertheless highly tunable. In this paper, we focus on the objective validation of the plausibility of the model and the study of the influence of its main parameters.

In section 2, we propose a set of constraints for evaluating the capacity of an attentional model to be integrated in an intelligent computer vision system. In section 3, we briefly present the bottom-up visual attention model proposed in [START_REF] Da Silva | Evaluation of preys / predators systems for visual attention simulation[END_REF]. In section 4, we evaluate the model objectively through three quantified measures.

COMPUTATIONAL ATTENTION AND COMPUTER VISION

Computer vision needs

As previously mentioned, it is necessary to get some criteria in order to classify or evaluate algorithms. Usually, processing time or robustness are used. Nevertheless, we can closely adapt a set of constraints to computer vision applications. We have named this set PAIRED, it is composed of the following elements:

• Plausible when compared to human behavior;

• Adaptable to varying contexts;

• Invariant through different transforms (rotation, translation and scale);

• Rapid to compute the focus of attention;

• Extensible concerning its ability to take into account new characteristics;

• Dynamic and capable of producing results at any time.

Once these criteria presented, we illustrate their importance in several vision applications in the following Table 1. One dot means a weak constraint opposed to a strong one represented by three dots.

In the next sub-section, we briefly present two classical families of attention model in computer vision and put them in regards to the PAIRED criteria.

Classical models in computer vision

Central map base models

In the 80's, Anne Treisman proposed the Feature-Integration Theory [START_REF] Treisman | A Feature-Integration Theory of Attention[END_REF]. She stated that attention is encoded by an internal central map (named master map of locations, saliency map, etc.). Many works inherits this works, like hierarchical or algorithmic models.

Hierarchical models

These models compute several feature maps from an image and progressively combine them until providing a unique map usually named saliency map. One of the most important models belonging to this class is the one of Laurent Itti [START_REF] Itti | A model of saliency-based visual attention for rapid scene analysis[END_REF]. Many reasons explain its popularity among which we found :

• it is one of the first model of attention;

• it is based on an influential theory [START_REF] Treisman | A Feature-Integration Theory of Attention[END_REF];

• its biologically inspired architecture is efficient and simple to understand;

• thanks to the Neuromorphic Visionl Toolkit1 (iNVT), researchers have source code and software to either improve the model or compare with their own

• since its presentation in 1998, it is regularly improved.

We describe this model in more details in section 3 since it is the base of the hierarchical part of Perreira's model.

Algorithmic models.

These models are usually built for a specific application. For example, in the field of computer vision, an interesting approach is proposed by Aziz et Mertsching [START_REF] Aziz | Fast and robust generation of feature maps for region-based visual attention[END_REF]. They use the notion of proto-objects: images are segmented into regions of uniforms colors on which different features are computed (contrast, color, size, symmetry orientation). This results in a very fast model which may however be very sensitive to quality of the initial color segmentation process.

Distributed models

Another attention modeling way can be found in distributed models. These are based on works done by neuroscientists and connectionists and are usually neuromimetic [START_REF] Rolls | Invariant visual object recognition: a model, with lighting invariance[END_REF]. Experimental studies may then be as close as possible of in vivo experiments through specific signals measures. Most of related works are inspired by biased competition model proposed by Desimone and Duncan [START_REF] Desimone | Neural mechanisms of selective visual attention[END_REF] in the middle of 90's. For instance, Deco, Stringer and Rolls [START_REF] Deco | A Neurodynamical cortical model of visual attention and invariant object recognition[END_REF] and Ji and Weng [START_REF] Ji | Where-What Network 1 : Where and What Assist Each Other Through Top-down Connections[END_REF] propose to deeply mimic pathways information processing through two channels; the "what "pathway, responsible for recognition, and the "where" pathway that takes into account spatial attention.

Conclusion

Considering interesting properties of fidelity, invariance dynamic and adaptability of the distributed models and rapidity and extensibility of hierarchical models, an hybrid approach between these two model families seems to be promising. This solution has been partially developed through connectionist centralized models [START_REF] Ahmad | VISIT: An efficient computational model of human visual attention[END_REF][START_REF] Vitay | Biomimetic Neural Learning for Intelligent Robots[END_REF], actually they combined a hierarchical approach to produce a saliency map and a distributed model to generate the focus of attention. One of the main limitations is that competition between features (color, intensity and orientation) is done by the hierarchical part of the model. As a consequence, saliency sources are not competed. It is the main reason why we think that an "ideal" model (if it exists) do not have to represent saliency through a centralized map. We exploited this idea in [START_REF] Da Silva | Evaluation of preys / predators systems for visual attention simulation[END_REF] where we proposed that conspicuity maps should compete via a dynamical system (prey predator system for instance) and thus, be able to provide a dynamical focus of attention.

BOTTOM-UP COMPUTATIONNAL ATTENTION AND DYNAMICAL SYSTEMS

In [START_REF] Da Silva | Evaluation of preys / predators systems for visual attention simulation[END_REF], we proposed a new method which allows studying the temporal evolution of the visual focus of attention. We modified the classical algorithm proposed by Itti [START_REF] Itti | A model of saliency-based visual attention for rapid scene analysis[END_REF], in which the first part of his architecture relies on the extraction of three conspicuity maps based on low level computation. These three conspicuity maps are representative of the three main human perceptual channels: color, intensity and orientation. These low level computations are optimized following works presented in [START_REF] Frintrop | A real-time visual attention system using integral images[END_REF]. Actually the way to accelerate computation, (i.e. the use of integral images [START_REF] Viola | Robust real-time object detection[END_REF]) is reused and extended to all maps. The second part of Itti's architecture proposes a medium level system which allows merging conspicuity maps and then simulates a visual attention path on the observed scene. The focus is determined by a winner takes all and an inhibition of return algorithms. We have substituted this second part by a competitive dynamical system, in order to introduce a temporal parameter, which allows generating saccades, fixations and more realistic paths (figure 1). Preys-predators based dynamical systems are particularly well adapted for such a task. The main reasons are:

• preys-predators systems are dynamic, they intrinsically include time evolution of their activities. Thus, visual attention focus, seen as a predator, can evolve dynamically;

• without any objective (top down information or pregnancy), choosing a method for conspicuity maps fusion is hard. A solution consists in developing a competition between conspicuity maps and waiting for a nat-ural balance in the preys / predators system, reflecting the competition between emergence and inhibition of elements that engage or do not engage our attention;

• discrete dynamic systems can have a chaotic behavior. Despite the fact that this property is not often interesting, it is an important one for an attention model. Actually, it allows the emergence of original paths and an exploration of the visual scene, even in non salient areas, reflecting something like curiosity.

In our original article, we proposed a subjective validation, with different improvements, of Laurent Itti's visual attention model. In this article, we present an objective evaluation using three different measures that confirm our first conclusions about the plausibility of the method.

ORIGINAL MODEL PLAUSIBILITY

Comparison to existing models

We have performed an objective validation, related to plausibility, of the model introduced in [START_REF] Da Silva | Evaluation of preys / predators systems for visual attention simulation[END_REF]. This validation consists in checking the plausibility of the system, i.e. checking if it is apparently reasonably valid, and truthful. All measures were done on two image databases. The first one is proposed by Bruce2 [START_REF] Bruce | Saliency, attention, and visual search: An information theoretic approach[END_REF]. It is made up of 120 color images which represent streets, gardens, vehicles or buildings, more or less salients. The second one, proposed by Le Meur3 [START_REF] Le Meur | A coherent computational approach to model bottom-up visual attention[END_REF], contains 26 color images. They represent sport scenes, animals, building, indoor scenes or landscapes. For both databases, eye movements recordings were performed during a free viewing task.

In [START_REF] Da Silva | Evaluation of preys / predators systems for visual attention simulation[END_REF], we have presented a subjective evaluation of our model. We confirm this evaluation by a more "classical" objective evaluation. Cross-correlation, Kullback-Leibler divergence and normalized scanpath saliency were used to compare 6 algorithms to an eye-tracking ground-truth. The models evaluated were :

• two naïve models. "AllEqual" correspond to a constant saliency map, consider all points as equally salient. "Gaussian" model considers the central part of the image as the most salient area. Saliency is distributed using a centered gaussian distribution, scaled in order to cover all the image;

• Le Meur model [START_REF] Le Meur | A coherent computational approach to model bottom-up visual attention[END_REF], in its "coherent normalization" flavor;

• the AIM model of Bruce and Tsotsos [START_REF] Bruce | Saliency, attention, and visual search: An information theoretic approach[END_REF];

• the NVT model of Itti [START_REF] Itti | A model of saliency-based visual attention for rapid scene analysis[END_REF].

• Perreira's model (with fast retinal blur, see [START_REF] Da Silva | Evaluation of preys / predators systems for visual attention simulation[END_REF] for details).

All models were tested using their default parameters.

Figure 2 shows some sample saliency maps and heatmaps whereas table 4 is a summary of the performance of each algorithm over all the images of the two test databases. The analysis of the latter table leads to the following remarks :

• Kullback-Leibler divergence is sensitive to maps normalization : the "AllEqual" model seems to perform better than Itti's model whereas it obtains a null score with the two other measures;

• the "AllEqual" model is (quite unsurprisingly) the worst performer;

• despite its simplicity, the "Gaussian" model is quite a plausible model. This central preference is well known bias when evaluating computational attention models over eye-tracking data. It may be due to the type of images included in the databases, the experimental protocol, the photographer bias (which tends to center it's subject in the picture), or a real attentional bias against the central position;

• Model's performances are comparable to other state of the art models and even outperform them on Bruce database (NSS measure).

Influence of parameters

In the previous section, we have shown that our dynamical attention model is as plausible as other state of the art models. However, this model (and in particular its dynamical system) depends on some parameters. Table 3 summarizes the influences of some of these parameters on the plausibility of the model. The following conclusions can be drawn :

• using a retinal filter during the generation of feature and conspicuity maps improves plausibility significantly. This tends to prove that each new attentional focus depends on the location of the previous attentional focus; • using central biasing in an attention model can improve significantly its plausibility, but this bias is partly due to the experimental protocol [START_REF] Tatler | The central fixation bias in scene viewing : Selecting an optimal viewing position independently of motor biases and image feature distributions[END_REF];

• the dynamical system used in our attention model needs some diffusion in order to work correctly, but adding more diffusion does not improve plausibility;

• similarly, noise is an important factor for the plausibility of the model. However, the influence of noise on the repeatability of the system (variation in behavior between different runs) is still an open question.

CONCLUSION

In this article, we have presented an objective and quantified validation of a new computational attention model dedicated to intelligent vision. In a first part, we propose a new set of constraints (named PAIRED) for the evaluation of an attention model to a specific application. Considering the evaluation of classical models of visual attention, in regards of these constrains, we propose to take advantage of the qualities of both most popular model types (hierarchical and distributed). This initial bottom-up hybrid model, subjectively validated in [START_REF] Da Silva | Evaluation of preys / predators systems for visual attention simulation[END_REF] is now objectively validated through three measures. These measures are used on two set of images. Each of them provides oculometric measures that represent ground truth. We show that our dynamic model is plausible and that some of its parameters have a great influence on its plausibility. An extension of this work will be to study how feedback mechanisms can be exploited in order to reuse the outputs of a real computer vision system.

Fig. 1 .

 1 Fig. 1. Architecture of our bottom-up visual attention model. This diagram has been adapted from [5].

Fig. 2 .

 2 Fig. 2. Example of heatmaps and saliency maps generated by different computational attention models.

Fig. 4 .

 4 Fig. 4. Comparison to ground-truth : comparision of different algorithms. (Please note that for KLD, lower values mean more plausibility),

Table 1 .

 1 PAIRED constraints balance for some vision applications.

	Plausible Adaptable Invariant	Rapid	Extensible	Dynamic
	Attention modelling
	ergonomics/advert
	Vision
	CBIR
	Image processing
	Plausible Adaptable	Invariant	Rapid	Extensible	Dynamic
	Target
	Distributed
	Hierarchical
	Algorithmic

Table 2 .

 2 Set of constraints put in regards to computer vision applications. First row corresponds to our objective. Criteria reached or exceeded by models families are represented in green.

Available at : http://ilab.usc.edu/toolkit/.

Bruce database is available at http://www-sop.inria.fr/ members/Neil.Bruce/

Le Meur database is available at http://www.irisa.fr/ temics/staff/lemeur/visualAttention/