Introduction
Let us recall first two questions about effective constructions in Commutative Algebra and Algebraic Geometry where residue currents play a central role, both as a discovery tool and in the proofs. These are the Hilbert's Nullstellensatz and the construction of Arakelov measures.
The main idea of our work on this subject was to use the analytic theory of multidimensional residues and residue currents to find key identities and several explicit constructions. What we used repeatedly was the fact that residues could also be computed by analytic continuation of associated zeta functions. We refer to the short monograph [BGVY] for the details. We recall here just a few points.
Let us assume that f 1 , . . . , f n , g are holomorphic functions near the origin of C n , and assume that {f 1 = · · · = f n = 0} = {0}, then the residue of the meromorphic function g/f 1 · · · f n at z = 0, as defined by Poincaré, is given by
dζ, (1.1) dζ = dζ 1 ∧ . . . ∧ dζ n , where |f | = ε, for ε = (ε 1 . . . ε n ), denotes the cycle {|f 1 | = ε 1 , . . . , |f n | = ε n }. When the Jacobian J of the f j does not vanish at 0 we have
as expected. This definition of the residue in several variables was introduced by Jacobi at least for polynomials in [Ja2] . It has been extended in [CH] to define residue currents : namely, if we replace gdζ by a smooth compactly supported (n, 0) differential form ϕ, then the limit in (1.1) still exists provided 1 , ..., n approach 0 in an admissible way, and we may define a (0, n) current ∂(1/f ) as ∂ 1 f , ϕ := lim where c n = 0 is an absolute constant.
The way residues help in finding identities depend on the Abel-Jacobi vanishing theorem and its generalizations. The following result is due to Jacobi [Ja2] : let P 1 , . . . , P n be polynomials in C n without any common zeros at infinity and let Q be another polynomial satisfying the inequality deg Q ≤ deg P 1 + · · · + deg P n − n − 1 ; (1.5)
where
denotes the total sum of local residues
at all common zeroes α of P 1 , ..., P n in the affine space C n . In other words, the sum of all the residues of the meromorphic function Q/P 1 · · · P n vanishes. When all the zeros are simple we obtain Jacobi's original statement α
Q(α) J(α)
= 0.
We refer to [Gr] , [Ku2] , and [EGH] for interesting geometric applications of this theorem. A problem one often finds in trying to apply Jacobi's theorem is that given an ideal I in C[X 1 , ..., X n ] defining a zero-dimensional variety in C n , it may not be possible to find P 1 , . . . P n in I without common dwiley@example.net. What we can do is construct in some elementary way (that is essentially without using elimination theory) polynomials P 1 , . . . P n in such an ideal I so that the map ζ → P (ζ) = (P 1 (ζ), . . . , P n (ζ))
is a proper map of C n into itself. The properness condition is equivalent to a Lojasiewicz type inequality: there are constants K > 0, γ > 0, and δ > 0 with the property that if |z| > K then
Such a δ is called a Lojasiewicz exponent for P .
In the situation of Hilbert's Nullstellensatz, where we have a collection of polynomials p 1 , . . . , p m in C[X 1 , ..., X n ] without common zeros, we can find polynomials P 1 , . . . P n in the ideal generated by them that define a proper map and also have their degrees and size of their coefficients controlled by degrees and size of the coefficients of the original polynomials p j . In particular, it follows that there is an effective generalization of the vanishing theorem of Abel-Jacobi. Namely, there is a proper affine function θ : IN ). Note that a proper map usually has zeroes at ∞, this is the point that makes this statement a strong generalization of the Abel-Jacobi theorem. The proofs of (1.9) given in [BY1] , [BGVY] depend very heavily on the properties of residue currents.
In [BY1] we used the method just sketched to compute residues in C n and the generalized Abel-Jacobi theorem to obtain effective estimates on the solvability of the Nullstellensatz for polynomials p 1 , . . . , p M ∈ Z[X 1 , ..., X n ] without common zeros in C n . This was based on the previous work of Brownawell [Br1, Br2] , J. Heintz and its collaborators [CGH1] , J. Kollár [Ko] , who proved that in the above situation there exist polynomials q 1 , . . . , q M ∈ Z[X 1 , ..., X n ] and r 0 ∈ Z\{0} such that the equation
(1.10)
is satisfied while max
By itself, this bound does not produce sufficiently good estimates on the complexity of deciding whether the Bézout equation is solvable. One needs to obtain also a priori estimates on the (logarithmic) size of an "optimal" solution q 1 , . . . , q M , r 0 . In fact, it was shown in [BY1] 
where K(n) is a computable constant and h(q j ) denotes the maximum of the logarithms of the absolute values of the coefficients of q j . One can also replace Z by an arbitrary integral domain A equipped with a size function, irrespective of the characteristic of the corresponding quotient field K and corresponding algebraic closure K [BY4] . Though one may substitute algebraic tools from residue calculus [BY3, BY4] to analytic tools, such analytic methods provided [BY6] some insight respect to the following result recently obtained by M. Hickel [Hi] : let I be an ideal in C[X 1 , ..., X n ], I its integral closure in this ring, p 1 , . . . , p m a system of generators of I, and let
; then one has the following alternative :
• if n > m, for every p ∈ I there exist q j such that
The use of (1.4) to compute residues plays also a role in a subject that is of interest in Algebraic Geometry, Mathematical Physics, and Number Theory. We refer for example to the construction and the properties derived thereby of the so-called Green currents [BY8, BY5] . These currents appear in the work of Arakelov on intersection theory and its applications to Physics [Ar] , as well in work of Faltings [Fa] , Bost, Gillet, and Soulé [BGS] , and the very interesting lecture of McMullen [Mc] relating them to the Fermat's last theorem.
As we pointed it out in this introduction, Abel-Jacobi vanshing theorem, together with various connected tools from residue calculus in C n , plays a major role in our approach towards algebraic intersection or division problems. What we do in the body of the paper is to extend this theorem to the case where the underlying space is not C n . In fact, we replace C n by a q-dimensional irreducible algebraic subvariety W and, assuming that W is not included in the variety V (P ) of common zeros of a family P of m polynomials P 1 , ..., P m in C[X 1 , ..., X n ], we shall attach to this polynomial family P a family of non-trivial W -restricted residual currents in D
, with support on |W |. These currents (constructed using analytic ideas) inherit most of the properties that are fulfilled in the case W = C n . When the set |W | ∩ V (P ) is discrete and m = q, we prove that, for every point α ∈ |W | ∩ V (P ), the W -restricted analytic residue of a (q, 0)-form Rdζ I , R ∈ C[X 1 , ..., X n ], at the point α is the same as the residue on W (completion of W in Proj C[X 0 , ..., X n ]) at the point α in the sense of Serre (q = 1) or Kunz-Lipman (1 < q < n) of the q-differential form (R/P 1 · · · P q )dζ I . We present a restricted version of the affine version of Jacobi's residue formula (1.6) and obtain applications of this formula to higher dimensional analogues of the Reiss-Wood relations, corresponding to situations where the Zariski closures of |W | and V (P ) intersect at infinity in an arbitrary way. We expect this extended Jacobi residue to have as many useful applications to effective constructions in algebraic varieties as our previous work had for the Nullstellensatz.
Preliminaries
Let Γ be a complete integral curve embedded as a closed subscheme in Proj C[X 0 , ..., X n ] and C(Γ) its function field. Following the exposition of Hübl and Kunz of the Serre's approach [HK2] , the residue of a meromorphic (1, 0)-differential form ω ∈ Ω 1 C(Γ)/C at the point α ∈ Γ is defined as follows : let M 1 , . . . , M d be the minimal prime ideals of the completion O Γ,α of the local ring of Γ at α and let R j , j = 1, ..., d, be the integral closures of the "branches"
, of the curve Γ at the point α. Then R j is isomorphic to an algebra of formal power series C[[t j ]] and in C((t j )) the differential (1, 0)-form ω can be written as
where a j k ∈ C, k ≥ k j , are complex numbers which are independent of the parameters t j . Define
It was pointed by G. Biernat in [Bi] that, if f 1 , ..., f n are n germs of holomorphic functions in n variables (with Jacobian determinant J f ∈ O n ) such that (f 1 , ..., f n−1 ) define a germ of curve γ (with branches parametrized respectively by ϕ 1 , ..., ϕ d ) and dim [γ ∩ {f n J f = 0}] = 0, then, for any h ∈ O n , the Grothendieck residue
(with the orientation for the cycle
In particular, if ω denotes the (1, 0)-meromorphic differential form [Lej] ). This suggests a natural relation between the approaches developed by Serre-Hübl-Kunz and the analytic residue approach developed by Coleff-Herrera [CH] (which precisely allows the transposition of the definition of the Grothendieck residue in the complete intersection case to the setting of currents).
The analytic approach we use to define restricted residual currents on a qdimensional reduced analytic space Y ⊂ U , where U is an open subset of C n , will be described in section 3 as follows : if f 1 , ..., f m are m functions holomorphic in U , then the map 
is holomorphic at the origin. Its value at 0 defines, up to a multiplicative constant, a residual regular holonomic (n − q, n − q + k)-current which is supported by Y ∩ V (f ) ; regular holonomiticity is here understood in the sense of Björk ([Bj2] , chapter 9). Properties of such currents are similar to those introduced above in the case q = n. Proposition 3.1 will summarize the different properties of such restricted residual currents. The main case of interest for us will be the case where m ≤ q and dim (Y ∩ V (f )) ≤ q − m, that is, when f 1 , ..., f m define a complete intersection in Y. In this case, the restricted residue current corresponding to I := {1, ..., m} is the Coleff-
introduced in [CH] . It is not surprising that residual restricted currents in such a complete intersection setting obey the transformation law for multidimensional residue calculus ( [BGVY] , chapter 6), which we will prove (and use next) in the case m = q. If f 1 = P 1 , ..., f q = P q are polynomials and W is an affine q-dimensional algebraic subvariety of the affine scheme A n C such that dim (V (P ) ∩ |W |) = 0, we will prove in section 4 that the total sum of restricted residues
vanishes as soon as the degree of Q is sufficiently small, under a properness assumption on the restriction of (P 1 , ..., P q ) to |W |. We will thus transpose to the restricted case an Abel-Jacobi formula proved in the case q = n and W = A n C in [VY] . Let again W be a q-dimensional irreducible algebraic subvariety in the affine scheme
We will prove in section 5, thanks to the algebraic residue theorem in [L2] and the properties of restricted residual currents that were pointed out in previous sections, that the residue at a closed point α in |W | ∩ V (P ) (in the sense of Hübl or Lipman [L1] ) of the differential form ω (viewed as an element in Ω q C(W)/C ) equals
where ψ denotes a test-function with compact support in some small neighborhood of α, such that ψ ≡ 1 near α. The result is clear when α is a smooth point of W , it will follow from the algebraic residue formula combined with a perturbation argument in the case α is a singular point of W . As a consequence of the fact that the analytic and algebraic approaches lead to the same restricted residual objects, we will extend in section 5 (with an algebraic formulation) to such a restricted context the affine Jacobi's theorem obtained in the non-restricted case W = A n C in [VY] . 
We will derive (in sections 5 and 6) some consequences of this result in the spirit of Cayley-Bacharach's theorem and Wood's results [W] . The key point here (compare to the framework of [HK2] or [Ku2] ) is that the properness assumption along |W | (2.2) which is satisfied by the polynomial map P := (P 1 , ..., P q ) does not imply that the Zariski closures of |W | and
have an empty common intersection on the dwiley@example.net.
Restricted residual currents
We begin this section by recalling some basic facts about currents on analytic manifolds, especially integration currents on analytic sets or Coleff-Herrera currents and their "multiplication" with integration currents. We inspire ourselves on [Bj1] , [BY8] , [BY5] , and [Meo] .
We start with some basic facts about integration on a q-dimensional irreducible analytic subset Y in U ⊂ C n [Le] . The subset Y reg of regular points of Y is a q-dimensional complex manifold. The set of singular points Y sing is an analytic subset of U with complex dimension dim Y sing < q. Therefore for any smooth
It is known ([Bj1] , [Bj2] ) that this current [Y ] is a regular holonomic current, which implies, for each point z 0 in U ∩ Y , for any distribution coefficient T [Y ] of the integration current [Y ] , the existence of a Bernstein-Sato relation
is a real analytic function (and not a holomorphic one). Nevertheless, the existence of Bernstein-Sato relations of the form (3.1) remains valid here since f 2 has the particular form
where the T i 1 ,...,i n−q are (0, n−q) currents which are regular holonomic because of Meo, Bj1] ). One can then proceed in
with blocks of variables (ζ, ζ) and profit from the fact that formally ∂ ζ and ∂ ζ can be considered as derivations respect to independent sets of variables. Consider then the function of one complex variable defined by
This function (which is a D (n−q,n−q)
(U )-current valued function) is well defined and holomorphic in {λ ∈ C ; Re λ > 0}. Thanks to the Bernstein-Sato relations (3.1), it can be continued to the whole complex plane as a meromorphic function. The poles of this meromorphic extension are strictly negative rational numbers. Furthermore, there is a true pole at any point
In fact, we will need a more precise result, where the construction of the meromorphic continuation of (3.2) play a role. What we need is formulated in the following proposition.
., m} and for any ordered subset
I ⊂ {1, ..., m} with cardinal k ≤ min(q, m), the D (n−q,n−q+k) -valued map λ → λc k f 2(λ−k−1) [Y ] ∧ ∂ f 2 ∧ k l=1 (−1) l−1 f i l k j=1 j =l df i j (which is holomorphic in Re λ > k + 1) can
be continued as a meromorphic map to the whole complex plane, with no pole at
and that ν is the (Hilbert-Samuel) multiplicity of the ideal O U,y generated by g 1 , ..., g N at a generic point y ∈ Y . Let d = n − q. One can conclude from [Meo] that [Y ] coincides with the value at µ = 0 of the meromorphic D
In fact, in the general situation where (g 1 , ..., g N ) define a q-purely dimensional cycle Z (non necessarily irreducible) in U , the integration current (with multiplicities) on Z can be expressed as the value at λ = 0 of some meromorphic D (d,d) (U )-valued function which can be made explicit in terms of g 1 , ..., g N (see Theorem 3.1 in [BY5] for a proof in the algebraic case). Let I ⊂ {1, ..., m} with cardinal k ≤ min(q, m) and, for Re λ > k + 1,
In order to prove the proposition, we can localize the problem and assume that the origin belongs to Y ∩ V (f ). As in our previous work (see for example [BY8] , pages 32-33, or [BY5] , page 208) we construct an analytic n-
and
), such that in local coordinates on X (centered at a point x), one has, in the corresponding local chart U x around x,
where the u j , j = 1, ..., m and the v k , k = 1, ..., N , are non vanishing holomorphic functions in U x , at least one of the monomials t α j , j = 1, ..., m divides all of them (we will denote this monomial as t α ), and at least one of the monomials t β k , k = 1, ..., N divides all of them (we will denote this monomial as t β ). When ϕ is a (q, q − k)-test form with support in V , one has, for Re λ >> 0,
(the right hand side being continued as a meromorphic function of µ which has no pole at µ = 0). For λ fixed with Re λ >> 0, one can rewrite for Re µ >> 0 the integral
as a sum of integrals of the form
where ρ is a test-function in U x which corresponds to a partition of unity for π * (Supp ϕ). We know from Lemmas 2.1 and 2.2 in [BY5] that
is a positive ∂ and ∂-closed current θ U x in U x , which implies that, as soon as Re λ >> 0,
On the other hand, in U x and for Re λ >> 0, a straightforward computation leads to
, where ϑ and are smooth differential forms in U x (with respective types (0, k) and (0, k − 1)) and a is a strictly positive real analytic function in U x . It follows from Stokes' theorem that
One can see also that, for Re µ >> 0,
where b is a strictly positive real analytic function in U x , η (0,1) , η (1, 0) , υ are smooth differential forms in U x with respective types (0, 1) , (1, 0) and
Such a function of λ can be continued to a meromorphic function in the whole complex plane, with no pole at λ = 0 (using Stokes' theorem). The assertion of the proposition follows, since for Re λ >> 0,
is a sum of integrals of the form (3.4). ♦ Keeping the above notation one obtains the following corollary.
Corollary 3.1 Under the conditions of Proposition 3.1, the residual current defined by (3.3) has the following properties
one has
then one has
Proof. Let us now suppose that h satisfies (3.6). If we do not perform integration by parts as in (3.5), we have, for Re λ >> 0,
Condition (3.6) implies that there exists some positive constant κ such that, for any l = 1, ..., s with t i l coprime with t
which gives the first assertion of the corollary since 
has in fact no antiholomorphic singularity (therefore has a meromorphic extension which is polefree at the origin). It follows that for such h, one has again
which proves the remaining assertion of the corollary since again
is a sum of integrals of the form (3.4). ♦ When k = m ≤ q, we will use the simplified notation
The transformation law for residual currents can be transposed to the case of restricted residual currents. Since we deal in this paper with restricted residual currents supported by discrete sets, we state the transformation law in this particular setting. One has the following proposition :
Then, one has the following equality between restricted residual currents :
where
Proof. In order to prove this equality, we just need to prove it when U is a neighborhood (V ) whose coefficients are holomorphic in a neighborhood of α. Let ϕ be such a test form. Since
where
one has, by Stokes' theorem, that
Similarly, if we introduce
and
Let, for ξ ∈ [0, 1] and j = 1, ..., q,
Note that we have
Therefore, one has, since
this concludes the proof of the proposition. ♦ As a consequence of this result, we will state in the algebraic context the following analogue of the global transformation law. We need first some additional of notation. Assume that W is a q-dimensional irreducible algebraic subvariety in the affine space A n C (the integration current on |W | without multiplicities taken into account being denoted as [W ] ) and that P 1 , ..., P q are q elements in
.., n}, we will denote as
the result of the action of the W -restricted current
is an effective q-dimensional algebraic cycle in the affine space C n and P 1 , ..., P q are q polynomials such that W j ∩ V (P ) is discrete for any j = 1, ..., M , we will also denote as Another key point about the restricted residual current in the discrete context is the following annihilator property :
Proof. We give here a self-contained proof of the above proposition. Actually, because of the properties quoted in Corollary 3.1, it is enough to show that if α ∈ V (P ) ∩ Y and ϕ is a test-function with support arbitrarily small about α with ϕ = 1 in some neighborhood v α of α, then, for any function h ∈ C ∞ (U ) which is holomorphic on v α , for any ordered subset I = {i 1 , ..., i q } ⊂ {1, ..., n}, one has, for j = 1, ..., q,
One can use Stokes' formula (as in the proof of Proposition 3.2) and write
where
One can see at once that 
as it is defined in [CH] . The proof of this claim can be carried out as it was done in the non restricted case in [PTY] , section 5. Since the proof of this fact is rather tedious, we will not give it here. A consequence of this result is that, when f 1 , ..., f m (m ≤ q) define a complete intersection on Y , one has for k = 1, ..., m,
(see [CH] 
then formula (3.8) remains valid with m instead of q.
An Abel-Jacobi formula in the restricted case (analytic approach)
One of the key facts about restricted residual currents (as defined through the analytic approach described in section 3) is that they satisfy (in the 0-dimensional complete intersection setting) Abel-Jacobi's formula, exactly as in the non-restricted case (see [VY] ). Such a result will be, together with the validity of the transformation law in the restricted context) a crucial fact in order to compare our analytic approach and the algebraic one.
Proposition 4.1 Let W be a q-dimensional irreducible affine algebraic subvariety of the affine scheme A n C (0 < q < n) and P 1 , ..., P q be q polynomials in C[X 1 , ..., X n ] such that there exist strictly positive rational numbers δ 1 , ..., δ q and two constants K > 0, κ > 0 with :
Before we give the proof of this result, let us state an important corollary : 
.., q, denotes the homogeneization of the polynomial
Proof of Corollary 4.1. Assume that
where
is a finite set in C n ; this implies (through a compacity argument) that there exists K, κ > 0 such that, for any (ζ 0 , .
Condition (4.1) with δ j = D j , j = 1, ..., q, holds if we restrict to the affine space C n . The statement (4.4) follows then from (4.2). ♦ We remark that a proposition similar to Proposition 4.1 was proved in the non restricted case (W = A n C ) in [VY] . Unfortunately, the proof which is given there (and depends heavily on resolution of singularities on the analytic manifold IP n (C)) cannot immediately be transposed to the restricted case (since the Zariski closure |W| of |W | in IP n (C) is not a smooth manifold anymore). Instead, we will follow an alternative approach (applicable also for the case q = n), based on an argument in the affine space (and not in its compactification IP n (C)), which was proposed by Haï Zhang in [Z] . Our task has been to adapt this argument to the restricted case.
Note that, if z = Aw is a linear change of variables in C n , one has, for any element in D
Therefore, we do not loose generality is we assume that I = {1, ..., q} and that the projection Π :
is a proper map from |W | to C q (coordinates can be choosen in such a way that Noether normalization theorem applies respect to any (q, n−q) splitting ζ = (ζ , ζ ) of the set of variables (ζ 1 , ..., ζ n ), see for example [Fo, Ru] ). , j = 1, ..., q .
Similarly, for the polynomials P 1 , . . . , P q , one can define, in the affine open set
, j = 1, ..., q . 
These functions s
j , j = 1, ..., q, extend (provided N >> 1) to C 1 functions in C n \ V (P ), satisfying q j=1 s j (ζ)P j (ζ) = 1 , ζ ∈ C n \ V (P ) .
Let finally
Since the restriction S | |W | is a proper map (all δ j 's, j = 1, ..., q, being strictly positive) and V (P ) ∩ |W | is a discrete (hence finite) algebraic subset of C n (this follows also from (4.1)), there exists R 0 such that for R > R 0 , ϕ R ≡ 1 in a neighborhood of |W | ∩ V (P ). Therefore, if
one has (see for example formula (3.9))
for any R > R 0 , where dζ = q l=1 dζ l . It follows from an homotopy argument similar to the one which is developed in the proof of Proposition 3.2 that
(4.9)
For any order J ⊂ {1, ..., q}, let
then one can write
In order to prove formula (4.4), it is enough to prove that for any J ⊂ {1, ..., q}, one has, as soon as deg
(4.10)
Since the restriction of P = (P 1 , ..., P q ) to each connected sheet F (above the ζ -space) of the 2q-dimensional real manifold |W | reg is proper, the map
is a R 2q -valued proper map, with topological degree d J ,F . Moreover, condition (4.6) implies that, for R > K,
where C = C(θ, Q) is a positive constant. It follows then from the properness of all maps F J ,F and from the positivity of the differential form
which proves the conclusion (4.10) we need. The proof of Proposition 4.1 is therefore completed. ♦
Analytic versus algebraic approach
Let X be an integral C-variety of dimension q and D 1 ,...,D q be q Cartier divisors on X such that |D 1 | ∩ · · · ∩ |D q | is finite. If ω is a meromorphic form in Ω q C(X )/C which has a simple pole along D 1 + · · · + D q , one may define (see [Hu] , page 621) the local residue of ω at any closed point α in
where η ∈ ω q C(X )/C,α and f j = 0, j = 1, ..., q, is a local equation for D j at α then
When X is smooth, this definition agrees with the definition in [GH] , chapter 5, section 1. (See [L1] , Appendix A). Adding the hypothesis that X is Ccomplete, one has (see Proposition 12.2, page 108, in [L2] )
which is known as residue theorem on X (it extends the classical residue theorem on a complete integral curve in its algebraic formulation, see [Se] ).
Such a residue theorem holds in our analytic setting (and is essentially a consequence of Stokes' formula). Namely, if W is an integral algebraic q-
is finite and included in C n , then ( [W ] being understood as the integration current free of multiplicities),
On the other hand, the transformation law holds for our analytic restricted residue (see Corollary 3.2). Such a transformation law remains valid (in its local formulation) for restricted residue symbols defined through the algebraic approach (see Theorem 2.4 in [HK1] ).
Finally, the local residue symbol
where
C(X )/C,α and f j = 0, j = 1, ..., q, is a local equation for D j at α, equals to 0 as soon as η = f j η for some η ∈ ω q C(X )/C,α (see also [HK1] , section 2). The same annihilation property is satisfied by the restricted residual current (Proposition 3.3).
Our goal in this section is to profit from the fact that both restricted residual objects (defined through the algebraic or analytic approach) satisfy the transformation law, the residue formula, the annihilation property, in order to show that they coincide. Therefore, we are able to give an algebraic formulation of the Proposition 4.1, which is the Theorem 2.1 stated in our preliminaries section.
In order to do that, we will need the following technical lemma :
one has :
• any point α ∈ |W | ∩ V ( P (N,u,t) ) but 0 belongs to |W | reg ;
• the set
Proof. Since |W | is irreducible and q-dimensional, one has dim |W | sing < q ; one can find an algebraic affine hypersurface H := {ζ ∈ C n ; H(ζ) = 0} (with Zariski closure |H|) such that |W | sing ⊂ H and dim (|W| ∩ |H|) < q. ..., q, and N ≥ N 0 . Assume also that N ≥ ρ P,W (0), where ρ P,W (0) is the order of vanishing of P at the origin (along |W |). j = 1, ..., q, k = 1, . .., n, be a (q, n) matrix with generic complex entries,
Therefore, for such a generic choice of u (u = u 0 ) (this choice will be refined later), for any t ∈ C * , the polynomials
tersects |W| only at points in C n (note that 0 is one of these points). The algebraic set |W | ∩ Z (N,u 0 ,t) can be described as has been conveniently chosen, one can pick up t = 0 such that the map P (N,u 0 ,t) satisfies the assertions of the lemma. ♦
We can now relate the analytic and algebraic approaches for restricted residual symbols. 
with arbitrary small support around α satisfying ϕ ≡ 1 in a neighborhood of α, one has
Proof. One can assume for the sake of simplicity that α = 0. Let M be the maximal ideal (X 1 , ..., X n ) in the local algebra O C[X 1 ,...,X n ],0 and (I(W )) 0 the localization at 0 of the radical ideal
It follows from the validity of the transformation law and the annihilating property in the algebraic context that, if If the u jk , j = 1, ..., q, k = 1, ..., n are generic (see for example the construction in the proof of Lemma 5.1), the algebraic set V ( P ) ∩ |W| ∩ C n is discrete (hence finite). We can then conclude from (5.2) and (5.3) that in order to prove (5.1), it is not restrictive to assume that the algebraic set
is finite, what we will do from now on.
The same argument as above shows that, in order to prove (5.1), one can replace P j , j = 1, ..., q, by the polynomial 
whenever ϕ is a test-function in D(C n ) with arbitrary small support around the points α ∈ V (P ) ∩ |W|, such that ϕ ≡ 1 in a neighborhood of each of these points (ϕ α will denote next ϕ θ α , where θ α is a test-function with support arbitrary small around α and θ α ≡ 1 in a neighborhood of α). If α is any point in V (P ) ∩ |W| distinct from 0, W is smooth about α (Lemma 5.1, first assertion) and we know in this case that 5) since the construction of our restricted residual currents corresponds to the construction proposed in [GH] , chapter 5, section 1 (this is a consequence of the classical relation between Bochner-Martinelli and Cauchy kernels), which is known to fit with the algebraic approach in the smooth case (as it was recalled at the beginning of this section). Formula (5.1) follows then from (5.4) and from the identifications (5.5). ♦ Proof of Theorem 2.1. We may now transpose to the algebraic context the analytic result stated in Proposition 4.1. This gives the statement of the Theorem 2.1 of our introduction, provided we remember that we have 
intersects transversally γ 1 ,...,γ d at the respective points p 1 (α, β) 
for (α, β) close to (0, 0). The (necessary and sufficient) condition he gave can be formulated as follows : • there exists strictly positive constants κ, K such that
(necessarily distinct and close to the p j0 ) and one has ..., n , (6.8) where the h (l) k are germs of holomorphic functions in α about the origin and
condition (6.7) implies that, when (α, β) is sufficiently close to (0, 0), one has
This shows that for (α, β) close to (0, 0), the only points in 
for (α, β) such that (6.9) holds. It is immediate to check (use for example formula (4.7)) that for such (α, β), one has, for any multi
Then it follows from (6.9) that the right-hand side of (6.10) (hence the lefthand side) equals identically 0 when
This proves that, when (α, β) is close to (0, 0) and
is a polynomial expression in β = (β 1 , ..., β q−m ) with total degree at most ρ+1 (the coefficients being holomorphic functions in α). The second assertion of the proposition is proved. ♦ Remark. Note that we recover here as a particular case the necessity of Wood's and Q(X) := B(X n−m+1 , ..., X n ; X 1 , ..., X n−m ), one can check that the collection of all polynomials P l,α , l = n − m + 1, ..., n, together with the polynomial Q fits with the assertions (6.12) and (6.13). ♦ In the particular case m = 1, one can be more precise and repeat Wood's argument in order to obtain the following : .. = h P m = 0} corresponding to the complete intersection V (P ) that interpolates the pieces γ j does not hit |H ∞ |∩|L 0,0 |). We do not have the answer to that question yet. Nevertheless, Proposition 6.2 can be seen as an attempt to settle a converse to Proposition 6.1 in general.

{'paper_id': '8179002', 'title': 'Analytic residues along algebraic cycles', 's2_url': 'https://api.semanticscholar.org/CorpusID:8179002', 'source': 'S2ORC'}