MULTIDIMENSIONAL RESIDUES AND IDEAL MEMBERSHIP

Let I(f) be a zero-dimensional ideal in C[z1, . . . , zn] defined by a mapping f . We compute the logarithmic residue of a polynomial g with respect to f . We adapt an idea introduced by Aizenberg to reduce the computation to a special case by means of a limiting process. We then consider the total sum of local residues of g w.r.t. f . If the zeroes of f are simple, this sum can be computed from a finite number of logarithmic residues. In the general case, you have to perturb the mapping f . Some applications are given. In particular, the global residue gives, for any polynomial, a canonical representative in the quotient space C[z]/I(f).


Introduction
We present some algebraic applications of the theory of multidimensional residues in C n .The logarithmic residues and the local (or Grothendieck) residues have been studied by many authors.In particular, we consider some ideas of Aizenberg, Tsikh and Yuzhakov (see [3] or [6] for a survey).
Let I(f ) be a zero-dimensional ideal in C[z 1 , . . ., z n ] defined by a polynomial mapping f .In Section 2 we consider the problem of computing the logarithmic residue of a polynomial g with respect to f .In the special case when the principal part of every component f i is a power z ki i , we give a method in order to simplify the computation.We reduce it to the application, to only one special polynomial, of a linear functional introduced by Aizenberg [1] and to the finding of the projection of g onto a finite-dimensional subspace of C[z 1 , . . ., z n ].We also give a description of the radical of I.
In the general case, we adapt an idea introduced by Aizenberg to reduce to the special case by means of a limiting process (Proposition 2 and Theorem 1).
In Section 3 we consider the total sum of local residues of a polynomial with respect to the mapping f .If all the zeroes of f are simple, we show that this sum can be computed from a finite number of logarithmic residues.In the general case, you have to perturb the mapping f to get a similar result (Theorem 2).
In Section 4 we say something about the applications of these results.In particular, we show (Proposition 3) how the total sum of residues gives, for any polynomial, a canonical representative of its class in the quotient space C[z]/I(f ).
We wish to acknowledge the hospitality of the Mathematics Department of the Trento University.

Logarithmic Residues
counted with their multiplicities.Let z (1) , . . ., z (N ) be these (possibly repeated) points.Given a polynomial g ∈ C[z], we want to compute the logarithmic residue of g with respect to the mapping f = (f 1 , . . ., f n ), that is the sum 2.2.We first consider the special case when f i = z ki i + P i , i = 1, . . ., n, where the total degree of P i is less than k i .In this situation, the logarithmic residue is given by an explicit formula introduced by Aizenberg (see [1], [3], [4], [6]), which can be derived from the application of the Leray-Koppelman integral representation formula for holomorphic functions (see for example [4] Section 3) on a pseudoball in C n : where J is the Jacobian determinant of the mapping f and N is the linear functional on the polynomials in z 1 , . . ., z n and 1/z 1 , . . ., 1/z n that assigns to each polynomial its free term.We show that the computation of LRes f (g) can be simplified by exploiting the decomposition is the N -dimensional space of the polynomials in C[z] with degree less than k i with respect to z i for every i = 1, . . ., n.This follows from the particular form of the polynomials f i .In fact, it can be easily seen that f 1 , . . ., f n is a Gröbner basis (not necessarily reduced) of the ideal I with respect to any degree ordering.
Let z α denote the monomial Then we get the following result.
Proposition 1.The logarithmic residue of g ∈ C[z] with respect to f is given by the linear functional • , K K evaluated on the (unique) projection g 0 of g in Two possible choices for the kernel K 0 (z, ζ) are the following: Remark.The second kernel is a Hefer determinant of the mapping It is the determinant of the polynomial matrix (P ij (z, ζ)) defined by the Hefer expansions Remark.If K 0 have integer coefficients, then the coefficients of K(z) are integer polynomial expressions in the coefficients of the f i .If the f i have integer, rational or real coefficients respectively, the same holds for K(z).
where the second subspace is formed by the polynomials g ∈ C k−1 [z] such that LRes f (g) = 0. Then the set of polynomials vanishing on V (f ), that is the radical ideal Rad I, decomposes as 2.4.Now we return to the general case.Let f = (f 1 , . . ., f n ) be a polynomial mapping with a discrete zero set V (f ) = {z (1) , . . ., z (N ) }.
We use an idea introduced by Aizenberg to reduce the general case to the previous case.
If, for some i, the polynomial f i has the special form considered in section 2.1, with principal part z ki j , we set f j = f i .For the remaining indices, we set Let a = (a 1 , . . ., a n ) be a vector of complex parameters and g = g + i a i z i .For any fixed value of µ, f has the special form considered in 2.1.Then we can compute the logarithmic residues LRes f ((g ) l ), l = 1 . . ., M .These are polynomial expressions in µ, a 1 , . . ., a n .From Newton's formula, we can find the elementary symmetric functions σ l g (µ) in the quantities g (z µ ), . . ., g (z ).
Remark.In general, the number N is not known in advance.It can be determined from the previous limiting processes, by counting how many ratios σ M −h−l g (µ) which have the same µ-degree as σ M −h g (µ).In particular, σ 1 g = LRes f (g).We have proved the following result.Theorem 1.The logarithmic residue of any g ∈ C[z] with respect to f can be computed from

Local Residues
Now we consider the total sum of local residues of a polynomial g ∈ C[z] with respect to the polynomial mapping f = (f 1 , . . ., f n ).In general, if f = (f 1 , . . ., f n ) is a holomorphic mapping with an isolated zero a in a closed neighbourhood U a of a, the local (or Grothendieck) residue at a of a holomorphic function g on U a with respect to f is the integral . ., n}, with i > 0 such that Γ a (f ) is relatively compact in U a (see for example [5]).
Since f has a finite number of isolated zeroes, we can consider the global residue Res f (g) = a∈V (f ) res a,f (g) of the local residues of g ∈ C[z] with respect to f .Remark.If g = h • J, where J is the Jacobian determinant of the mapping f , the local residue coincides with the logarithmic residue of h at a. Then LRes If f has the special form f i = z ki i + P i , with deg(P i ) < k i , the global residue Res f (g) can be computed from the explicit formula of Aizenberg [1].
In the general case, Yuzhakov introduced in [9] an algorithm to reduce the problem to the special case, by applying the transformation formula for the local residue and the generalized resultants.We proceed in a different way.We obtain Res f (g) from the computation of a finite number of (global) logarithmic residues, which can be found with the method of Section 2.

3.2.
In the case that the zeroes z (1) , . . ., z (N ) of f are all simple, then Res . We can now apply the following lemma, which generalizes Newton's formulas (for a proof, see for example [7]).
Then we obtain the following formula: where σ 1 g•J k and σ l J can be found from Proposition 2.

3.3.
If not all the zeroes of f are simple, f can be perturbed.We consider f − w, where w is a small complex n-tuple.For generic values of w, the Jacobian J does not vanish at the zeroes of f − w.Let z (1) (w), . . ., z (N ) (w) be the elements of V (f − w).In [6], Section 6.2, Tsikh showed that the sum is a holomorphic function in w on a small neighbourhood of 0. Then φ(0) is the sum of the local residues of g at the zeroes of f .As a result, we obtain the following theorem.
Theorem 2. The global residue Res f (g) of any g ∈ C[z] with respect to f is equal to ψ(0), where ψ(w) is the holomorphic function given by Here σ l J (w) are the elementary symmetric functions in J(z (1) (w)),. . .,J(z (N ) (w)), which can be found from the logarithmic residues LRes f −w (J l ), l = 1, . . ., N .

Applications
4.1.The global residues and the total logarithmic residues have well known applications.They give a method for eliminating variables which does not use resultants.For any i = 1, . . ., n, from LRes f a univariate polynomial in I(f ) ∩ C[z i ] of degree N can be computed.It preserves multiplicities of the zeroes of f (for this method, see [4] Section 21).
From Res f a membership criterion for the ideal I(f ) can be deduced.In [8], Tsikh applied Lasker-Noether Theorem and got the following: where Note that from P ij (z, ζ) and P ij (ζ, z) we can get a Hefer determinant which is symmetric in z and ζ.

Let g, h ∈ C[z] and g
From the membership criterion above we get that g 0 = h 0 if and only if the difference g − h ∈ I(f ), that is g and h define the same class in the N -dimensional quotient space C[z]/I(f ).
If we apply the transformation formula for the global residue (see [8]) to the Hefer expansion of f , we get, for any polynomial p, Res be a polynomial which belongs to C k−1 [z] for any fixed ζ and to C k−1 [ζ] for any fixed z and has the following property: (*) the set {K α (ζ)} defined by the decomposition K 0