Multidimensional residues and ideal membership

Publicacions Matem`atiques, Vol 42 (1998), 143–152. MULTIDIMENSIONAL RESIDUES AND IDEAL MEMBERSHIP Alessandro Perotti Abstract Let I(f) be a zero-dimensional ideal in C[z1,... ,z n ] 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 gw.r.t. f.If the zeroes of fare 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 Cn. 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[z1,... ,z n] defined by a polynomial mapping f. In Section 2 we consider the problem of computing the logarithmic residue of a polynomial gwith respect to f. In the special case when the principal part of every component fiisapowerz k i i, we give a method in order to simplify the computation. We reduce it to The author is a member of the G.N.S.A.G.A. of C.N.R. Keywords. Multidimensional Residues, Local Residues, Integral Representations. 1991 Mathematics subject classifications: Primary: 32A27; Secondary: 32C30, 32A25. 144 A. Perotti the application, to only one special polynomial, of a linear functional introduced by Aizenberg [1] and to the finding of the projection of gonto a finite-dimensional subspace of C[z1,... ,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 fare 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 fto 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. 2. Logarithmic Residues 2.1. Let I=I(f)=(f 1 ,... ,f n) be a zero-dimensional polynomial ideal in C[z]=C[z 1 ,... ,z n]. This means that the zero set V(f)= V(f 1 ,... ,f n) is a discrete algebraic variety in Cn, with at most deg(f1)···deg(fn) points, 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 gwith respect to the mapping f=(f 1 ,... ,f n), that is the sum LResf(g)= N X ν=1 g(z(ν)). 2.2. We first consider the special case when fi=zki i+Pi,i=1,... ,n, where the total degree of Piis less than ki. 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 Cn: LResf(g)=N gJ z1···z n zk 1 1···zk n n deg(g) X |α|=0 (−1)|α|ÃP1 zk1 1!α1 ···µP n zk n n¶α n  Multidimensional Residues 145 where Jis the Jacobian determinant of the mapping fand Nis the linear functional on the polynomials in z1,... ,z nand 1/z1,... ,1/zn that assigns to each polynomial its free term. We show that the computation of LResf(g) can be simplified by exploiting the decomposition C[z]=C k−1 [z]⊕I, where Ck−1[z]isthe N-dimensional space of the polynomials in C[z] with degree less than ki with respect to zifor every i=1,... ,n. This follows from the particular form of the polynomials fi. In fact, it can be easily seen that f1,... ,f n is a Gr¨obner basis (not necessarily reduced) of the ideal Iwith respect to any degree ordering. Let zαdenote the monomial zα1 1···zα n n. Let K0(z,ζ)∈C[z,ζ]bea polynomial which belongs to Ck−1[z] for any fixed ζand to Ck−1[ζ] for any fixed zand has the following property: (∗) The set {Kα(ζ)}defined by the decomposition K0(z,ζ)= P α K α (ζ)z α is a basis of Ck−1[ζ]. Let K(z) = LResf(K0)=PαLResf(Kα)zα. Consider the non-degenerate bilinear form on C[z] defined for any p=Pαaαzα,q=Pαbαzα by hp, qiK=X α,β mα,βaαbβ where M=(m α,β) is the transition matrix from the basis {Kα}to the basis {zβ}0≤βi<ki. 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 h·,Ki Kevaluated on the (unique) projection g0of gin Ck−1[z]. Proof: If g=g0+g1∈Ck−1[z]⊕Iand g0=Pαaαzα=Pα,β mα,βaαKβ, then LResf(g) = LResf(g0)=Pα,β mα,βaαLResf(Kβ)=hg 0 ,Ki K. Two possible choices for the kernel K0(z,ζ) are the following: (i) K0(z,ζ)=P 0≤α i <kiQi(ziζi)αi, with associated form hp, qi= Pαaαbα; (ii) K0(z,ζ)=Qi (ζki i−zki i) (ζi−zi), with associated form hp,qi=Pαaαbk−α−1, where k−α−1 is the multiindex (k1−α1−1,... ,k n−α n−1). Remark. The second kernel is a Hefer determinant of the mapping Q=f−P=(z k 1 1,... ,zk n n). It is the determinant of the polynomial 146 A. Perotti matrix (Pij(z,ζ)) defined by the Hefer expansions Qi(ζ)−Qi(z)=X j P ij(z,ζ)(ζj−zj). Remark. If K0have integer coefficients, then the coefficients of K(z) are integer polynomial expressions in the coefficients of the fi. If the fihave integer, rational or real coefficients respectively, the same holds for K(z). 2.3. Let K0(z,ζ) be the kernel given in (i). If the polynomials fi have real coefficients, then hK, KiKis a real number greater than N2, since K(0) = N. It follows the decomposition C[z]=hKi⊕ ³C k−1 [z]∩hKi⊥´⊕I, where the second subspace is formed by the polynomials g∈Ck−1[z] such that LResf(g) = 0. Then the set of polynomials vanishing on V(f), that is the radical ideal Rad I, decomposes as Rad I= (Rad I∩Ck−1[z]) ⊕I with Rad I∩Ck−1[z]=ng∈hKi ⊥∩C k−1 [z]:(g l ) 0∈hKi ⊥ for every l=2,... ,No. Here (gl)0denotes the component of glin Ck−1[z]. Remark. Since hK0(a, ζ),K(ζ)i K=K(a), if Kis not the constant N we get that K0(a, ζ)∈hKi ⊥∩C k−1 [ζ] if and only if K(a)=0. 2.4. Now we return to the general case. Let f=(f 1 ,... ,f n)bea polynomial mapping with a discrete zero set V(f)={z (1),... ,z(N)}. Let ki= deg(fi) for i=1,... ,n. Then N≤k1···k n. We use an idea introduced by Aizenberg to reduce the general case to the previous case. If, for some i, the polynomial fihas the special form considered in Section 2.1, with principal part zki j, we set f0 j=fi. For the remaining indices, we set f0 i=zki+1 i+µfi,µ∈C. Let I0 µbe the ideal generated by f0 1,... ,f0 n. It has zero set V(f0) containing M= deg(f0 1)···deg(f0 n) points (with multiplicities), which we shall denote by z(1) µ,... ,z(M) µ.If fis not in the special form, than M>N. Multidimensional Residues 147 Let g∈C[z]. Let a=(a 1 ,... ,a n) be a vector of complex parameters and g0=g+Piaizi. For any fixed value of µ,f0has the special form considered in 2.1. Then we can compute the logarithmic residues LResf0((g0)l), l=1... ,M. These are polynomial expressions in µ, a1,... ,a n. From Newton’s formula, we can find the elementary symmetric functions σl g0(µ) in the quantities g0(z(1) µ),... ,g0(z(M) µ). It follows from Rouch´e’s principle (see [4, Section 2]) that Nelements of V(f0) tend to the points in V(f)asµ→∞, while the other M−N points tend to ∞. After reordering, we can assume that z(1) µ,... ,z(N) µ have limits z(1),... ,z(N)respectively. Let us denote by σl g0,l=1,... ,N, the elementary symmetric functions in g0(z(1)),... ,g0(z(N)). The polynomial g0can vanish identically (with respect to a) only in the point 0 and in this case g(0)=0. If 0∈V(f), then 0 ∈V(f0) with the same multiplicity h. Assume that z(1) µ=0,...,z(h) µ= 0. Let us denote by σ−l g0,l=1,... ,N −h, the elementary symmetric functions in g0(z(h+1))−1,... ,g0(z(N)) −1. Proposition 2. (i) σl g= lima→0σl g0for every l=1,... ,N; (ii) σl g0= limµ→∞ σM−N+l g0(µ) σM−N g0(µ)for every l=1,... ,N. Proof: (i) is immediate, since σl g0depends polynomially from a; for (ii), we adapt the arguments given in [4, Section 21.3]. If 0 /∈V(f) then σM g0(µ)6≡ 0. For all awith the exception of a set of complex dimension n−1, the ratios σM−l g0(µ)¡σM g0(µ)¢−1tend to 0 for l=N+1,... ,M, and to σ−l g0for l=1,... ,N. But the functions σl g0(µ) are polynomials in C(a)[µ] and therefore the ratios σM−l g0(µ)¡σM g0(µ)¢−1have limit in C(a), as µ→∞, equal to 0 for l=N+1,... ,M, and equal to σ−l g0for l=1,... ,N. Then σM−N+l g0(µ)³σM−N g0(µ)´−1tends to σ−N+l g0³σ−N g0´−1=σl g0for every l=1,... ,N. If 0 ∈V(f) with multiplicity h, then σl g0(µ)≡0 for l=M−h+ 1,... ,M, while σM−h g0(µ)6≡ 0. The ratios σM−h−l g0(µ)³σM−h g0(µ)´−1 tend to 0 for l=N−h+1,... ,M −h, and to σ−l g0for l=1,... ,N−h. In particular, σM−N g0(µ)³σM−h g0(µ)´−1has limit σ−N+h g06≡ 0, hence σM−N g0(µ)6≡ 0. 148 A. Perotti It remains to note that σl g0=σ−N+h+l g0³σ−N+h g0´−1for every l=1,... ,N −h. Remark. In general, the number Nis not known in advance. It can be determined from the previous limiting processes, by counting how many ratios σM−h−l g0(µ)³σM−h g0(µ)´−1tend to 0. Equivalently, it is the number of functions σM−h−l g0(µ) which have the same µ-degree as σM−h g0(µ). In particular, σ1 g= LResf(g). We have proved the following result. Theorem 1. The logarithmic residue of any g∈C[z]with respect to fcan be computed from LResf(g) = lim a→0lim µ→∞ σM−N+1 g0(µ) σM−N g0(µ). 3. 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 ain a closed neighbourhood Uaof a, the local (or Grothendieck) residue at aof a holomorphic function gon Uawith respect to fis the integral resa,f (g)= 1 (2πi)nZΓa(f) gdz 1∧···∧dzn f1···f n where Γa(f)isthen-chain = {z∈Ua:|fi(z)|=²i,i=1,... ,n}, with ²i>0 such that Γa(f) is relatively compact in Ua(see for example [5]). 3.1. Let I=(f 1 ,... ,f n) be a zero-dimensional polynomial ideal in C[z]. Since fhas a finite number of isolated zeroes, we can consider the global residue Resf(g)=Pa∈V(f)resa,f (g) of the local residues of g∈C[z] with respect to f. Remark. If g=h·J, where Jis the Jacobian determinant of the mapping f, the local residue coincides with the logarithmic residue of h at a. Then LResf(h) = Resf(h·J). If fhas the special form fi=zki i+Pi, with deg(Pi)<k i , the global residue Resf(g) can be computed from the explicit formula of Aizenberg [1]. Multidimensional Residues 149 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 Resf(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 fare all simple, then Resf(g)=PN ν=1 g(z(ν)) J(z(ν)). We can now apply the following lemma, which generalizes Newton’s formulas (for a proof, see for example [7]). Lemma 1. Let σl(a)denote the l-th elementary symmetric function of mscalars a1,...,a m.Ifb 1 ,... ,b mare scalars different from zero, the sum σ1¡a b¢=a1 b1+···+a m b mis given by σ1³a b´= m−1 X k=0 (−1)kσ1(abk)·σm−k−1(b) σm(b). Then we obtain the following formula: Resf(g)= N−1 X k=0 (−1)kσ1 g·Jk·σN−k−1 J σN J where σ1 g·Jkand σl Jcan be found from Proposition 2. 3.3. If not all the zeroes of fare simple, fcan be perturbed. We consider f−w, where wis a small complex n-tuple. For generic values of w, the Jacobian Jdoes 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 φ(w)= N X ν=1 g(z(ν)(w)) J(z(ν)(w)) is a holomorphic function in won a small neighbourhood of 0. Then φ(0) is the sum of the local residues of gat the zeroes of f. As a result, we obtain the following theorem. Theorem 2. The global residue Resf(g)of any g∈C[z]with respect to fis equal to ψ(0), where ψ(w)is the holomorphic function given by ψ(w)= N−1 X k=0 (−1)kLResf−w(g·Jk)·σN−k−1 J(w) σN J(w). 150 A. Perotti Here σl J(w) are the elementary symmetric functions in J(z(1)(w)),... ,J(z(N)(w)), which can be found from the logarithmic residues LResf−w(Jl), l=1,... ,N. 4. 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 LResfa univariate polynomial in I(f)∩C[zi] of degree Ncan be computed. It preserves multiplicities of the zeroes of f(for this method, see [4, Section 21]). From Resfa membership criterion for the ideal I(f) can be deduced. In [8], Tsikh applied Lasker-Noether Theorem and got the following: g∈I(f)⇔Resf(g(ζ)H(z,ζ))=0, where H(z,ζ) is a Hefer determinant of f. Remark. A polynomial Hefer determinant of fcan be computed from the Hefer expansions fi(ζ)−fi(z)=X j P ij(z,ζ)(ζj−zj) where Pij(z,ζ)=fi(ζ1,... ,ζ j,z j+1,... ,z n)−f i(ζ 1,... ,ζ j−1,z j,... ,z n) ζ j−z j . Note that from Pij(z,ζ) and Pij (ζ,z) we can get a Hefer determinant which is symmetric in zand ζ. 4.2. Let g,h ∈C[z] and g0(ζ) = Resf(g(z)H(z,ζ)), h0(ζ) = Resf(h(z)H(z,ζ)). From the membership criterion above we get that g0=h0if and only if the difference g−h∈I(f), that is gand hdefine 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, Multidimensional Residues 151 Resz−ζp(z) = Resf−f(ζ)(p(z)H(z,ζ)). It follows that for any a∈V(f), Resf(p(z)H(z,a)) = p(a). In particular, we get ResfH(z,a)=1. From this we can deduce that Resf(g0(z)H(z,ζ)) = Resf(g(z)H(z,ζ)) = g0(ζ). For simplicity, assume that the zeroes of fare simple. Then Resf(g0(z)H(z,ζ)) = X ν g0(zν)H(zν,ζ) J(zν) =X ν,µ g(zµ)H(zν,zµ)H(zν,ζ) J(zν)J(zµ) =X µ g(zµ) J(zµ)Resf(H(z,zµ)H(z,ζ)) = Resf(g(z)H(z,ζ)) = g0(ζ). As a result, we get the following proposition. Proposition 3. Let g∈C[z],g0(ζ) = Resf(g(z)H(z,ζ)). Then g−g0∈I(f), that is g0represents gin the quotient space C[z]/I(f).In particular, Resfg= Resfg0. Note added in proof. The paper by E. Cattani, A. Dickenstein, B. Sturmfels, Computing multidimensional residues, Algorithms in Algebraic Geometry and Applications (L. Gonzales-Vega and T. Recio, eds.), Progress in Mathematics, Vol. 143, Birkh¨auser Verlag, Basel, 1996, pp. 135–164, contains interesting relations between global residues and Gr¨obner bases and other references about these problems. References 1. L. A. Aizenberg, On a formula for the generalized multidimensional logarithmic residue and solutions of systems of nonlinear equations, Dokl. Akad. Nauk SSSR 234 (1977), 505–508; English transl., Soviet Math. Dokl. 18 (1977), 691–695. 2. L. A. Aizenberg and A. K. Tsikh, Application of the multidimensional logarithmic residue to systems of nonlinear algebraic equations, Sibirsk. Mat. Zh. 20 (1979), 699–707; English transl., Siberian Math. J. 20 (1979), 485-491.