Let I(f) be a zero-dimensional ideal in C[z1; : : : ; zn] de¯ned 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 ¯nite number of logarithmic residues. In the general case, you have to perturb the mapping f.Some applications are given. In particular, the global residu

Perotti, A.

Perotti, Alessandro

Diposit Digital de Documents de la UAB

           Publicacions Matemµatiques, Vol 42 (1998), 143{152.MULTIDIMENSIONAL RESIDUESAND IDEAL MEMBERSHIPAlessandro PerottiAbstractLet I(f) be a zero-dimensional ideal in C[z1; : : : ; zn] deflned by amapping f . We compute the logarithmic residue of a polynomial gwith respect to f . We adapt an idea introduced by Aizenberg toreduce the computation to a special case by means of a limitingprocess.We then consider the total sum of local residues of g w.r.t. f . Ifthe zeroes of f are simple, this sum can be computed from a flnitenumber of logarithmic residues. In the general case, you have toperturb the mapping f .Some applications are given. In particular, the global residuegives, for any polynomial, a canonical representative in the quo-tient space C[z]=I(f).IntroductionWe present some algebraic applications of the theory of multidimen-sional residues in Cn. The logarithmic residues and the local (orGrothendieck) residues have been studied by many authors. In partic-ular, 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; : : : ; zn] deflned by a poly-nomial mapping f . In Section 2 we consider the problem of computingthe logarithmic residue of a polynomial g with respect to f . In the spe-cial case when the principal part of every component fi is a power zkii ,we give a method in order to simplify the computation. We reduce it toThe 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 classiflcations: Primary: 32A27; Secondary: 32C30, 32A25.      144 A. Perottithe application, to only one special polynomial, of a linear functional in-troduced by Aizenberg [1] and to the flnding of the projection of g onto aflnite-dimensional subspace of C[z1; : : : ; zn]. We also give a descriptionof the radical of I.In the general case, we adapt an idea introduced by Aizenberg toreduce to the special case by means of a limiting process (Proposition 2and Theorem 1).In Section 3 we consider the total sum of local residues of a polynomialwith respect to the mapping f . If all the zeroes of f are simple, weshow that this sum can be computed from a flnite number of logarithmicresidues. In the general case, you have to perturb the mapping f to geta 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 residuesgives, for any polynomial, a canonical representative of its class in thequotient space C[z]=I(f).We wish to acknowledge the hospitality of the Mathematics Depart-ment of the Trento University.2. Logarithmic Residues2.1. Let I = I(f) = (f1; : : : ; fn) be a zero-dimensional polynomialideal in C[z] = C[z1; : : : ; zn]. This means that the zero set V (f) =V (f1; : : : ; fn) is a discrete algebraic variety in Cn, with atmost deg(f1) ¢ ¢ ¢deg(fn) points, counted with their multiplicities. Letz(1); : : : ; z(N) be these (possibly repeated) points. Given a polynomialg 2 C[z], we want to compute the logarithmic residue of g with respectto the mapping f = (f1; : : : ; fn), that is the sumLResf (g) =NX”=1g(z(”)):2.2. We flrst consider the special case when fi = zkii +Pi, i = 1; : : : ; n,where the total degree of Pi is less than ki. In this situation, the log-arithmic residue is given by an explicit formula introduced by Aizen-berg (see [1], [3], [4], [6]), which can be derived from the application ofthe Leray-Koppelman integral representation formula for holomorphicfunctions (see for example [4, Section 3]) on a pseudoball in Cn:LResf (g) = N0@gJ z1 ¢ ¢ ¢ znzk11 ¢ ¢ ¢ zknndeg(g)Xjfij=0(¡1)jfijˆP1zk11!fi1¢ ¢ ¢µPnzknn¶fin1A       Multidimensional Residues 145where J is the Jacobian determinant of the mapping f and N is thelinear functional on the polynomials in z1; : : : ; zn and 1=z1; : : : ; 1=znthat assigns to each polynomial its free term.We show that the computation of LResf (g) can be simplifled by ex-ploiting the decomposition C[z] = Ck¡1[z] ' I, where Ck¡1[z] is theN -dimensional space of the polynomials in C[z] with degree less than kiwith respect to zi for every i = 1; : : : ; n. This follows from the particularform of the polynomials fi. In fact, it can be easily seen that f1; : : : ; fnis a Gro˜bner basis (not necessarily reduced) of the ideal I with respectto any degree ordering.Let zfi denote the monomial zfi11 ¢ ¢ ¢ zfinn . Let K0(z; ‡) 2 C[z; ‡] be apolynomial which belongs to Ck¡1[z] for any flxed ‡ and to Ck¡1[‡] forany flxed z and has the following property:(⁄) The set fKfi(‡)g deflned by the decomposition K0(z; ‡) =PfiKfi(‡)zfi is a basis of Ck¡1[‡].Let K(z) = LResf (K0) =Pfi LResf (Kfi)zfi. Consider the non-degen-erate bilinear form on C[z] deflned for any p =Pfi afizfi, q =Pfi bfizfibyhp; qiK =Xfi;flmfi;flafibflwhere M = (mfi;fl) is the transition matrix from the basis fKfig to thebasis fzflg0•fli<ki .Then we get the following result.Proposition 1. The logarithmic residue of g 2 C[z] with respect to fis given by the linear functional h¢;KiK evaluated on the (unique) pro-jection g0 of g in Ck¡1[z].Proof: If g=g0+g12Ck¡1[z]'I and g0 =Pfi afizfi=Pfi;flmfi;flafiKfl ;then LResf (g) = LResf (g0) =Pfi;flmfi;flafi LResf (Kfl) = hg0;KiK .Two possible choices for the kernel K0(z; ‡) are the following:(i) K0(z; ‡) =P0•fii<kiQi(zi‡i)fii , with associated form hp; qi =Pfi afibfi;(ii) K0(z; ‡)=Qi(‡kii ¡zkii )(‡i¡zi) , with associated form hp;qi=Pfiafibk¡fi¡1;where k¡ fi¡ 1 is the multiindex (k1 ¡ fi1 ¡ 1; : : : ; kn ¡ fin ¡ 1).Remark. The second kernel is a Hefer determinant of the mappingQ = f ¡ P = (zk11 ; : : : ; zknn ). It is the determinant of the polynomial    146 A. Perottimatrix (Pij(z; ‡)) deflned by the Hefer expansionsQi(‡)¡Qi(z) =XjPij(z; ‡)(‡j ¡ zj):Remark. If K0 have integer coe–cients, then the coe–cients of K(z)are integer polynomial expressions in the coe–cients of the fi. If thefi have integer, rational or real coe–cients respectively, the same holdsfor K(z).2.3. Let K0(z; ‡) be the kernel given in (i). If the polynomials fihave real coe–cients, then hK;KiK is a real number greaterthan N2, since K(0) = N . It follows the decomposition C[z] = hKi '‡Ck¡1[z]\hKi?·' I, where the second subspace is formed by the poly-nomials g 2 Ck¡1[z] such that LResf (g) = 0. Then the set of poly-nomials vanishing on V (f), that is the radical ideal Rad I, decomposesasRad I = (Rad I \Ck¡1[z])' IwithRad I \Ck¡1[z] =ng 2 hKi? \Ck¡1[z] : (gl)0 2 hKi?for every l = 2; : : : ; No:Here (gl)0 denotes the component of gl in Ck¡1[z].Remark. Since hK0(a; ‡);K(‡)iK = K(a), if K is not the constant Nwe get that K0(a; ‡) 2 hKi? \Ck¡1[‡] if and only if K(a) = 0.2.4. Now we return to the general case. Let f = (f1; : : : ; fn) be apolynomial mapping with a discrete zero set V (f) = fz(1); : : : ; z(N)g.Let ki = deg(fi) for i = 1; : : : ; n. Then N • k1 ¢ ¢ ¢ kn.We use an idea introduced by Aizenberg to reduce the general case tothe previous case.If, for some i, the polynomial fi has the special form considered inSection 2.1, with principal part zkij , we set f0j = fi. For the remainingindices, we set f 0i = zki+1i + „fi, „ 2 C. Let I 0„ be the ideal generatedby f 01; : : : ; f0n. It has zero set V (f0) containing M = deg(f 01) ¢ ¢ ¢deg(f 0n)points (with multiplicities), which we shall denote by z(1)„ ; : : : ; z(M)„ . Iff is not in the special form, than M > N .      Multidimensional Residues 147Let g 2 C[z]. Let a = (a1; : : : ; an) be a vector of complex pa-rameters and g0 = g +Pi aizi. For any flxed value of „, f0 has thespecial form considered in 2.1. Then we can compute the logarithmicresidues LResf 0((g0)l), l = 1 : : : ;M . These are polynomial expressionsin „; a1; : : : ; an. From Newton’s formula, we can flnd the elementarysymmetric functions ¾lg0(„) in the quantities g0(z(1)„ ); : : : ; g0(z(M)„ ).It follows from Rouch¶e’s principle (see [4, Section 2]) that N elementsof V (f 0) tend to the points in V (f) as „ ! 1, while the other M ¡Npoints tend to 1. After reordering, we can assume that z(1)„ ; : : : ; z(N)„have limits z(1); : : : ; z(N) respectively.Let us denote by ¾lg0 , l = 1; : : : ; N , the elementary symmetric func-tions in g0(z(1)); : : : ; g0(z(N)). The polynomial g0 can vanish identically(with respect to a) only in the point 0 and in this case g(0) = 0. If0 2 V (f), then 0 2 V (f 0) with the same multiplicity h. Assume thatz(1)„ = 0, : : : , z(h)„ = 0. Let us denote by ¾¡lg0 , l = 1; : : : ; N ¡ h, theelementary symmetric functions in g0(z(h+1))¡1; : : : ; g0(z(N))¡1.Proposition 2. (i) ¾lg = lima!0 ¾lg0 for every l = 1; : : : ; N ;(ii) ¾lg0 = lim„!1¾M¡N+lg0 („)¾M¡Ng0 („)for every l = 1; : : : ; N .Proof: (i) is immediate, since ¾lg0 depends polynomially from a;for (ii), we adapt the arguments given in [4, Section 21.3]. If 0 =2 V (f)then ¾Mg0 („) 6· 0. For all a with the exception of a set of complex dimen-sion n¡1, the ratios ¾M¡lg0 („)¡¾Mg0 („)¢¡1 tend to 0 for l = N+1; : : : ;M ,and to ¾¡lg0 for l = 1; : : : ; N . But the functions ¾lg0(„) are polynomials inC(a)[„] and therefore the ratios ¾M¡lg0 („)¡¾Mg0 („)¢¡1 have limit in C(a),as „ ! 1, equal to 0 for l = N + 1; : : : ;M , and equal to ¾¡lg0 forl = 1; : : : ; N .Then ¾M¡N+lg0 („)‡¾M¡Ng0 („)·¡1tends to ¾¡N+lg0‡¾¡Ng0·¡1= ¾lg0 forevery l = 1; : : : ; N .If 0 2 V (f) with multiplicity h, then ¾lg0(„) · 0 for l = M ¡ h +1; : : : ;M , while ¾M¡hg0 („) 6· 0. The ratios ¾M¡h¡lg0 („)‡¾M¡hg0 („)·¡1tend to 0 for l=N ¡ h + 1; : : : ;M ¡ h, and to ¾¡lg0 for l= 1; : : : ; N¡h.In particular, ¾M¡Ng0 („)‡¾M¡hg0 („)·¡1has limit ¾¡N+hg0 6· 0, hence¾M¡Ng0 („) 6· 0.      148 A. PerottiIt remains to note that ¾lg0 = ¾¡N+h+lg0‡¾¡N+hg0·¡1for everyl = 1; : : : ; N ¡ h.Remark. In general, the number N is not known in advance. Itcan be determined from the previous limiting processes, by countinghow many ratios ¾M¡h¡lg0 („)‡¾M¡hg0 („)·¡1tend to 0. Equivalently, itis the number of functions ¾M¡h¡lg0 („) which have the same „-degree as¾M¡hg0 („).In particular, ¾1g = LResf (g). We have proved the following result.Theorem 1. The logarithmic residue of any g 2 C[z] with respectto f can be computed fromLResf (g) = lima!0lim„!1¾M¡N+1g0 („)¾M¡Ng0 („):3. Local ResiduesNow we consider the total sum of local residues of a polynomialg 2 C[z] with respect to the polynomial mapping f = (f1; : : : ; fn). Ingeneral, if f = (f1; : : : ; fn) is a holomorphic mapping with an isolatedzero a in a closed neighbourhood Ua of a, the local (or Grothendieck)residue at a of a holomorphic function g on Ua with respect to f is theintegralresa;f (g) =1(2…i)nZ¡a(f)g dz1 ^ ¢ ¢ ¢ ^ dznf1 ¢ ¢ ¢ fnwhere ¡a(f) is the n-chain = fz 2 Ua : jfi(z)j = †i; i = 1; : : : ; ng, with†i > 0 such that ¡a(f) is relatively compact in Ua (see for example [5]).3.1. Let I = (f1; : : : ; fn) be a zero-dimensional polynomial ideal inC[z]. Since f has a flnite number of isolated zeroes, we can considerthe global residue Resf (g) =Pa2V (f) resa;f (g) of the local residues ofg 2 C[z] with respect to f .Remark. If g = h ¢ J , where J is the Jacobian determinant of themapping f , the local residue coincides with the logarithmic residue of hat a. Then LResf (h) = Resf (h ¢ J).If f has the special form fi = zkii + Pi, with deg(Pi) < ki, the globalresidue Resf (g) can be computed from the explicit formula of Aizen-berg [1].      Multidimensional Residues 149In the general case, Yuzhakov introduced in [9] an algorithm to reducethe problem to the special case, by applying the transformation formulafor the local residue and the generalized resultants. We proceed in a dif-ferent way. We obtain Resf (g) from the computation of a flnite numberof (global) logarithmic residues, which can be found with the method ofSection 2.3.2. In the case that the zeroes z(1); : : : ; z(N) of f are all simple, thenResf (g) =PN”=1g(z(”))J(z(”)). We can now apply the following lemma, whichgeneralizes Newton’s formulas (for a proof, see for example [7]).Lemma 1. Let ¾l(a) denote the l-th elementary symmetric functionof m scalars a1, : : : , am. If b1; : : : ; bm are scalars difierent from zero,the sum ¾1¡ab¢= a1b1 + ¢ ¢ ¢+ ambm is given by¾1‡ab·=m¡1Xk=0(¡1)k ¾1(abk) ¢ ¾m¡k¡1(b)¾m(b):Then we obtain the following formula:Resf (g) =N¡1Xk=0(¡1)k¾1g¢Jk ¢ ¾N¡k¡1J¾NJwhere ¾1g¢Jk and ¾lJ can be found from Proposition 2.3.3. If not all the zeroes of f are simple, f can be perturbed. Weconsider f ¡ w, where w is a small complex n-tuple. For generic val-ues of w, the Jacobian J does not vanish at the zeroes of f ¡ w. Letz(1)(w); : : : ; z(N)(w) be the elements of V (f ¡ w). In [6, Section 6.2],Tsikh showed that the sum`(w) =NX”=1g(z(”)(w))J(z(”)(w))is a holomorphic function in w on a small neighbourhood of 0. Then`(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 Resf (g) of any g 2 C[z] with respectto f is equal to ˆ(0), where ˆ(w) is the holomorphic function given byˆ(w) =N¡1Xk=0(¡1)kLResf¡w(g ¢ Jk) ¢ ¾N¡k¡1J (w)¾NJ (w):     150 A. PerottiHere ¾lJ(w) are the elementary symmetric functions inJ(z(1)(w)); : : : ; J(z(N)(w));which can be found from the logarithmic residues LResf¡w(J l),l = 1; : : : ; N .4. Applications4.1. The global residues and the total logarithmic residues have wellknown applications. They give a method for eliminating variables whichdoes not use resultants. For any i = 1; : : : ; n, from LResf a univariatepolynomial in I(f) \ C[zi] of degree N can be computed. It preservesmultiplicities of the zeroes of f (for this method, see [4, Section 21]).From Resf a membership criterion for the ideal I(f) can be deduced.In [8], Tsikh applied Lasker-Noether Theorem and got the following:g 2 I(f), Resf (g(‡)H(z; ‡)) = 0;where H(z; ‡) is a Hefer determinant of f:Remark. A polynomial Hefer determinant of f can be computedfrom the Hefer expansionsfi(‡)¡ fi(z) =XjPij(z; ‡)(‡j ¡ zj)where Pij(z; ‡)=fi(‡1; : : : ; ‡j ; zj+1; : : : ; zn)¡fi(‡1; : : : ; ‡j¡1; zj ; : : : ; zn)‡j ¡ zj :Note that from Pij(z; ‡) and Pij(‡; z) we can get a Hefer determinantwhich is symmetric in z and ‡.4.2. Let g; h 2 C[z] andg0(‡) = Resf (g(z)H(z; ‡));h0(‡) = Resf (h(z)H(z; ‡)):From the membership criterion above we get that g0 = h0 if and only ifthe difierence g ¡ h 2 I(f), that is g and h deflne the same class in theN -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 151Resz¡‡ p(z) = Resf¡f(‡)(p(z)H(z; ‡)). It follows that for any a 2 V (f),Resf (p(z)H(z; a)) = p(a). In particular, we get Resf H(z; a) = 1.From this we can deduce thatResf (g0(z)H(z; ‡)) = Resf (g(z)H(z; ‡)) = g0(‡):For simplicity, assume that the zeroes of f are simple. ThenResf (g0(z)H(z; ‡)) =X”g0(z”)H(z” ; ‡)J(z”)=X”;„g(z„)H(z” ; z„)H(z” ; ‡)J(z”)J(z„)=X„g(z„)J(z„)Resf (H(z; z„)H(z; ‡))= Resf (g(z)H(z; ‡)) = g0(‡):As a result, we get the following proposition.Proposition 3. Let g 2 C[z], g0(‡) = Resf (g(z)H(z; ‡)). Theng¡ g0 2 I(f), that is g0 represents g in the quotient space C[z]=I(f). Inparticular, Resf g = Resf g0.Note added in proof. The paper by E. Cattani, A. Dickenstein,B. Sturmfels, Computing multidimensional residues, Algorithms in Al-gebraic Geometry and Applications (L. Gonzales-Vega and T. Recio,eds.), Progress in Mathematics, Vol. 143, Birkha˜user Verlag, Basel, 1996,pp. 135{164, contains interesting relations between global residues andGro˜bner bases and other references about these problems.References1. L. A. Aizenberg, On a formula for the generalized multidimen-sional logarithmic residue and solutions of systems of nonlinearequations, Dokl. Akad. Nauk SSSR 234 (1977), 505{508; Englishtransl., Soviet Math. Dokl. 18 (1977), 691{695.2. L. A. Aizenberg and A. K. Tsikh, Application of the multi-dimensional logarithmic residue to systems of nonlinear algebraicequations, Sibirsk. Mat. Zh. 20 (1979), 699{707; English transl.,Siberian Math. J. 20 (1979), 485-491.      152 A. Perotti3. L. A. Aizenberg, A. K. Tsikh and A. P. Yuzhakov, Multidi-mensional Residues and Applications, in \Several Complex VariablesII," (G. M. Khenkin and A. G. Vitushkin, eds.), Encyclopaedia ofMath. Sc. 8, Springer-Verlag, Berlin, Heidelberg, New-York, 1990,pp. 1{58.4. L. A. Aizenberg and A. P. Yuzhakov, \Integral Representa-tions and Residues in Multidimensional Complex Analysis," Nauka,Novosibirsk, 1979; English transl., Translations of MathematicalMonographs 58, Amer. Math. Soc., Providence, 1983.5. P. Griffiths and J. Harris, \Principles of Algebraic Geometry,"Wiley, New York, Chichester, 1978.6. A. K. Tsikh, \Multidimensional Residues and Their Applications,"Nauka, Novosibirsk, 1988; English transl., Translations of Mathe-matical Monographs 103, Amer. Math. Soc., Providence, 1992.7. A. K. Tsikh, \Residues with respect to holomorphic mappings andtheir applications," Krasnoyarsk Univ. Press, Krasnoyarsk, 1986,(Russian).8. A. K. Tsikh and A. P. Yuzhakov, Properties of the total sum ofresidues with respect to a polynomial map and applications of them,Sibirsk. Mat. Zh. 25 (1984), 207{213; English transl., Siberian Math.J. 25 (1984), 677{682.9. A. P. Yuzhakov, On the computation of the total sum of residuesrelative to a polynomial mapping in Cn, Dokl. Akad. Nauk SSSR275 (1984), 817{820; English transl., Soviet Math. Dokl. 29 (1984),321{324.Dipartimento di MatematicaUniversitµa di MilanoVia Saldini 50I-20133 MilanoITALYe-mail: perotti@vmimat.mat.unimi.itPrimera versi¶o rebuda el 30 de gener de 1997,darrera versi¶o rebuda el 4 de novembre de 1997

Multidimensional residues and ideal membership

Let $I(f)$ be a zero-dimensional ideal in $\bold C[z_1,\ldots,z_n]$ 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 $g$ 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 $\bold C[z]/I(f)$

Revistes Catalanes amb Accés Obert

           
Publicacions Matemµatiques, Vol 42 (1998), 143{152.
MULTIDIMENSIONAL RESIDUES
AND IDEAL MEMBERSHIP
Alessandro Perotti
Abstract
Let I(f) be a zero-dimensional ideal in C[z1; : : : ; zn] deflned 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 flnite
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 quo-
tient space C[z]=I(f).
Introduction
We present some algebraic applications of the theory of multidimen-
sional residues in Cn. The logarithmic residues and the local (or
Grothendieck) residues have been studied by many authors. In partic-
ular, 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; : : : ; zn] deflned by a poly-
nomial mapping f . In Section 2 we consider the problem of computing
the logarithmic residue of a polynomial g with respect to f . In the spe-
cial case when the principal part of every component fi is a power zkii ,
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 classiflcations: Primary: 32A27; Secondary: 32C30, 32A25.
      
144 A. Perotti
the application, to only one special polynomial, of a linear functional in-
troduced by Aizenberg [1] and to the flnding of the projection of g onto a
flnite-dimensional subspace of C[z1; : : : ; zn]. 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 flnite 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 Depart-
ment of the Trento University.
2. Logarithmic Residues
2.1. Let I = I(f) = (f1; : : : ; fn) be a zero-dimensional polynomial
ideal in C[z] = C[z1; : : : ; zn]. This means that the zero set V (f) =
V (f1; : : : ; fn) 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 2 C[z], we want to compute the logarithmic residue of g with respect
to the mapping f = (f1; : : : ; fn), that is the sum
LResf (g) =
NX
”=1
g(z(”)):
2.2. We flrst consider the special case when fi = zkii +Pi, i = 1; : : : ; n,
where the total degree of Pi is less than ki. In this situation, the log-
arithmic residue is given by an explicit formula introduced by Aizen-
berg (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
0@gJ z1 ¢ ¢ ¢ zn
zk11 ¢ ¢ ¢ zknn
deg(g)X
jfij=0
(¡1)jfij
ˆ
P1
zk11
!fi1
¢ ¢ ¢
µ
Pn
zknn
¶fin1A
       
Multidimensional Residues 145
where J is the Jacobian determinant of the mapping f and N is the
linear functional on the polynomials in z1; : : : ; zn and 1=z1; : : : ; 1=zn
that assigns to each polynomial its free term.
We show that the computation of LResf (g) can be simplifled by ex-
ploiting the decomposition C[z] = Ck¡1[z] ' I, where Ck¡1[z] is the
N -dimensional space of the polynomials in C[z] with degree less than ki
with respect to zi for every i = 1; : : : ; n. This follows from the particular
form of the polynomials fi. In fact, it can be easily seen that f1; : : : ; fn
is a Gro˜bner basis (not necessarily reduced) of the ideal I with respect
to any degree ordering.
Let zfi denote the monomial zfi11 ¢ ¢ ¢ zfinn . Let K0(z; ‡) 2 C[z; ‡] be a
polynomial which belongs to Ck¡1[z] for any flxed ‡ and to Ck¡1[‡] for
any flxed z and has the following property:
(⁄) The set fKfi(‡)g deflned by the decomposition K0(z; ‡) =P
fiKfi(‡)z
fi is a basis of Ck¡1[‡].
Let K(z) = LResf (K0) =
P
fi LResf (Kfi)z
fi. Consider the non-degen-
erate bilinear form on C[z] deflned for any p =
P
fi afiz
fi, q =
P
fi bfiz
fi
by
hp; qiK =
X
fi;fl
mfi;flafibfl
where M = (mfi;fl) is the transition matrix from the basis fKfig to the
basis fzflg0•fli<ki .
Then we get the following result.
Proposition 1. The logarithmic residue of g 2 C[z] with respect to f
is given by the linear functional h¢;KiK evaluated on the (unique) pro-
jection g0 of g in Ck¡1[z].
Proof: If g=g0+g12Ck¡1[z]'I and g0 =
P
fi afiz
fi=
P
fi;flmfi;flafiKfl ;
then LResf (g) = LResf (g0) =
P
fi;flmfi;flafi LResf (Kfl) = hg0;KiK .
Two possible choices for the kernel K0(z; ‡) are the following:
(i) K0(z; ‡) =
P
0•fii<ki
Q
i(zi‡i)
fii , with associated form hp; qi =P
fi afibfi;
(ii) K0(z; ‡)=
Q
i
(‡kii ¡zkii )
(‡i¡zi) , with associated form hp;qi=
P
fiafibk¡fi¡1;
where k¡ fi¡ 1 is the multiindex (k1 ¡ fi1 ¡ 1; : : : ; kn ¡ fin ¡ 1).
Remark. The second kernel is a Hefer determinant of the mapping
Q = f ¡ P = (zk11 ; : : : ; zknn ). It is the determinant of the polynomial
    
146 A. Perotti
matrix (Pij(z; ‡)) deflned by the Hefer expansions
Qi(‡)¡Qi(z) =
X
j
Pij(z; ‡)(‡j ¡ zj):
Remark. If K0 have integer coe–cients, then the coe–cients of K(z)
are integer polynomial expressions in the coe–cients of the fi. If the
fi have integer, rational or real coe–cients respectively, the same holds
for K(z).
2.3. Let K0(z; ‡) be the kernel given in (i). If the polynomials fi
have real coe–cients, then hK;KiK is a real number greater
than N2, since K(0) = N . It follows the decomposition C[z] = hKi '‡
Ck¡1[z]\hKi?
·
' I, where the second subspace is formed by the poly-
nomials g 2 Ck¡1[z] such that LResf (g) = 0. Then the set of poly-
nomials 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] =
n
g 2 hKi? \Ck¡1[z] : (gl)0 2 hKi?
for every l = 2; : : : ; N
o
:
Here (gl)0 denotes the component of gl in Ck¡1[z].
Remark. Since hK0(a; ‡);K(‡)iK = K(a), if K is not the constant N
we get that K0(a; ‡) 2 hKi? \Ck¡1[‡] if and only if K(a) = 0.
2.4. Now we return to the general case. Let f = (f1; : : : ; fn) be a
polynomial mapping with a discrete zero set V (f) = fz(1); : : : ; z(N)g.
Let ki = deg(fi) for i = 1; : : : ; n. Then N • k1 ¢ ¢ ¢ kn.
We use an idea introduced by Aizenberg to reduce the general case to
the previous case.
If, for some i, the polynomial fi has the special form considered in
Section 2.1, with principal part zkij , we set f
0
j = fi. For the remaining
indices, we set f 0i = z
ki+1
i + „fi, „ 2 C. Let I 0„ be the ideal generated
by f 01; : : : ; f
0
n. It has zero set V (f
0) containing M = deg(f 01) ¢ ¢ ¢deg(f 0n)
points (with multiplicities), which we shall denote by z(1)„ ; : : : ; z
(M)
„ . If
f is not in the special form, than M > N .
      
Multidimensional Residues 147
Let g 2 C[z]. Let a = (a1; : : : ; an) be a vector of complex pa-
rameters and g0 = g +
P
i aizi. For any flxed value of „, f
0 has the
special form considered in 2.1. Then we can compute the logarithmic
residues LResf 0((g0)
l), l = 1 : : : ;M . These are polynomial expressions
in „; a1; : : : ; an. From Newton’s formula, we can flnd the elementary
symmetric functions ¾lg0(„) in the quantities g
0(z(1)„ ); : : : ; g0(z
(M)
„ ).
It follows from Rouch¶e’s principle (see [4, Section 2]) that N elements
of V (f 0) tend to the points in V (f) as „ ! 1, while the other M ¡N
points tend to 1. After reordering, we can assume that z(1)„ ; : : : ; z(N)„
have limits z(1); : : : ; z(N) respectively.
Let us denote by ¾lg0 , l = 1; : : : ; N , the elementary symmetric func-
tions in g0(z(1)); : : : ; g0(z(N)). The polynomial g0 can vanish identically
(with respect to a) only in the point 0 and in this case g(0) = 0. If
0 2 V (f), then 0 2 V (f 0) with the same multiplicity h. Assume that
z
(1)
„ = 0, : : : , z
(h)
„ = 0. Let us denote by ¾¡lg0 , l = 1; : : : ; N ¡ h, the
elementary symmetric functions in g0(z(h+1))¡1; : : : ; g0(z(N))¡1.
Proposition 2. (i) ¾lg = lima!0 ¾
l
g0 for every l = 1; : : : ; N ;
(ii) ¾lg0 = lim„!1
¾M¡N+lg0 („)
¾M¡Ng0 („)
for every l = 1; : : : ; N .
Proof: (i) is immediate, since ¾lg0 depends polynomially from a;
for (ii), we adapt the arguments given in [4, Section 21.3]. If 0 =2 V (f)
then ¾Mg0 („) 6· 0. For all a with the exception of a set of complex dimen-
sion n¡1, the ratios ¾M¡lg0 („)
¡
¾Mg0 („)
¢¡1 tend to 0 for l = N+1; : : : ;M ,
and to ¾¡lg0 for l = 1; : : : ; N . But the functions ¾
l
g0(„) are polynomials in
C(a)[„] and therefore the ratios ¾M¡lg0 („)
¡
¾Mg0 („)
¢¡1 have limit in C(a),
as „ ! 1, equal to 0 for l = N + 1; : : : ;M , and equal to ¾¡lg0 for
l = 1; : : : ; N .
Then ¾M¡N+lg0 („)
‡
¾M¡Ng0 („)
·¡1
tends to ¾¡N+lg0
‡
¾¡Ng0
·¡1
= ¾lg0 for
every l = 1; : : : ; N .
If 0 2 V (f) with multiplicity h, then ¾lg0(„) · 0 for l = M ¡ h +
1; : : : ;M , while ¾M¡hg0 („) 6· 0. The ratios ¾M¡h¡lg0 („)
‡
¾M¡hg0 („)
·¡1
tend to 0 for l=N ¡ h + 1; : : : ;M ¡ h, and to ¾¡lg0 for l= 1; : : : ; N¡h.
In particular, ¾M¡Ng0 („)
‡
¾M¡hg0 („)
·¡1
has limit ¾¡N+hg0 6· 0, hence
¾M¡Ng0 („) 6· 0.
      
148 A. Perotti
It remains to note that ¾lg0 = ¾
¡N+h+l
g0
‡
¾¡N+hg0
·¡1
for every
l = 1; : : : ; N ¡ h.
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 ¾M¡h¡lg0 („)
‡
¾M¡hg0 („)
·¡1
tend to 0. Equivalently, it
is the number of functions ¾M¡h¡lg0 („) which have the same „-degree as
¾M¡hg0 („).
In particular, ¾1g = LResf (g). We have proved the following result.
Theorem 1. The logarithmic residue of any g 2 C[z] with respect
to f can be computed from
LResf (g) = lim
a!0
lim
„!1
¾M¡N+1g0 („)
¾M¡Ng0 („)
:
3. Local Residues
Now we consider the total sum of local residues of a polynomial
g 2 C[z] with respect to the polynomial mapping f = (f1; : : : ; fn). In
general, if f = (f1; : : : ; fn) is a holomorphic mapping with an isolated
zero a in a closed neighbourhood Ua of a, the local (or Grothendieck)
residue at a of a holomorphic function g on Ua with respect to f is the
integral
resa;f (g) =
1
(2…i)n
Z
¡a(f)
g dz1 ^ ¢ ¢ ¢ ^ dzn
f1 ¢ ¢ ¢ fn
where ¡a(f) is the n-chain = fz 2 Ua : jfi(z)j = †i; i = 1; : : : ; ng, with
†i > 0 such that ¡a(f) is relatively compact in Ua (see for example [5]).
3.1. Let I = (f1; : : : ; fn) be a zero-dimensional polynomial ideal in
C[z]. Since f has a flnite number of isolated zeroes, we can consider
the global residue Resf (g) =
P
a2V (f) resa;f (g) of the local residues of
g 2 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 LResf (h) = Resf (h ¢ J).
If f has the special form fi = zkii + Pi, with deg(Pi) < ki, the global
residue Resf (g) can be computed from the explicit formula of Aizen-
berg [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 dif-
ferent way. We obtain Resf (g) from the computation of a flnite 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
Resf (g) =
PN
”=1
g(z(”))
J(z(”))
. We can now apply the following lemma, which
generalizes Newton’s formulas (for a proof, see for example [7]).
Lemma 1. Let ¾l(a) denote the l-th elementary symmetric function
of m scalars a1, : : : , am. If b1; : : : ; bm are scalars difierent from zero,
the sum ¾1
¡
a
b
¢
= a1b1 + ¢ ¢ ¢+ ambm is given by
¾1
‡a
b
·
=
m¡1X
k=0
(¡1)k ¾
1(abk) ¢ ¾m¡k¡1(b)
¾m(b)
:
Then we obtain the following formula:
Resf (g) =
N¡1X
k=0
(¡1)k
¾1g¢Jk ¢ ¾N¡k¡1J
¾NJ
where ¾1g¢Jk and ¾
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 val-
ues 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
`(w) =
NX
”=1
g(z(”)(w))
J(z(”)(w))
is a holomorphic function in w on a small neighbourhood of 0. Then
`(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 Resf (g) of any g 2 C[z] with respect
to f is equal to ˆ(0), where ˆ(w) is the holomorphic function given by
ˆ(w) =
N¡1X
k=0
(¡1)kLResf¡w(g ¢ J
k) ¢ ¾N¡k¡1J (w)
¾NJ (w)
:
     
150 A. Perotti
Here ¾lJ(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(J l),
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 LResf a univariate
polynomial in I(f) \ C[zi] of degree N can be computed. It preserves
multiplicities of the zeroes of f (for this method, see [4, Section 21]).
From Resf a membership criterion for the ideal I(f) can be deduced.
In [8], Tsikh applied Lasker-Noether Theorem and got the following:
g 2 I(f), Resf (g(‡)H(z; ‡)) = 0;
where H(z; ‡) is a Hefer determinant of f:
Remark. A polynomial Hefer determinant of f can be computed
from the Hefer expansions
fi(‡)¡ fi(z) =
X
j
Pij(z; ‡)(‡j ¡ zj)
where Pij(z; ‡)=
fi(‡1; : : : ; ‡j ; zj+1; : : : ; zn)¡fi(‡1; : : : ; ‡j¡1; zj ; : : : ; zn)
‡j ¡ zj :
Note that from Pij(z; ‡) and Pij(‡; z) we can get a Hefer determinant
which is symmetric in z and ‡.
4.2. Let g; h 2 C[z] and
g0(‡) = Resf (g(z)H(z; ‡));
h0(‡) = Resf (h(z)H(z; ‡)):
From the membership criterion above we get that g0 = h0 if and only if
the difierence g ¡ h 2 I(f), that is g and h deflne 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¡‡ p(z) = Resf¡f(‡)(p(z)H(z; ‡)). It follows that for any a 2 V (f),
Resf (p(z)H(z; a)) = p(a). In particular, we get Resf H(z; a) = 1.
From this we can deduce that
Resf (g0(z)H(z; ‡)) = Resf (g(z)H(z; ‡)) = g0(‡):
For simplicity, assume that the zeroes of f are simple. Then
Resf (g0(z)H(z; ‡)) =
X
”
g0(z”)H(z” ; ‡)
J(z”)
=
X
”;„
g(z„)H(z” ; z„)H(z” ; ‡)
J(z”)J(z„)
=
X
„
g(z„)
J(z„)
Resf (H(z; z„)H(z; ‡))
= Resf (g(z)H(z; ‡)) = g0(‡):
As a result, we get the following proposition.
Proposition 3. Let g 2 C[z], g0(‡) = Resf (g(z)H(z; ‡)). Then
g¡ g0 2 I(f), that is g0 represents g in the quotient space C[z]=I(f). In
particular, Resf g = Resf g0.
Note added in proof. The paper by E. Cattani, A. Dickenstein,
B. Sturmfels, Computing multidimensional residues, Algorithms in Al-
gebraic Geometry and Applications (L. Gonzales-Vega and T. Recio,
eds.), Progress in Mathematics, Vol. 143, Birkha˜user Verlag, Basel, 1996,
pp. 135{164, contains interesting relations between global residues and
Gro˜bner bases and other references about these problems.
References
1. L. A. Aizenberg, On a formula for the generalized multidimen-
sional 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 multi-
dimensional logarithmic residue to systems of nonlinear algebraic
equations, Sibirsk. Mat. Zh. 20 (1979), 699{707; English transl.,
Siberian Math. J. 20 (1979), 485-491.
      
152 A. Perotti
3. L. A. Aizenberg, A. K. Tsikh and A. P. Yuzhakov, Multidi-
mensional Residues and Applications, in \Several Complex Variables
II," (G. M. Khenkin and A. G. Vitushkin, eds.), Encyclopaedia of
Math. Sc. 8, Springer-Verlag, Berlin, Heidelberg, New-York, 1990,
pp. 1{58.
4. L. A. Aizenberg and A. P. Yuzhakov, \Integral Representa-
tions and Residues in Multidimensional Complex Analysis," Nauka,
Novosibirsk, 1979; English transl., Translations of Mathematical
Monographs 58, Amer. Math. Soc., Providence, 1983.
5. P. Griffiths and J. Harris, \Principles of Algebraic Geometry,"
Wiley, New York, Chichester, 1978.
6. A. K. Tsikh, \Multidimensional Residues and Their Applications,"
Nauka, Novosibirsk, 1988; English transl., Translations of Mathe-
matical Monographs 103, Amer. Math. Soc., Providence, 1992.
7. A. K. Tsikh, \Residues with respect to holomorphic mappings and
their applications," Krasnoyarsk Univ. Press, Krasnoyarsk, 1986,
(Russian).
8. A. K. Tsikh and A. P. Yuzhakov, Properties of the total sum of
residues with respect to a polynomial map and applications of them,
Sibirsk. Mat. Zh. 25 (1984), 207{213; English transl., Siberian Math.
J. 25 (1984), 677{682.
9. A. P. Yuzhakov, On the computation of the total sum of residues
relative to a polynomial mapping in Cn, Dokl. Akad. Nauk SSSR
275 (1984), 817{820; English transl., Soviet Math. Dokl. 29 (1984),
321{324.
Dipartimento di Matematica
Universitµa di Milano
Via Saldini 50
I-20133 Milano
ITALY
e-mail: perotti@vmimat.mat.unimi.it
Primera versi¶o rebuda el 30 de gener de 1997,
darrera versi¶o rebuda el 4 de novembre de 1997


A. Perotti

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