Multivariate Subresultants

Transcription

1 Multvarate Subresultants Marc Chardn Équpe de Calcul Formel Centre de Mathématques École Polytechnque F Palaseau cedex e-mal : chardn@polytechnque.fr Abstract In ths text, we wll ntroduce the natural generalzaton of the so-called subresultants of two polynomals n one varable, to the case of s n homogeneous polynomals n n varables. As a specal case, we wll of course recover the multvarate resultant. A frst attempt n ths drecton was done by L. González-Vega n [G-V]. If P 1,..., P s are homogeneous polynomals of k[x 1,..., X n ] wth d = deg P > 0, and s n we defne a polynomal ν S n the coeffcents of the P s attached to the followng data: () the numbers n and s and the s-tuple d = (d 1,..., d s ), () a postve nteger ν, and () a set S of H d (ν) monomals of degree ν, where H d (ν) s the Hlbert functon of a complete ntersecton gven by s homogeneous polynomals n n varables of degrees d 1,..., d s. The unversal property of ν S s the followng. If ψ s the canoncal specalzaton homomorphsm from the unversal rng Z[coeff. of the P s] to k sendng each coeffcent on ts value, then f k s a feld: ψ( ν S ) 0 f and only f I ν + k S = k[x 1,..., X n ] ν, where I ν s the degree ν part of the deal generated by the P s. We recover the resultant, takng r = n, ν > d d n n and S = (H d (ν) = 0 for such a ν). Moreover, we prove that for any monomal X β S of degree ν we have a unversal relaton: ψ( ν S) X β + X α S ±ψ( ν S {X β } {X α } ) Xα I ν. In the partcular case r = n = 2, ν = d 1 + d 2 j 1, S j = {X2 ν, X2 ν 1 X 1,..., X ν j+1 2 X j 1 1 } (S 0 = ) and X β = X ν j 2 X j 1, the left sde of ths relaton s the so-called subresultant of order j of P 1 and P 2. So, the classcal subresultants are also a specal case of these general objects. I The tools Let A be a noetheran and factoral doman, k ts quotent feld and R = A[X 1,..., X n ]. If P 1,..., P s are homogeneous polynomals n R, let I be the deal generated by the P s, d the degree of P and K the assocated Koszul complex: K := 0 s R s d s s 1 R s d s 1 d 2 1 R s d 1 R 0 p d p (e 1 e p ) := ( 1) k+1 P k e 1 ê k e p k=1 wth the notaton R s = e 1 R e s R. Wth partal support of the ECC Esprt BRA project POSSO 1

2 If we put on the modules K p = p R s the natural graduaton deg(x α e 1 e p ) = α +d 1 + +d p, ths complex of R-modules s graded, ts dfferental s of degree zero. Moreover, f the P s forms a regular sequence, then H p (K) = 0 for p > 0 (see e.g. [Se] IV Prop. 2, orgnal proof n [Hu]). We wll wrte K ν p and d ν p for the degree ν parts of the modules and dfferentals. Notce that K ν p, and therefore H p (K ν ), are A-modules of fnte type. If M s a torson A-module of fnte type we wll denote by dv(m) the dvsor assocated to M: dv(m) = P Ass(M) ht(p)=1 length(m P ) P. If I s an deal of A, and [I] the prncpal part of I (.e. the gcd of generators of I), then dv(r/i) = e P f [I] = Pe s the decomposton nto rreducble factors of the prncpal deal [I]. A good reference for these concepts s [Bo] Chap. 7, 4. Proposton 1. Let C be a complex of fntely generated free A-modules (A factoral and noetheran): C : 0 C n n n 1 Cn C1 C0 0 ( ) ap φ and suppose that we have a decomposton C = E +1 E, E n+1 = E 0 = 0, p = p where φ p s an njectve endomorphsm of E p. Then the complex has only torson homology, and we have: ( 1) dv(h (C)) = ( 1) dv(det φ +1 ), n partcular, ( 1) dv(det φ +1 ) s ndependent of the decomposton. Let us defne for k = n 1,..., 0, n 1 k := (det φ +1 ) ( 1) k =k the element 0 s, up to an nvertble element of A, determned by the homology of C. Moreover, f H (C) = 0 for > 0, the cokernel of 1 beng equal to H 0 (C), we have: and k A for k = n 1,..., 0. [Coker( 1 )] = [H 0 (C)] = 0, Ths s proved for nstance n [De] p. 5 and n ths exact settng n [Ch1]. We wll now assume that s n, and therefore we have the followng classcal result (see e.g. [Jo1] pp. 6 8) : Proposton 2. Let P 1,..., P s be generc homogeneous polynomals n n s varables, P = α =d U,α X α A[X 1,..., X n ] where A = Z[U,α ] =1,...,s. Then H p (K ν ) = 0 for all ν and all p > 0. α =d II Ascendng and descendng decompostons of the Koszul complex Let us frst descrbe two general technques of decomposton. If F s a bounded exact sequence of fnte dmensonal vector spaces over a feld k : 0 F n n 1 n Fn F1 F0 0, 2 b p c p

3 fxng a base B for each F, then we may construct a decomposton of the F s n the two followng ways: Ascendng decomposton : choose a maxmal non zero mnor of 1 (there exsts one because 1 s onto). Ths choce splts B 1 nto two parts: B 1 = B 1 B 0 where #B 0 = #B 0, recursvely, for 2, the co-restrcton of to the module A B 1 s onto, because of the choce of B 2 and of the exactness of F. So there exsts a maxmal non zero mnor of, and choosng one splts B nto B B 1 where #B 1 = #B 1, remark that n s a square (nvertble) matrx as n =0 ( 1) dm F = 0. And we have the dual descendng decomposton : choose a maxmal non zero mnor of n (there exsts one because n s nto). Ths choce splts B n 1 nto two parts: B n 1 = B n B n 1 where #B n = #B n, recursvely, the restrcton n of n to the module A B n s nto, so we can terate the process by choosng at each step a maxmal mnor of n and we get a decomposton B n 1 = B n B n 1 wth #B n = #B n, for the same reason as above 1 s a square nvertble matrx. We wll now apply theses technques to the Koszul complex. Let us note M ν the set of monomals of degree ν n the varables X 1,..., X n. Let us also recall that, f s n, the Hlbert functon of the deal generated by a regular sequence (P 1,..., P s ), of homogeneous polynomals over a feld s gven by : s H d (ν)t ν =1 = (1 T d ) (1 T ) n ν 0 where d = (d 1,..., d s ) s the s-tuple of the degrees of the P s. In the generc stuaton of proposton 2, f we choose a set S of H d (ν) monomals of degree ν that generates H 0 (K ν ) A k (where k = Frac(A)), we get the exact complex of fnte dmensonal k-vector spaces K S ν A k, where: K S ν : 0 K ν n d s K ν d s 1 d n 1 2 K ν 1 where ϕ S s the co-restrcton of d ν 1 to A M ν S. Notce that ϕ S s onto ff k[x 1,..., X n ] ν = k S + I ν. We wll fx as a base of K ν p the set B p = 1 1 < < p n ϕ S A Mν S 0 X α M ν (d1 + +d p ) Xα e 1 e p. Applyng to ths complex one of the two technques of decomposton descrbed above, we get a decomposton of K ν satsfyng the condtons of proposton 1 (notce that φ : A t A t s njectve ff φ A k s an somorphsm). The element S = n =1 (det φ ) ( 1)+1 A comng from the decomposton s, up to the sgn, ndependent of the choces made to perform the decomposton. From proposton 1 and 2, we know that : Defnton of the Subresultants. For every set S of H d (ν) monomals of degree ν such that dv(coker ϕ S ) s a torson module, there exsts a polynomal S A such that dv( S ) = dv(coker ϕ S ), where ϕ S s the co-restrcton of d ν 1 to the A-module generated by monomals of degree ν that are not n S. Ths defnes, up to the sgn, the polynomal S A called the S-subresultant of the generc polynomals, n the non-generc case the S-subresultant s defned as the mage of the generc S-subresultant by the canoncal specalsaton homomorphsm. If dv(coker ϕ S ) s not a torson module, we set S = 0. 3

4 Before fxng the sgn of S, let us do some useful remarks : 1) If you fx any decomposton of K S ν (wth S M ν and #S = H d (ν)) satsfyng the condtons of proposton 1, t s a good decomposton for every S of the same type whch generates H 0 (K ν A k). Indeed, the complexes K S ν and K S ν do not dffer except for the last morphsm and the last vector space. In other words, ϕ S s onto ff ϕ S s onto (here the star correspond to the decomposton made for KS ν ). 2) Every non zero mnor of d ν 1 of sze #M ν H d (ν), gves a set S wth #S = H d (ν) (the set of monomals correspondng to the erased columns) such that S generates H 0 (K ν A k). Therefore extendng ths frst step of decomposton for K S ν by the ascendng decomposton technque, we can conclude that S dvdes ths mnor. That s, S dvdes all maxmal mnors of ϕ S. 3) From [De] p. 20, for each the degree of S n the set of varables U,α s Hˆd(ν d ) wth the notaton ˆd = (d 1,..., d 1, d +1,..., d s ). In order to fx the sgn of S, let us ntroduce some notatons : R d (ν) = {Xα 1 1 Xn αn N d (ν) = s =1 M ν α j < d j, j < }. X α R d (ν d) Xα e B1 ν and W d (ν) = k N d (ν). Lemma 1. The restrcton d 1 of d ν 1 to W d (ν) s nto and ts mage s of corank H d (ν) n K ν 0 A k. It s suffcent to prove ths for a specalzaton, namely P = X d. The njectvty s a straghtforward consequence of the defnton of R d, and an easy reducton argument shows that d 1(W d (ν)) = I ν wth I = (X d 1 1,..., Xds s ). As I s a complete ntersecton, the concluson follows. From ths lemma, we can choose a decomposton of K ν (vald for every S such that ϕ S s onto by the preceedng remark 1) such that B 0 = N d (ν), such a decomposton s descrbed n [De] p. 10. If we consder the set of monomals of degree ν that are not n I ν, the map ϕ S gves a bjecton between the elements of N d (ν) and M ν S, and ths map preserves the lexcographcal orderng. From that, det ϕ S = 1, f we order lexcographcally the monomals n both lnes and columns (the lexcographcal orderng n the columns s for the order e 1 > X 1 > e 2 > > e n > X n ). We fx the sgn of S by mposng that : 1) for ths set of monomals and these polynomals S = 1 (as 1 S = 1 we must have S = 1 = ±1), 2) the reference decomposton s the one, that naturally extends ths frst step, explaned by Demazure n [De] p. 10 (or any one that extends ths frst step), 3) both lnes and columns are ordered lexcographcally n a coherent manner as above. III The unversal property of the subresultants In order to prove ths we need a classcal proposton : Proposton 3. Let P 1,..., P s be generc homogeneous polynomals n n s varables, P = A[X 1,..., X n ] where A = Z[U,α ] =1,...,s. The deal I of A[X 1,..., X n ] generated by α =d α =d U,α X α the P s s a prme deal f s < n. In the case s = n, I = I N where N s (X 1,..., X n )-prmary and the deal I = r>0 I : (X 1,..., X n ) r s prme ; moreover, the prme deal I A s prncpal, generated by the resultant of the P s. We wll just recall here the sketch of a geometrcal proof, and refer to the work of Jouanolou [Jo1] for detaled arguments. The deal I defnes a subscheme X of P = P n 1 P where P = P (n+d 1 1 n 1 ) 1 P (n+ds 1 n 1 ) 1 that has a natural structure of vector bundle over P n 1 (the fber over a pont x s the product of the hyperplanes P (x) = 0), ths mples that X s smooth and rreducble. Now, as I s gven by a regular sequence, I s unmxed. The only non geometrc homogeneous deal of codmenson n n A[X 1,..., X n ] s the deal M = (X 1,..., X n ); ths mples that I s prme for s < n and the case s = n except the fact that I A s prncpal, whch s the basc property of the resultant (t s a consequence of the bratonalty of the projecton of X on ts mage V (I A) P, see [Jo2] p. 14). 4

5 Theorem 1. Wth the notaton of part II, f T s a subset of M ν wth H d (ν) + 1 elements, then : X α T ε α,t T {X α }X α I, where ε α,t = ±1 wll be precsed below. N.B. For n = 2, ν = d 1 +d 2 1 j and 0 j mn{d 1, d 2 }, settng T j = {X2 ν, X2 ν 1 X 1,..., X ν j 2 X j 1 }, the frst member of the relaton above s the classcal subresultant of order j of P 1 and P 2 (X 2 s the homogenzaton varable f one wants to recover the non homogeneous defnton). Corollary. Gven S M ν wth #S = H d (ν), f X β M ν s not n S, settng S = S {X β }, we get : where ε α,β = ±1. S X β X α S ε α,β S {X α } X α mod(i ν ), If the S-subresultant s nvertble n a specfc example, we therefore get a unversal formula whch lfts any monomal of degree ν to a lnear combnaton of elements of S modulo the deal generated by the polynomals. We now prove theorem 1. Let M be the sub-matrx of d 1 ( correspondng to the decomposton fxed n II) where we have erased the columns correspondng to T. For every element X β e n N d (ν), the matrx M β, obtaned by erasng from M the lne correspondng to X β e s a square matrx, let s call D β, ts determnant affected wth the sgn that corresponds to lookng at t as a cofactor of ϕ S for one fxed S n T wth #S = #T 1. Then we have : D β, P = D β, c (β,),α X α X β e N d (ν) X β e N d (ν) X α M ν = c (β,),α D β, X α M ν X α = X α T = 1 X β e N d (ν) ε α,t det ϕ T {X α } Xα X α T ε α,t T {X α }X α as the other terms vansh n the sum X β e N d (ν) c (β,),αd β,, because they correspond to developments, relatvely to one column, of determnants that have two columns n common. Remark that ε α,t = 1 ff the sgn affected to det M β, as a cofactor of ϕ T {X α } s the same as the one of D β,. So, we know that 1 X α T ε α,t T {X α }X α I. As 1 A and A I = {0}, 1 I, and the proposton 4 drectly gves the concluson for s < n. For the case s = n we remark that for the decomposton we fxed, 1 does not depend on the coeffcents of P n (see [De] p. 15 corollare 2, or [Ch1], III, remark 7) and therefore s not n I A, and the concluson s now clear from proposton 3. Let us now gve the central result of ths artcle: Theorem 2. For all ν and all S M ν wth #S = H d (ν), the polynomal S defned n part II, verfes the followng property. If k s a feld, (P 1,..., P s ) a s-tuple of homogeneous polynomals n n varables wth coeffcents n k, d = deg P and I = (P 1,..., P s ), denotng by ψ the canoncal specalzaton homomorphsm from A to k, we have : 1) ψ( S ) 0 H I (ν) = H d (ν) = dm k (k S + I ν )/I ν. 1 ) ψ( S ) 0 k S + I ν = k[x 1,..., X n ] ν. 2) {ψ( S ) = 0, S M ν, #S = H d (ν)} H I (ν) H d (ν). 3) For each, S s homogeneous n the set of varables U α, of degree Hˆd(ν d ). 5

6 Corollary. When s = n, S = = Res(P 1,..., P n ), for all ν > n =1 (d 1). Ths corollary s the basc result of [De] and the hstory of ths result goes back to Cayley. It also follows from 1) and 3) as the resultant s rreducble, has the same degree as and s zero ff 1) s verfed. Before provng ths theorem, let us do some easy remarks: 2) s a consequence of 1). 3) s already proved (remark 3, followng the defnton of S ). Therefore t remans to prove 1) and 1 ). : If S 0, then, by the corollary of theorem 1, (k[x 1,..., X n ]/I) ν s generated by the mages of the elements of S, therefore H I (ν) = dm k (k S + I ν )/I ν #S = H d (ν), as one always has H I (ν) H d (ν) (the Hlbert functon s the corank of d ν 1 and therefore can only ncrease n a specalzaton), we are done. : If S = 0, then by the remark 2, followng the defnton of S, all the maxmal mnors of the specalzaton ϕ S of ϕ S vanshes. Therefore ϕ S s not onto. In other words, k S + I ν k[x 1,..., X n ] ν, and therefore dm k (k S + I ν )/I ν < H I (ν). It s clear from the proof that 1) may be replaced by 1 ). IV An example As an llustraton we wll treat the example of three polynomals of degrees 2,3,3, lookng for the subresultants of degree (d 1) = 5, and then compute two specfc cases. So let, P 1 = a 1 X a 2 X 1 X 2 + a 3 X 1 X 3 + a 4 X a 5 X 2 X 3 + a 6 X 2 3, P 2 = b 1 X b 2 X 2 1 X 2 + b 3 X 2 1 X 3 + b 4 X 1 X b 5 X 1 X 2 X 3 + b 6 X 1 X b 7 X b 8 X 2 2 X 3 + b 9 X 2 X b 0 X 3 3, P 3 = c 1 X c 2 X 2 1 X 2 + c 3 X 2 1 X 3 + c 4 X 1 X c 5 X 1 X 2 X 3 + c 6 X 1 X c 7 X c 8 X 2 2 X 3 + c 9 X 2 X c 0 X 3 3. The matrx of d 5 2 s : ( ) b1 b 2 b 3 b 4 b 5 b 6 b 7 b 8 b 9 b 0 a 1 0 a 2 a a 4 a 5 a c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 c 9 c 0 0 a a 2 a a 4 a 5 a 6 we remark that ths matrx s of rank 2 unless P 1 = 0, and P 2 and P 3 are proportonal. The reference decomposton chooses the submatrx: ( a1 0 0 a 1 ), and the matrx of d 1 s : 6

7 a 1 a 2 a 3 a 4 a 5 a a 1 0 a 2 a 3 0 a 4 a 5 a a 1 0 a 2 a 3 0 a 4 a 5 a a a 2 a a 4 a 5 a a a 2 a a 4 a 5 a a a 2 a a 4 a 5 a a a 2 a a 4 a 5 a a a 2 a a 4 a 5 a a a 2 a a 4 a 5 a a a 2 a a 4 a 5 a 6 0 b 1 0 b 2 b 3 0 b 4 b 5 b 6 0 b 7 b 8 b 9 b b 1 0 b 2 b 3 0 b 4 b 5 b 6 0 b 7 b 8 b 9 b c 1 0 c 2 c 3 0 c 4 c 5 c 6 0 c 7 c 8 c 9 c c 1 0 c 2 c 3 0 c 4 c 5 c 6 0 c 7 c 8 c 9 c b b 2 b b 4 b 5 b b 7 b 8 b 9 b b b 2 b b 4 b 5 b b 7 b 8 b 9 b b b 2 b b 4 b 5 b b 7 b 8 b 9 b c c 2 c c 4 c 5 c c 7 c 8 c 9 c c c 2 c c 4 c 5 c c 7 c 8 c 9 c c c 2 c c 4 c 5 c c 7 c 8 c 9 c 0 where the columns corresponds to degree fve monomals, ordered lexcographcally, X 5 1, X 4 1 X 2, X 4 1 X 3, X 3 1 X 2 2, X 3 1 X 2 X 3,... Before computng two specfc examples, let us do some remarks concernng the case s = n and ν = δ = n =1 (d 1), so that H d (ν) = 1: 0) If P k[x 1,..., X n ], k a feld, then H I (δ) 1 mples δ+1 = Res(P 1,..., P n ) = 0, 1) If there exsts X α M δ such that X α 0, then for X β M δ, X β = 0 X β I Z(I) {X β = 0}, 2) If H I (δ + 1) 0, H I (δ) = 1 X δ 0 [n]. Hence, the deals of A = Z[U,α ] generated the resultant and respectvely by the polynomals ( X δ ) [n] and the polynomals ( X α) α =δ have the same radcal. We wll see further the geometrcal meanng of theses deals. As a frst example, let us take the three followng polynomals: P 1 = X 2 + 3XY + 3XZ + 2Y 2 7Y Z + Z 2, P 2 = 7X 3 + X 2 Y + 4X 2 Z + 8XY 2 XY Z + XZ 2 Y 3 + 6Y 2 Z + Y Z 2 + 5Z 3, P 3 = 3X 3 X 2 Y + 9X 2 Z + 9XY 2 3XY Z + XZ 2 Y 3 + 4Y 2 Z + Y Z 2 + 3Z 3, and ν = δ = 5, so that H (2,3,3) (5) = 1, we have the followng results : X 5 1 = = , X 5 2 = = , X 5 3 = = Remark that these numbers have only 3 as a common factor, so the reducton modulo p of these equatons defnes ether the empty varety or a sngle pont, except for p = 3. And we can compute the resultant : 6 = =

8 Therefore the redutons mod p defne a sngle pont for p {73, , } and at least two ponts (but n fact exactly two ponts by the proposton 3 of [Ch1]) for p = 3. As a second example we wll take polynomals that have a common zero, namely (0 : 0 : 1): P 1 = X X 1 X 2 + 2X X 2 X 3, P 2 = 3X X 2 1 X 2 + 3X 2 1 X 3 2X 1 X X 1 X 2 3 X X 2 X 2 3, P 3 = X 3 1 X 2 1 X 2 X 2 1 X 3 + 4X 1 X 2 X 3 + 2X 1 X X 2 2 X 2 3 X 2 X 3 3. Here, we have X 5 3 = As = , assumng the dentfcaton of X 5 3 wth the reduced subresultant correspondng to the sngle pont (0 : : 0 : 1) proved n [Ch2], we know that the reductons modulo p of the equatons defne a sngle reduced pont (namely (0 : 0 : 1)) f and only f p {3, 13, 47, 4139}. The dentfcaton of X δ n wth the reduced resultant assocated to the sngle pont (0 : : 0 : 1), and classcal propertes of the Koszul complex show that H I (δ) = 1 ff I defnes the empty varety or a sngle pont (proof n [Ch2]). Ths gves the geometrcal nterpretaton mentoned above and prove that, n these two stuatons (empty varety or sngle pont gven by n polynomals) H I only depends on d = (d 1,..., d n ). Fnal remarks and open questons. Except f n = 2 (or n = 3 and d 1 = d 2 = d 3 = 2), these objects don t produce (at least for suffcently general polynomals of the form P = GQ ) the gcd of the P s (ths can be seen by calculaton of H d (ν)). For whch S do we have S 0 n the generc case? We conjecture that every S s good f s = n and ν n 1 =1 (d 1) f d n = nf d. What about the rreducblty of these polynomals and of the varety defned by H I (ν) H d (ν)? We only know, up to now that, n the case s = n, the resultant (.e. the case ν = δ + 1) and the polynomals X δ are rreducble. References [Bo] N. Bourbak, Algèbre Commutatve Chaptres 1 à 9, Masson 1983 et [Ch1] M. Chardn, The Resultant va a Koszul Complex, Computatonal Algebrac Geometry (Proc. of MEGA 92), Brkhäuser, [Ch2] M. Chardn, Formules à la Macaulay pour les sous-résultants en pluseurs varables et applcaton au calcul d un résultant rédut, prépublcaton du Centre de Mathématques de l Ecole Polytechnque, décembre [De] M. Demazure, Une défnton constructve du résultant, Notes Informelles de Calcul Formel 2, prépublcaton du Centre de Mathématques de l École Polytechnque, [G-V] L. González-Vega, Une théore des sous-résultants pour les polynômes en pluseurs varables, C. R. Acad. Sc. Pars, t. 313, Sére I, p , [Gr] W. Gröbner, Modern Algebrasche Geometre, Sprnger-Verlag, Wen und Innsbruck, [Hu] A. Hurwtz, Über de Träghetsformen enes algebraschen Moduls, Annal d Mathematca pura ed applcata (3) 20, 1913, pp [Jo1] J.-P. Jouanolou, Idéaux résultants, Adv. n Maths, Vol. 37 N 3 (1980), pp [Jo2] J.-P. Jouanolou, Le formalsme du résultant, Adv. n Maths, Vol. 90 N 2 (1991). [Jo3] J.-P. Jouanolou, Aspects nvarants de l élmnaton, Publcaton de l IRMA 457/P-263, Unversté de Strasbourg, [Ma] F. S. Macaulay, The Algebrac Theory of Modular Systems, Stechert-Hafner Servce Agency, New-York and London, 1964, (Orgnally publshed n 1916 by Cambrdge Unversty Press). [Se] J.-P. Serre, Algèbre locale et multplctés, Lecture Notes n Mathematcs 11,