Multivariate Subresultants
Marc Chardin †
Équipe de Calcul Formel
Centre de Mathématiques
École Polytechnique
F–91128 Palaiseau cedex
e-mail : [email protected]
Abstract
In this text, we will introduce the natural generalization of the so-called subresultants of two polynomials
in one variable, to the case of s ≤ n homogeneous polynomials in n variables. As a special case, we will of
course recover the multivariate resultant.
A first attempt in this direction was done by L. González-Vega in [G-V].
If P1 , . . . , Ps are homogeneous polynomials of k[X1 , . . . , Xn ] with di = deg Pi > 0, and s ≤ n we define a
polynomial ∆νS in the coefficients of the Pi ’s attached to the following data: (i) the numbers n and s and the
s-tuple d = (d1 , . . . , ds ), (ii) a positive integer ν, and (iii) a set S of Hd (ν) monomials of degree ν, where
Hd (ν) is the Hilbert function of a complete intersection given by s homogeneous polynomials in n variables
of degrees d1 , . . . , ds .
The universal property of ∆νS is the following. If ψ is the canonical specialization homomorphism from
the universal ring Z[coeff. of the Pi ’s] to k sending each coefficient on its value, then if k is a field: ψ(∆νS ) 6= 0
if and only if Iν + khSi = k[X1 , . . . , Xn ]ν , where Iν is the degree ν part of the ideal generated by the Pi ’s.
We recover the resultant, taking r = n, ν > d1 + · · · + dn − n and S = ∅ (Hd (ν) = 0 for such a ν).
Moreover, we prove that for any monomial X β 6∈ S of degree ν we have a “universal” relation:
ψ(∆νS ) X β +
X
±ψ(∆νS∪{X β }−{X α } ) X α ∈ Iν .
X α ∈S
In the particular case r = n = 2, ν = d1 + d2 − j − 1, Sj = {X2ν , X2ν−1 X1 , . . . , X2ν−j+1 X1j−1 } (S0 = ∅) and
X β = X2ν−j X1j , the left side of this relation is the so-called subresultant of order j of P1 and P2 . So, the
classical subresultants are also a special case of these general objects.
I The tools
Let A be a noetherian and factorial domain, k its quotient field and R = A[X1 , . . . , Xn ].
If P1 , . . . , Ps are homogeneous polynomials in R, let I be the ideal generated by the Pi ’s, di the degree
of Pi and K the associated Koszul complex:
ds−1
d
d
d
s
2
1
0 → ∧s Rs −→
∧s−1 Rs −→ · · · −→
∧1 Rs −→
R→0
p
X
dp (ei1 ∧ · · · ∧ eip ) :=
(−1)k+1 Pik ei1 ∧ · · · ∧ ec
ik ∧ · · · ∧ eip
K :=
k=1
with the notation Rs = e1 R ⊕ · · · ⊕ es R.
† With partial support of the ECC Esprit BRA project POSSO
1
Vp s
If we put on the modules Kp =
R the natural graduation deg(X α ei1 ∧· · ·∧eip ) = |α|+di1 +· · ·+dip ,
this complex of R-modules is graded, its differential is of degree zero. Moreover, if the Pi ’s forms a regular
sequence, then Hp (K) = 0 for p > 0 (see e.g. [Se] IV Prop. 2, original proof in [Hu]).
We will write Kpν and dνp for the degree ν parts of the modules and differentials. Notice that Kpν , and
therefore Hp (Kν ), are A-modules of finite type.
If M is a torsion A-module of finite type we will denote by div(M ) the divisor associated to M :
div(M ) =
X
length(MP ) P.
P∈Ass(M )
ht(P)=1
P If I is an ideal
Q ofei A, and [I] the principal part of I (i.e. the gcd of generators of I), then div(R/I) =
e
P
if
[I]
=
i i i
i Pi is the decomposition into irreducible factors of the principal ideal [I].
A good reference for these concepts is [Bo] Chap. 7, §4.
Proposition 1.— Let C be a complex of finitely generated free A-modules (A factorial and noetherian):
∂n−1
∂
∂
∂
n
2
1
C : 0 −→ Cn −→
Cn−1 −→ · · · −→
C1 −→
C0 −→ 0
µ
and suppose that we have a decomposition Ci = Ei+1 ⊕ Ei , En+1 = E0 = 0, ∂p =
ap
bp
φp
cp
¶
where φp is
an injective endomorphism of Ep .
Then the complex has only torsion homology, and we have:
X
(−1)i div(Hi (C)) =
X
i
(−1)i div(det φi+1 ),
i
P
in particular, i (−1)i div(det φi+1 ) is independent of the decomposition.
Let us define for k = n − 1, . . . , 0,
∆k :=
n−1
Y
(det φi+1 )(−1)
i−k
i=k
the element ∆0 is, up to an invertible element of A, determined by the homology of C. Moreover, if Hi (C) = 0
for i > 0, the cokernel of ∂1 being equal to H0 (C), we have:
[Coker(∂1 )] = [H0 (C)] = ∆0 ,
and ∆k ∈ A for k = n − 1, . . . , 0.
This is proved for instance in [De] p. 5 and in this exact setting in [Ch1].
We will now assume that s ≤ n, and therefore we have the following classical result (see e.g. [Jo1] pp.
6–8) :
2.— Let P1 , . . . , Ps be generic homogeneous polynomials in n ≥ s variables, Pi =
P Proposition
α
U
X
∈
A[X
1 , . . . , Xn ] where A = Z[Ui,α ] i=1,...,s . Then Hp (Kν ) = 0 for all ν and all p > 0.
|α|=di i,α
|α|=di
II Ascending and descending decompositions of the Koszul complex
Let us first describe two general techniques of decomposition. If F is a bounded exact sequence of finite
dimensional vector spaces over a field k :
∂
∂n−1
∂
∂
n
2
1
0 −→ Fn −→
Fn−1 −→ · · · −→
F1 −→
F0 −→ 0,
2
fixing a base Bi for each Fi , then we may construct a decomposition of the Fi ’s in the two following ways:
Ascending decomposition :
• choose a maximal non zero minor of ∂1 (there exists one because ∂1 is onto). This choice splits B1
into two parts: B1 = B10 ∪ B000 where #B000 = #B0 ,
0
• recursively, for i ≥ 2, the co-restriction ∂i∗ of ∂i to the module AhBi−1
i is onto, because of the choice
and of the exactness of F. So there exists a maximal non zero minor of ∂i∗ , and choosing one splits
of
0
00
00
= #Bi−1
,
where #Bi−1
Bi into Bi0 ∪ Bi−1
Pn
∗
• remark that ∂n is a square (invertible) matrix as i=0 (−1)i dim Fi = 0.
00
Bi−2
And we have the dual descending decomposition :
• choose a maximal non zero minor of ∂n (there exists one because ∂n is into). This choice splits Bn−1
00
into two parts: Bn−1 = Bn0 ∪ Bn−1
where #Bn0 = #Bn ,
?
00
• recursively, the restriction ∂n−i
of ∂n−i to the module AhBn−i
i is into, so we can iterate the process
?
0
00
by choosing at each step a maximal minor of ∂n−i and we get a decomposition Bn−i−1 = Bn−i
∪ Bn−i−1
0
00
with #Bn−i
= #Bn−i
,
• for the same reason as above ∂1? is a square invertible matrix.
We will now apply theses techniques to the Koszul complex. Let us note Mν the set of monomials of
degree ν in the variables X1 , . . . , Xn .
Let us also recall that, if s ≤ n, the Hilbert function of the ideal generated by a regular sequence
(P1 , . . . , Ps ), of homogeneous polynomials over a field is given by :
X
Qs
Hd (ν)T ν =
ν≥0
− T di )
(1 − T )n
i=1 (1
where d = (d1 , . . . , ds ) is the s-tuple of the degrees of the Pi ’s.
In the generic situation of proposition 2, if we choose a set S of Hd (ν) monomials of degree ν that
generates H0 (Kν ) ⊗A k (where k = Frac(A)), we get the exact complex of finite dimensional k-vector spaces
KSν ⊗A k, where:
d
ds−1
d
ϕS
2
s
ν
−→ · · · −→
K1ν −→ AhMν − Si → 0
KSν : 0 → Knν −→
Kn−1
where ϕS is the co-restriction of dν1 to AhMν − Si.
Notice that ϕS is onto iff k[X1 , . . . , Xn ]ν = khSi + Iν .
S
S
We will fix as a base of Kpν the set Bp = 1≤i1 <···<ip ≤n X α ∈Mν−(d
i1 +···+dip )
X α ei1 ∧ · · · ∧ eip .
Applying to this complex one of the two techniques of decomposition described above, we get a decomposition of Kν satisfying the conditions of proposition 1 (notice that φ : At −→ At is injective iff φ ⊗A k is
an isomorphism).
Qn
i+1
The element ∆S = i=1 (det φi )(−1) ∈ A coming from the decomposition is, up to the sign, independent of the choices made to perform the decomposition.
From proposition 1 and 2, we know that :
Definition of the Subresultants.— For every set S of Hd (ν) monomials of degree ν such that
div(Coker ϕS ) is a torsion module, there exists a polynomial ∆S ∈ A such that div(∆S ) = div(Coker ϕS ),
where ϕS is the co-restriction of dν1 to the A-module generated by monomials of degree ν that are not in S.
This defines, up to the sign, the polynomial ∆S ∈ A called the S-subresultant of the generic polynomials, in
the non-generic case the S-subresultant is defined as the image of the generic S-subresultant by the canonical
specialisation homomorphism. If div(Coker ϕS ) is not a torsion module, we set ∆S = 0.
3
Before fixing the sign of ∆S , let us do some useful remarks :
1) If you fix any decomposition of KSν (with S ⊂ Mν and #S = Hd (ν)) satisfying the conditions of
proposition 1, it is a good decomposition for every S 0 of the same type which generates H0 (Kν ⊗A k). Indeed,
0
the complexes KSν and KSν do not differ except for the last morphism and the last vector space. In other
words, ϕS 0 is onto iff ϕ∗S 0 is onto (here the star correspond to the decomposition made for KSν ).
2) Every non zero minor of dν1 of size #Mν − Hd (ν), gives a set S with #S = Hd (ν) (the set of
monomials corresponding to the erased columns) such that S generates H0 (Kν ⊗A k). Therefore extending
this first step of decomposition for KSν by the ascending decomposition technique, we can conclude that ∆S
divides this minor. That is, ∆S divides all maximal minors of ϕS .
3) From [De] p. 20, for each i the degree of ∆S in the set of variables Ui,α is Hd̂i (ν − di ) with the
notation d̂i = (d1 , . . . , di−1 , di+1 , . . . , ds ).
In order to fix the sign of ∆S , let us introduce some notations :
• Rdi (ν) = {X1α1 · · · Xnαn ∈ Mν | αj < dj , ∀j < i}.
Ss S
• N d (ν) = i=1 X α ∈Rd (ν−di ) X α ei ⊆ B1ν and W d (ν) = khN d (ν)i.
i
Lemma 1.— The restriction d∗1 of dν1 to W d (ν) is into and its image is of corank Hd (ν) in K0ν ⊗A k.
It is sufficient to prove this for a specialization, namely Pi = Xidi . The injectivity is a straightforward
consequence of the definition of Rdi , and an easy reduction argument shows that d∗1 (W d (ν)) = Iν with
I = (X1d1 , . . . , Xsds ). As I is a complete intersection, the conclusion follows.
From this lemma, we can choose a decomposition of Kν (valid for every S such that ϕS is onto by the
preceeding remark 1) such that B000 = N d (ν), such a decomposition is described in [De] p. 10. If we consider
the set of monomials of degree ν that are not in Iν , the map ϕ∗S gives a bijection between the elements of
N d (ν) and Mν − S, and this map preserves the lexicographical ordering. From that, det ϕ∗S = 1, if we order
lexicographically the monomials in both lines and columns (the lexicographical ordering in the columns is
for the order e1 > X1 > e2 > · · · > en > Xn ). We fix the sign of ∆S by imposing that :
1) for this set of monomials and these polynomials ∆S = 1 (as ∆1 ∆S = 1 we must have ∆S = ∆1 = ±1),
2) the reference decomposition is the one, that naturally extends this first step, explained by Demazure
in [De] p. 10 (or any one that extends this first step),
3) both lines and columns are ordered lexicographically in a coherent manner as above.
III The universal property of the subresultants
In order to prove this we need a “classical” proposition :
3.— Let P1 , . . . , Ps be generic homogeneous polynomials in n ≥ s variables, Pi =
P Proposition
α
U
X
∈
A[X1 , . . . , Xn ] where A = Z[Ui,α ] i=1,...,s . The ideal I of A[X1 , . . . , Xn ] generated by
|α|=di i,α
|α|=di
the Pi ’s is S
a prime ideal if s < n. In the case s = n, I = I0 ∩ N where N is (X1 , . . . , Xn )-primary and the
0
ideal I = r>0 I : (X1 , . . . , Xn )r is prime ; moreover, the prime ideal I0 ∩ A is principal, generated by the
resultant of the Pi ’s.
We will just recall here the sketch of a geometrical proof, and refer to the work of Jouanolou [Jo1] for
detailed arguments.
n+d1 −1
n+ds −1
The ideal I defines a subscheme X of P = Pn−1 × P∨ where P∨ = P( n−1 )−1 × · · · × P( n−1 )−1
that has a natural structure of vector bundle over Pn−1 (the fiber over a point x is the product of the
hyperplanes Pi (x) = 0), this implies that X is smooth and irreducible. Now, as I is given by a regular
sequence, I is unmixed. The only “non geometric” homogeneous ideal of codimension ≤ n in A[X1 , . . . , Xn ]
is the ideal M = (X1 , . . . , Xn ); this implies that I is prime for s < n and the case s = n except the fact
that I0 ∩ A is principal, which is the basic property of the resultant (it is a consequence of the birationality
of the projection of X on its image V (I0 ∩ A) ⊂ P∨ , see [Jo2] p. 14).
4
Theorem 1.— With the notation of part II, if T is a subset of Mν with Hd (ν) + 1 elements, then :
X
εα,T ∆T −{X α } X α ∈ I,
X α ∈T
where εα,T = ±1 will be precised below.
N.B. For n = 2, ν = d1 +d2 −1−j and 0 ≤ j ≤ min{d1 , d2 }, setting Tj = {X2ν , X2ν−1 X1 , . . . , X2ν−j X1j } ,
the first member of the relation above is the classical subresultant of order j of P1 and P2 (X2 is the
“homogenization variable” if one wants to recover the non homogeneous definition).
Corollary.—Given S ⊆ Mν with #S = Hd (ν), if X β ∈ Mν is not in S, setting S 0 = S ∪ {X β }, we get :
X
εα,β ∆S 0 −{X α } X α mod(Iν ),
∆S X β ≡
X α ∈S
where εα,β = ±1.
If the S-subresultant is invertible in a specific example, we therefore get a universal formula which
lifts any monomial of degree ν to a linear combination of elements of S modulo the ideal generated by the
polynomials.
We now prove theorem 1. Let M be the sub-matrix of d∗1 (“∗” corresponding to the decomposition
fixed in II) where we have erased the columns corresponding to T . For every element X β ei in N d (ν), the
matrix Mβ,i obtained by erasing from M the line corresponding to X β ei is a square matrix, let’s call Dβ,i
its determinant affected with the sign that corresponds to looking at it as a cofactor of ϕ∗S for one fixed S
in T with #S = #T − 1. Then we have :
X
X
X
Dβ,i Pi =
Dβ,i
c(β,i),α X α
X β ei ∈N d (ν)
X β ei ∈N d (ν)
=
X
Xα
X α ∈M
=
X
ν
X α ∈Mν
X
c(β,i),α Dβ,i
X β ei ∈N d (ν)
εα,T det ϕ∗T −{X α } X α
X α ∈T
= ∆1
X
εα,T ∆T −{X α } X α
X α ∈T
P
as the other terms vanish in the sum X β ei ∈N d (ν) c(β,i),α Dβ,i , because they correspond to developments,
relatively to one column, of determinants that have two columns in common. Remark that εα,T = 1 iff the
sign affected to det Mβ,i as a cofactor of ϕ∗T −{X α } is the same as the one of Dβ,i .
P
So, we know that ∆1 X α ∈T εα,T ∆T −{X α } X α ∈ I. As ∆1 ∈ A and A ∩ I = {0}, ∆1 6∈ I, and the
proposition 4 directly gives the conclusion for s < n. For the case s = n we remark that for the decomposition
we fixed, ∆1 does not depend on the coefficients of Pn (see [De] p. 15 corollaire 2, or [Ch1], III, remark 7)
and therefore is not in I0 ∩ A, and the conclusion is now clear from proposition 3.
Let us now give the central result of this article:
Theorem 2.— For all ν and all S ⊆ Mν with #S = Hd (ν), the polynomial ∆S defined in part II,
verifies the following property.
If k is a field, (P1 , . . . , Ps ) a s-tuple of homogeneous polynomials in n variables with coefficients in k,
di = deg Pi and I = (P1 , . . . , Ps ), denoting by ψ the canonical specialization homomorphism from A to k,
we have :
1) ψ(∆S ) 6= 0 ⇔ HI (ν) = Hd (ν) = dimk (khSi + Iν )/Iν .
1’) ψ(∆S ) 6= 0 ⇔ khSi + Iν = k[X1 , . . . , Xn ]ν .
2) {ψ(∆S ) = 0, ∀S ⊆ Mν , #S = Hd (ν)} ⇔ HI (ν) 6= Hd (ν).
3) For each i, ∆S is homogeneous in the set of variables Uα,i of degree Hd̂i (ν − di ).
5
Corollary.—When s = n, ∆S = ∆∅ = Res(P1 , . . . , Pn ), for all ν >
Pn
i=1 (di
− 1).
This corollary is the basic result of [De] and the history of this result goes back to Cayley. It also follows
from 1) and 3) as the resultant is irreducible, has the same degree as ∆∅ and is zero iff 1) is verified.
Before proving this theorem, let us do some easy remarks:
• 2) is a consequence of 1).
• 3) is already proved (remark 3, following the definition of ∆S ).
Therefore it remains to prove 1) and 1’).
⇒ : If ∆S 6= 0, then, by the corollary of theorem 1, (k[X1 , . . . , Xn ]/I)ν is generated by the images of
the elements of S, therefore HI (ν) = dimk (khSi + Iν )/Iν ≤ #S = Hd (ν), as one always has HI (ν) ≥ Hd (ν)
(the Hilbert function is the corank of dν1 and therefore can only increase in a specialization), we are done.
⇐ : If ∆S = 0, then by the remark 2, following the definition of ∆S , all the maximal minors of the
specialization ϕS of ϕS vanishes. Therefore ϕS is not onto. In other words, khSi + Iν 6= k[X1 , . . . , Xn ]ν , and
therefore dimk (khSi + Iν )/Iν < HI (ν).
It is clear from the proof that 1) may be replaced by 1’).
IV An example
As an illustration P
we will treat the example of three polynomials of degrees 2,3,3, looking for the
subresultants of degree i (di − 1) = 5, and then compute two specific cases.
So let,
P1 = a1 X12 + a2 X1 X2 + a3 X1 X3 + a4 X22 + a5 X2 X3 + a6 X32 ,
P2 = b1 X13 + b2 X12 X2 + b3 X12 X3 + b4 X1 X22 + b5 X1 X2 X3 + b6 X1 X32 + b7 X23 + b8 X22 X3 + b9 X2 X32 + b0 X33 ,
P3 = c1 X13 + c2 X12 X2 + c3 X12 X3 + c4 X1 X22 + c5 X1 X2 X3 + c6 X1 X32 + c7 X23 + c8 X22 X3 + c9 X2 X32 + c0 X33 .
The matrix of d52 is :
µ
−b1 −b2 −b3 −b4 −b5 −b6 −b7 −b8 −b9 −b0 a1 0 a2 a3 0 0 a4 a5 a6 0 0 0
−c1 −c2 −c3 −c4 −c5 −c6 −c7 −c8 −c9 −c0 0 a1 0 0 a2 a3 0 0 0 a4 a5 a6
we remark that this matrix is of rank 2 unless P1 = 0, and P2 and P3 are proportional.
The reference decomposition chooses the submatrix:
µ
a1
0
0
a1
and the matrix of d∗1 is :
6
¶
,
¶

a1
 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

0
0
a2
a1
0
0
0
0
0
0
0
0
b1
0
c1
0
0
0
0
0
0
0
a3
0
a1
0
0
0
0
0
0
0
0
b1
0
c1
0
0
0
0
0
0
a4
a2
0
a1
0
0
0
0
0
0
b2
0
c2
0
b1
0
0
c1
0
0
a5
a3
a2
0
a1
0
0
0
0
0
b3
b2
c3
c2
0
b1
0
0
c1
0
a6
0
a3
0
0
a1
0
0
0
0
0
b3
0
c3
0
0
b1
0
0
c1
0
a4
0
a2
0
0
a1
0
0
0
b4
0
c4
0
b2
0
0
c2
0
0
0
a5
a4
a3
a2
0
0
a1
0
0
b5
b4
c5
c4
b3
b2
0
c3
c2
0
0
a6
a5
0
a3
a2
0
0
a1
0
b6
b5
c6
c5
0
b3
b2
0
c3
c2
0
0
a6
0
0
a3
0
0
0
a1
0
b6
0
c6
0
0
b3
0
0
c3
0
0
0
a4
0
0
a2
0
0
0
b7
0
c7
0
b4
0
0
c4
0
0
0
0
0
a5
a4
0
a3
a2
0
0
b8
b7
c8
c7
b5
b4
0
c5
c4
0
0
0
0
a6
a5
a4
0
a3
a2
0
b9
b8
c9
c8
b6
b5
b4
c6
c5
c4
0
0
0
0
a6
a5
0
0
a3
a2
b0
b9
c0
c9
0
b6
b5
0
c6
c5
0
0
0
0
0
a6
0
0
0
a3
0
b0
0
c0
0
0
b6
0
0
c6
0
0
0
0
0
0
a4
0
0
0
0
0
0
0
b7
0
0
c7
0
0
0
0
0
0
0
0
a5
a4
0
0
0
0
0
0
b8
b7
0
c8
c7
0
0
0
0
0
0
0
a6
a5
a4
0
0
0
0
0
b9
b8
b7
c9
c8
c7
0
0
0
0
0
0
0
a6
a5
a4
0
0
0
0
b0
b9
b8
c0
c9
c8
0
0
0
0
0
0
0
0
a6
a5
0
0
0
0
0
b0
b9
0
c0
c9

0
0 

0 

0 

0 

0 

0 

0 

0 

a6 

0 

0 

0 

0 

0 

0 

b0 

0 

0
c0
where the columns corresponds to degree five monomials, ordered lexicographically,
X15 , X14 X2 , X14 X3 , X13 X22 , X13 X2 X3 , . . .
Pn Before computing two specific examples, let us do some remarks concerning the case s = n and ν = δ =
i=1 (di − 1), so that Hd (ν) = 1:
0) If Pi ∈ k[X1 , . . . , Xn ], k a field, then HI (δ) 6= 1 implies ∆δ+1
= Res(P1 , . . . , Pn ) = 0,
∅
1) If there exists X α ∈ Mδ such that ∆X α 6= 0, then for X β ∈ Mδ ,
∆X β = 0 ⇔ X β ∈ I ⇒ Z(I) ⊆ {X β = 0},
2) If HI (δ + 1) 6= 0, HI (δ) = 1 ⇔ ∆Xiδ 6= 0 ∀i ∈ [n].
Hence, the ideals of A = Z[Ui,α ] generated the resultant and respectively by the polynomials (∆Xiδ )i∈[n]
and the polynomials (∆X α )|α|=δ have the same radical. We will see further the geometrical meaning of
theses ideals.
As a first example, let us take the three following polynomials:
P1 = −X 2 + 3XY + 3XZ + 2Y 2 − 7Y Z + Z 2 ,
P2 = −7X 3 + X 2 Y + 4X 2 Z + 8XY 2 − XY Z + XZ 2 − Y 3 + 6Y 2 Z + Y Z 2 + 5Z 3 ,
P3 = 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 following results :
∆X15 = 18012905909370 = 2 × 3 × 5 × 29 × 43 × 10651 × 5023,
∆X25 = −72512681354325 = −32 × 52 × 23 × 1123 × 1386377,
∆X35 = −91838873218842 = −2 × 3 × 31 × 103 × 532640111.
Remark that these numbers have only 3 as a common factor, so the reduction modulo p of these equations
defines either the empty variety or a single point, except for p = 3.
And we can compute the resultant :
∆6∅ = 2962182511891377 = 32 × 73 × 660131 × 6829931.
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Therefore the redutions mod p define a single point for p ∈ {73, 660131, 6829931} and at least two points
(but in fact exactly two points by the proposition 3 of [Ch1]) for p = 3.
As a second example we will take polynomials that have a common zero, namely (0 : 0 : 1):
P1 = X12 + 3X1 X2 + 2X22 + X2 X3 ,
P2 = 3X13 + X12 X2 + 3X12 X3 − 2X1 X22 + 3X1 X32 − X23 + 3X2 X32 ,
P3 = X13 − X12 X2 − X12 X3 + 4X1 X2 X3 + 2X1 X32 + X22 X32 − X2 X33 .
Here, we have ∆X35 = −22760361.
As 22760361 = 32 × 13 × 47 × 4139, assuming the identification of ∆X35 with the reduced subresultant
corresponding to the single point (0 : · · · : 0 : 1) proved in [Ch2], we know that the reductions modulo p of
the equations define a single reduced point (namely (0 : 0 : 1)) if and only if p 6∈ {3, 13, 47, 4139}.
The identification of ∆Xnδ with the reduced resultant associated to the single point (0 : · · · : 0 : 1), and
classical properties of the Koszul complex show that HI (δ) = 1 iff I defines the empty variety or a single
point (proof in [Ch2]). This gives the geometrical interpretation mentioned above and prove that, in these
two situations (empty variety or single point given by n polynomials) HI only depends on d = (d1 , . . . , dn ).
Final remarks and open questions.
– Except if n = 2 (or n = 3 and d1 = d2 = d3 = 2), these objects don’t produce (at least for sufficiently
general polynomials of the form Pi = GQi ) the gcd of the Pi ’s (this can be seen by calculation of Hd (ν)).
– For which S do we have ∆S 6= 0 in the generic case ? We conjecture that every S is good if s = n and
P
n−1
ν ≥ i=1 (di − 1) if dn = inf i di .
– What about the irreducibility of these polynomials and of the variety defined by HI (ν) 6= Hd (ν)? We
only know, up to now that, in the case s = n, the resultant (i.e. the case ν = δ + 1) and the polynomials
∆Xiδ are irreducible.
References
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[Ch2] M. Chardin, Formules à la Macaulay pour les sous-résultants en plusieurs variables et application au calcul d’un résultant réduit, prépublication du Centre de Mathématiques de l’Ecole Polytechnique,
décembre 1993.
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[G-V] L. González-Vega, Une théorie des sous-résultants pour les polynômes en plusieurs variables,
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[Jo1] J.-P. Jouanolou, Idéaux résultants, Adv. in Maths, Vol. 37 N◦ 3 (1980), pp. 212–238.
[Jo2] J.-P. Jouanolou, Le formalisme du résultant, Adv. in Maths, Vol. 90 N◦ 2 (1991).
[Jo3] J.-P. Jouanolou, Aspects invariants de l’élimination, Publication de l’IRMA 457/P-263, Université de Strasbourg, 1991.
[Ma] F. S. Macaulay, The Algebraic Theory of Modular Systems, Stechert-Hafner Service Agency,
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