and partition it inton+ 1subsets
S0 =
Now we can create a system of equations that generalizes system (5.2)
xα/xd00F0 = 0 for allxα ∈S0, (5.6) . . .
xα/xdnnFn = 0 for allxα ∈Sn.
This system has some special properties. Since Fi is a homogeneous polynomial of total degree di, the polynomial xα/xdiiFi is a homogeneous polynomial of total degree d for all i= 0, . . . , nand therefore can be written as a linear combination of monomials of degreedin n+ 1variablesx0, . . . , xn. There exist(n+d
d
)monomials of degreedinn+ 1variables.
The total number of equations in the system (5.6) is equal to the number of elements in sets S0, . . . Sn, which is also (n+d
d
) because these sets contain all monomials of degree din vari-ablesx0, . . . xn. Thus the system of polynomial equations (5.6) consists of(n+d
We can consider each monomial of total degreedin system (5.6) as an unknown and in this way get a system of(n+d
d
)linear equations in(n+d
d
)unknowns. Such system can be written in the form
MX=0, (5.7)
where Mis a (n+d
d
)×(n+d
d
) coefficient matrix and X is a vector of (n+d
d
) monomials. We denote the determinant of this coefficient matrixMasDn. This determinant has some interesting properties (see for example [39]) and is very close to the resultant of the initial system (5.2).
Macaulay [100] showed that the resultant is equal to this determinantDndivided by a minor, i.e. the determinant of a submatrix of the(n+d
d
)×(n+d
d
)matrixM. To obtain this submatrix we first need the following definition
Definition 1. A monomialxα = xα00. . . xαnn of total degreedis reduced ifxdii divides xαfor exactly onei.
We denote byM′ the submatrix of the coefficient matrixM(5.7) of the extended system (5.6) that we obtain by deleting all rows and columns corresponding to reduced monomialsxα, and byD′nits determinant. Then there holds the following theorem.
Theorem 8. The resultant of the system of homogeneous polynomial equations (5.2) is given by Res(F0, . . . , Fn) =±Dn
D′n (5.8)
wheneverD′n̸= 0.
Proof. The proof for this theorem can be found in Macaulay’s paper [100] or in [54].
Note that the setsS0, . . . , Snfrom (5.5) and the determinantsDn andD′nare dependent on the ordering of the variablesx0, . . . , xn. In this context, the indexnin the notationDnmeans that the variablexn comes last when creating sets S0, . . . , Sn. For different ordering of the variables with the last variablexi fori ∈ {0, . . . , n−1}we get different setsS0, . . . , Sn and slightly different extended system (5.6). For example, takingx0 last means thatS0 consists of the monomials of the form We will denoteDi the determinant of the coefficient matrixMof the extended system wherexi come last when creating setsS0, . . . , Sn. Theorem 8 holds also for this determinantDi and its minorDi′.
Theorem 8 is universal and can be used to obtain the resultant for any system ofn+1 homoge-neous polynomial equations inn+1unknowns (5.2). However, this theorem has some disadvan-tages. In general case, when coefficients of the polynomials in the system are not numerical, the computation of the resultant using formula (5.8) requires to divide two very large polynomials, which can be very time-consuming. On the other hand for numerical coefficients problems with vanishing ofDn′ appear. The problem whenDn′ = 0can be solved using Canny’s method which introduces a new variableuand considers the resultantRes
(
F0−uxd00, . . . , Fn−uxdnn )
. This method is described in [39].
In some cases, the resultant can be expressed as a single determinant like for the system ofn+ 1 linear homogeneous equations inn+ 1unknowns from Example 26 or in the case of Sylvester resultant (3.2) for two homogeneous equations in two unknowns. In these cases previously mentioned problems disappear. Therefore, it would be nice if this could be done for all resultants, i.e. all resultants could be expressed as a single determinant. Even though in some cases such formulas exist (e.g. for the resultant of three ternary quadrics which can be expressed by the6×6determinant [121] or for the resultant of three polynomials of degreer), most of the time we can’t express the resultant in the form of a single determinant. Moreover, it is not known if this is possible in general.
Besides the Macaulay’s formula from Theorem 8, there are other ways how to represent mul-tivariate resultants as quotients. The best known areBezoutiansorBezout’s resultants[44] and Dixon’s resultants[76]. The Macaulay formulation has however the advantage that the entries of the coefficient matrices are actually the coefficients of the initial polynomial equations, whereas in the case of Bezout or Dixon formulations the entries are polynomial functions of these coef-ficients [101].
Now we show the Macaulay’s method for computing resultants on a simple example.
Example 27. (Macaulay’s method for computing resultants) Consider a system of three homogeneous polynomial equations in three unknownsx0, x1, x2and symbolic coefficients
F0 = a0x0+a1x1+a2x2= 0, (5.11)
F1 = b00x20+b01x0x1+b02x0x2+b11x21+b12x1x2+b22x22 = 0, F2 = c00x20+c01x0x1+c02x0x2+c11x21+c12x1x2+c22x22 = 0.
The degrees of these equations ared0 = 1, d1 = 2, d3 = 2. Therefore,dfrom Equation (5.4) is d=∑3
i=0(di−1) + 1 = 3.
Now we take all monomials of degree3 in variablesx0, x1, x2. There is (2+3
3
) = 10 such monomials and these monomials arex30, x20x1, x20x2, x0x21, x0x22, x0x1x2, x31, x21x2, x1x22, x32. We partition these monomials into setsS0, S1andS3as described in (5.5). In this case
S0 = {
x30, x20x1, x20x2, x0x21, x0x22, x0x1x2
}, (5.12)
S1 = {
x31, x21x2
}, S2 = {
x1x22, x32} .
We use these sets to create the extended system of polynomial equations in the form (5.6). In this way we obtain the following system of ten equationsx20F0 =x0x1F0=x0x2F0 =x21F0 = x22F0 =x1x2F0=x1F1=x2F1=x1F2=x2F2 = 0.
These ten equations can be written in the matrix form
We denote the10×10coefficient matrix in (5.13) asMand its determinantD2.
To compute the resultant using Macaulay’s formula from Theorem 8 we need to compute the determinant of a submatrix of this coefficient matrix. We obtain this submatrix by deleting all rows and columns corresponding to reduced monomials, i.e. monomialsxα = xα00xα11xα22 of total degree3which are divisible by exactly onexdii,i∈ {0,1,2}andd0 = 1, d1 = 2, d2 = 2.
In this case reduced monomials are monomialsx30, x20x1, x20x2, x0x1x2, x31, x21x2, x1x22, x32. The only two remaining monomials are monomials x0x21 andx0x22, which correspond to the fourth and fifth column and row of the coefficient matrix in (5.13). Therefore, after deleting all rows and columns corresponding to the reduced monomials we obtain the2×2matrix
M′ =
Then the resultant of the system of three homogeneous polynomial equations in three un-knowns (5.11) is according to Theorem 8
Res1,2,2(F0, F1, F2) =±det(M)
det(M′) =±D2
D2′, (5.15)
wheneverD2̸= 0.