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Observe that if we replace r
i
by s
j
then f(X) and g(X) have a common factor X - s
j
.
By Theorem 10.4.3, R vanishes after this substitution and hence Fact 2 is proved.
Fact 2 and the fact that the ring of polynomials k[a
m
,b
n
,r
1
,...,r
m
,s
1
,...,s
n
] over
any field k is a prime factorization domain imply that the right-hand side of equation
(10.30) divides R. On the other hand, (10.33) implies that R is homogeneous of degree
mn with respect to r
i
's and s
j
's. Therefore,
m
n
'
'
n
n
m
(
)
Rcab
=
r s
-
(10.34)
i
j
i
=
1
j
=
1
for some constant c. To determine c, set all the r
i
to 0. In that case, a
1
= a
2
= ...= a
m
= 0 and so the determinant of the Sylvester matrix (10.25), namely, R, is a
m
b
m
. This
fact and (10.34) implies that
m
n
Ê
Á
ˆ
˜
'
m
nm
n
n
m
(
)
ab
=
cab
s
.
(10.35)
j
0
j
=
1
The right-hand side of (10.35) is just
mn
n
n
m
j
m
m
nm
-
()
1
ca b s
=
ca b .
0
Therefore, c = 1 and (10.30) is proved. Equalities (10.31) and (10.32) follow easily, and
so the theorem is proved.
Theorem 10.4.8 leads to another useful formula for evaluating resultants.
10.4.10. Corollary.
Given polynomials f(X), g(X), and h(X), then the resultants
satisfy the product identity
(
)
=
()( )
Rfgh
,
Rfh Rgh
,
,
.
Proof.
Exercise 10.4.2. For an alternate proof see [Seid68].
The resultant we have defined above addresses the problem of finding the common
zeros of two polynomials. There are times when one is interested in the common zeros
of a larger collection of polynomials. For properties and applications of the corre-
sponding
multipolynomial resultant
see, for example, [CoLO98].
10.5
More Polynomial Preliminaries
Polynomials are the glue which holds the various aspects of algebraic geometry
together. This section summarizes some additional important basic facts about poly-
nomials and the hypersurfaces they define.
Now, we saw in Section 10.2 that if one wanted to understand curves it was impor-
tant to look at these in projective space (in fact, complex projective space) since one
