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In-Depth Information
where we have n rows of a's and m rows of b's. The determinant of SM(f,g) is called
the
(Sylvester) resultant
of f and g and is denoted by R(f,g) (or R
X
(f,g) if we want to
emphasize the fact that f and g are polynomials in X in case a
i
and b
j
are themselves
polynomials in some other variables).
Note that the resultant is not a symmetric function, but it is easy to show from
basic properties of the determinant that R(g,f) = (-1)
mn
R(f,g).
10.4.1. Lemma.
If R = R(f,g) is the resultant of two polynomials f(X) and g(X) of
positive degree m and n, respectively, then
()()
+
()(
,
RGXfX FXgX
=
where F(X) and G(X) are polynomials with deg (F) < m and deg (G) < n.
Proof.
For each i, 1 £ i £ m + n - 1, multiply the ith column of the Sylvester matrix
(10.25) by X
m+n-i
and add the result to the last column. This will produce the matrix
n
-
1
()
()
L
L
L
Ê
aa
a a
0
Xf X
ˆ
mm
-
1
1
0
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
˜
˜
˜
˜
˜
˜
˜
˜
˜
˜
n
-
2
L
L
L
0
aa
aa
Xf X
mm
-
1
1
0
O
L
O
M
()
()
()
L
L
L
0
aa
f X
(10.26)
mm
-
1
m
-
1
L
L
L
bb
b b
0
XgX
n
n
-
1
1
0
m
-
2
L
L
L
0
bb
bb
XgX
n
n
-
1
1
0
M
O
O
L
M
Ë
¯
()
L
L
L
.
0
bb
gX
n
n
-
1
Let C
i,j
denote the cofactors of the matrix in (10.26). Since both of the matrices (10.25)
and (10.26) have the same determinant, expanding the determinant of (10.26) by the
last column gives that
()
=
n
-
1
()
n
-
2
()
++
()
Rfg
,
X
fXC
+
X
fXC
...
fXC
1
,
mn
+
2
,
mn
+
nmn
,
+
m
-
1
()
m
-
2
()
++
()
+
XgXC
+
X gXC
...
gXC
nmn
++
1
,
nmn
++
2
,
mnmn
+ +
,
()()
+
()
(
.
=
GXfX
FXg
Since the polynomials F(X) and G(X) have the desired properties, Lemma 10.4.1 is
proved.
10.4.2. Lemma.
If R = R(f,g) is the resultant of two polynomials f(X) and
g(X) of positive degree m and n, respectively, then R = 0 if and only if there
exist
nonzero
polynomials F(X) and G(X) with deg (F) < m and deg (G) < n, so
that
()()
+
()()
= 0.
GXfX
FXgX
(10.27)
