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and
n
n
-
1
.
..
+
b
,
GbZ b Z
=
+
+
(10.47b)
n
n
-
1
0
where a
i
and b
j
are homogeneous polynomials in
C
[X,Y] of degree m - i and n - j,
respectively. Now the fact that a
i
and b
j
are homogeneous polynomials in X and Y
means that a
i
(0,0) = b
j
(0,0) = 0 whenever i < m and j < n and so
()
=
(
)
=
()
=
(
)
=
F
p
F
001
,,
a
and
G
p
G
001
,,
b
.
m
n
Since
p
does not belong to either
C
1
or
C
2
, it follows that both constants a
m
and b
n
are nonzero. Therefore, by Theorem 10.4.8, the resultant R = R(F,G) of F and G
thought of as polynomials in Z is either 0 or a homogeneous polynomial in X and Y
of degree mn. The possibility that R is identically zero is impossible by Theorem 10.4.3
since F and G do not have any common factors. Therefore, R must be a nonzero homo-
geneous polynomial of degree mn.
Now assume that [x,y,z] lies on the intersection of
C
1
and
C
2
. Then F(x,y,z) =
G(x,y,z) = 0 and z is a common root of
(
)
m
(
)
m
-
1
++
(
)
=
axyz a
,
+
xyz
,
...
axy
,
0
(10.48a)
m
m
-
1
0
and
(
)
n
(
)
n
-
1
++
(
)
=
b xyz
,
+
b
xyz
,
...
b xy
,
0
.
(10.48b)
n
n
-
1
0
In other words, R(x,y) = 0. The converse is also true, that is, if R(x,y) = 0, then equa-
tions (10.48) have a common root. To summarize,
10.7.1. Lemma.
The resultant R of the polynomials F and G vanishes precisely
at those points
q
= (x,y) of
L
where the lines
L
q
contain an intersection point of
C
1
and
C
2
.
Our first result about the number of intersection points of two curves is the fol-
lowing:
10.7.2. Theorem.
Two plane curves of order m and n, respectively, that do not have
a common component have at most mn points in common.
Proof.
Suppose that the curves have mn + 1 points
p
1
,
p
2
,...,
p
mn+1
in common.
Choose the point
p
so that
p
does not lie on any line through any of the point pairs
p
i
and
p
j
, i π j. Then all the lines through
p
contain at most one intersection point
p
i
.
Let
q
i
be the point of
L
that is the central projection of
p
i
from
p
. The mn + 1 points
q
i
are all distinct and roots of the resultant R(F,G) (Lemma 10.7.1). This contradicts
the fact that the resultant has degree mn and proves the Theorem.
We are ready to define the multiplicity of intersection points. Let
C
1
and
C
2
be two
plane curves in
P
2
(
C
) of order m and n, respectively, that have no common component.
Let
p
be a point that does not lie on either
C
1
or
C
2
and let
L
be any line that does not
contain
p
. Choose homogeneous coordinates (X,Y,Z) for
P
2
(
C
) so that
p
has coordi-
