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Eventually (in Section 10.17) we shall make this equation look more elegant still.
If two hypersurfaces intersect transversally it will say that the degree of the intersec-
tion is equal to the product of the degrees of the hypersurfaces. We cannot do this
yet because the intersection, although a variety, may not be a hypersurface (defined
by a single polynomial) and we have only defined the degree of hypersurfaces so
far.
The definition of intersection multiplicity as presented has the advantage that it
leads relatively easily to generalizations of Bézout's theorem in higher dimensions.
However, as it stands, it is difficult to work with since zeros of resultants are not easy to
analyze. Therefore, the following theorem is useful. It allows one to compute the inter-
section multiplicities from the multiplicities of the individual curves in many cases.
10.7.6. Theorem.
Let
C
1
and
C
2
be two plane curves that have no component in
common and let
p
be an intersection point. Then
(
)
≥
() ( )
m
CC
,
m
C
m
C
p
12
p
1
p
2
with equality holding if and only if the tangents of
C
1
at
p
are disjoint from those of
C
2
at
p
, that is, the curves intersect transversally.
Proof.
See [BriK81].
Bèzout's theorem has a number of geometric applications. Among them are the
theorems of Pascal and Brianchon. Another application gets an estimate of the
number of singular points of a plane curve.
10.7.7. Theorem.
Let
C
be a plane curve of order n without multiple components.
Then
Â
()
(
()
-
)
£-
(
)
mm
CC
1
n
1 .
p
p
pC
Œ
Proof.
See [BriK81].
The result in Theorem 10.7.7 can be strengthened in the case of some special
curves.
10.7.8. Theorem.
Let
C
be an irreducible plane curve of order n. Then
Â
()
(
()
-
)
£-
(
)
(
)
mm
CC
1
nn
1
-
2 .
p
p
pC
Œ
Proof.
See [BriK81]. The inequality cannot be improved because equality is possi-
ble as the curve
x
nn
-1
+
=
0
shows (the origin is a point of multiplicity n - 1).
