Graphics Reference
In-Depth Information
Note that singular points of
C
(or f) are those points
p
for which
()
=
()
=
()
= 0.
f
p
f
p
f
p
X
Y
It should also be clear that if f has no terms of degree less than r and some of degree
r, then the origin is a point of multiplicity r and the tangents to
C
(or f) at the origin
are the components of the equation which equates the terms of f of degree r to 0.
10.6.3. Theorem.
If
C
and
D
are distinct irreducible curves, then every point
p
in
the intersection of
C
and
D
is a singular point of
C
»
D
. In particular, every point that
belongs to two distinct components of a curve is a singular point.
Proof.
Let f and g be minimal polynomials for
C
and
D
, respectively. It is easy to
see that fg is the minimal polynomial for
C
»
D
. Without loss of generality, we may
assume that
p
is the origin. Then
()
=
()
+
()
≥ 2,
ord fg
ord f
ord g
which clearly implies that the origin is a singular point for fg = 0.
10.6.4. Theorem.
A plane curve has at most a finite number of singularities.
Proof.
First assume that the curve is irreducible and defined by the irreducible poly-
nomial f(X,Y). Either X or Y must appear in f(X,Y). Without loss of generality assume
that it is X. Then f
X
π 0. Since f
X
has smaller degree than f, f does not divide f
X
. It follows
from Theorem 10.5.6 that f
X
has at most a finite number of zeros on the curve. If the
curve is reducible, the result follows from the irreducible case since distinct compo-
nents intersect in at most a finite number of points. Finally, the result applies to pro-
jective curves as well because a curve has only a finite number of infinite points.
10.6.5. Theorem.
If (x
0
,y
0
) is a simple point of a plane curve
C
defined by f(X,Y) =
0, then
(
)
(
)
+
(
)
(
)
=
fxy Xx
,
-
fxy Yy
,
-
0
0
X
00
0
Y
00
is the equation of the tangent line to
C
at (x
0
,y
0
).
Proof.
This is an easy consequence of equation (10.40) and the discussion above.
Below are some curves that show some of the possibilities in the behavior of a
curve at a point, in this case, the origin.
X
3
- X
2
+ Y
2
10.6.6. Example.
= 0
Analysis.
See Figure 10.10(a). The origin is an ordinary double point with tangent
lines X + Y = 0 and X - Y = 0.
X
3
+ X
2
+ Y
2
10.6.7. Example.
= 0
