Graphics Reference
In-Depth Information
with
(
) =
() =
2
gXY
,
DG
Y X X
+
-
1
.
Let W = V(g) Õ R 2 . Figure 10.7(b) shows W . The variety W is the counterexample we
are looking for. Note that G = H(g) and [1,0,0] ΠV(H(g)), but the topological closure
of W in P 2 is W » {[0,1,0]} (Exercise 10.3.4) and [1,0,0] is not in it. Although we do
not yet have the tools to prove this, it is the case that H( W ) = V(H(g)) (Exercise 10.5.3).
We have found a counterexample to Theorem 10.3.6(2). Actually, W serves as our
counterexample no matter what might have happened because if it were the case that
H( W ) π V(H(g)), then we would have violated Theorem 10.3.6(1) instead.
The reason that behavior as in Example 10.3.7 is impossible in complex projec-
tive space is that the algebraic closure of the complex numbers guarantees that equa-
tions have enough solutions to prevent the kind of isolated points that we found in
our example. Over the complex numbers the projective variety defined by g(X,Y) will
be a pinched sphere and what was our isolated point over the reals will be the pinched
point which is no longer isolated (Exercise 10.3.5).
We shall have more to say about the projective completion of a variety at the end
of Section 10.8 once we have a little more algebra behind us.
10.4
Resultants
This section introduces an important tool for determining if polynomials in one vari-
able have a common factor. It will be needed later in several contexts.
Definition.
Let
() =
m
m
-
1
fX
a X
+
a
X
+
...
+
a
,
m
m
-
1
0
() =
n
n
-
1
gX
b X
+
b
X
+
...
+
b
(10.24)
n
n
-
1
0
be two polynomials in D[X] of positive degrees m and n, respectively, where D is a
unique factorization domain. Define the Sylvester matrix SM(f,g) of f and g by
aa
L
L
a a
0
L
0
Ê
ˆ
mm
-
1
1
0
Á
Á
Á
Á
Á
Á
Á
Á
Á
˜
˜
˜
0
aa
L
L
aa
L
0
mm
-
1
1
0
O
L
O M
˜
˜
˜
˜
˜
˜
0
L
aa
L
L
a
mm
-
1
0
, () =
(10.25)
SM f g
bb
L
b b
0
L
L
0
n
n
-
1
1
0
0
bb
L
bb
L
L
0
n
n
-
1
1
0
M
O
O
L M
Ë
¯
0
L
L
bb
L
b
nn
-
1
0
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