Graphics Reference
In-Depth Information
Generalize (a) and show that for any finite set of points in
P
n
, there is a hyper-
plane in
P
n
that does not pass through these points.
(b)
Section 10.4
Consider the polynomial f(X) = (X - 1)
2
. Show that R(f,f ¢) = 0, thereby verifying
Corollary 10.4.5 in this case.
10.4.1.
10.4.2.
Prove Corollary 10.4.10.
Section 10.5
Show that the only varieties in
C
2
are
C
2
, a finite (possibly empty) set of points, or the
union of a plane curve and a finite (possibly empty) set of points. (
Hint:
Suppose that
a variety
X
is defined by polynomial equations
10.5.1.
ff
== ==
...
k
0
.
12
Let g be the greatest common factor of the f
i
and let f
i
= h
i
g. Determine the conditions
under which
X
is the union of two varieties, one of which is determined by g and the
other by the h
i
. Use Theorem 10.5.6.)
Show that f(X,Y) = X
2
+ Y
2
- r
2
, r Œ
R
, r π 0, is irreducible in
C
[X,Y] and hence
also in
R
[X,Y]. (
Hint:
Show that any factorization of f(X,Y) must consist of two
linear factors and then use Exercise 1.5.19(b).)
10.5.2.
(a)
Show that f(X,Y) = X
2
+ Y
2
+ r
2
, r Œ
R
, r π 0, is irreducible in
C
[X,Y]. (
Hint:
Con-
sider the transformation (x,y) Æ (
i
x,
i
y) in
C
2
.)
(b)
10.5.3.
Look back at Example 10.3.7 where we considered the polynomial
2
gXY
(
,
)
=
YX X
+
-
1
,
and the variety
W
= V(g) Õ
R
2
. Show the following
(a)
H(g) is irreducible in
R
[X,Y,Z].
The projective completion H(
W
) in
P
2
is just V(H(g)).
(b)
10.5.4.
Just because a set is described by transcendental functions does not automatically
mean that it is not an algebraic variety. Consider, for example, the circle { (cos t,sin t)
| t Œ
R
}. On the other hand, show that the graph of the sine function, { (t,sin t) | t Œ
R
}, is not an algebraic variety.
Section 10.6
10.6.1.
Consider the polynomial
(
)
=
2
2
2
fXY
,
XY XY
-
-
2
X
+
Y
+
2
X
-
4
Y
+
4
.
Compute deg
(1,2)
f.
