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In-Depth Information
10.10.8.
Use the deglex order on k[X,Y] and Algorithm 10.10.13 to find Gröbner bases for the
ideals <P> below assuming that Y < X:
(a)
P = { XY + X, X 2
+ Y }
P = { X 2 Y + X, X + Y }
(b)
Consider the ideal I =<X 2 Y - X - Y,XY 2 + Y> in k[X,Y]. Use a Gröbner basis to deter-
mine which, if any, of the polynomials below belongs to I. If it does, then express the
polynomial in terms of that Gröbner basis.
(a)
10.10.9.
f = X 3 Y + 2X 2 Y 2
+ XY 3
- X 2
- XY
f = X 3 Y + X 2 Y 2
- XY + X 2 Y - XY 2
- X 2
(b)
10.10.10.
Solve Exercise 10.9.1 using Gröbner bases.
Section 10.12
Let C be a plane curve in P 2 (k). Show that a parameterization of C defined in one
coordinate system will remain a parameterization when transformed to another coor-
dinate system.
10.12.1.
10.12.2.
Show that
2
È
Í
33
1
-
+
t
4
t
t
˘
˙
() =
g t
,
,
1
2
2
t
1
+
is a parameterization of the projective curve in P 2 ( C ) defined by
2 2 2
XY Z
4
+
9
-
36
=
0
.
Find its center.
Let g(t) be the parameterization in Exercise 10.12.2. Let h(t) = t + t 2 . Show by direct
computation that g h (t) =g(h(t)) has the same center as g(t).
10.12.3.
10.12.4.
Consider the irreducible curves below:
(a)
Y 2
- X 5
= 0
X 4
+ X 2 Y 2
- Y 2
(b)
= 0
(One way to see that this curve is irreducible is to note that it has a parameterization
Ê
Á
2
2
4
2
t
t
-
+
1
1
t
--
+
21
t
ˆ
˜ ˆ
() =
g t
,
.
˜
(
)
2
2
tt
1
What are their singular points? Find a sequence of quadratic transformations that
transform them into curves with only ordinary singularities.
Section 10.13
10.13.1.
Consider the affine conic defined by
(
) =
2
2
f X Y
,
56 5 4250
X
-
XY
+
Y
-
X
+
Y
+
=
.
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