Graphics Reference
In-Depth Information
using the resultant like in Examples 10.9.1. Compare your answer with what you
would get by simple elimination of t in the equations:
Section 10.10
10.10.1.
List the following monomials in three variables X
1
, X
2
, and X
3
in the degrevlex order:
3
2
2
3
3
X XXX XX XX X X
2
,
,
,
,
,
.
123
32
3
1
1
2
10.10.2.
Show that the degrevlex and deglex order are the same in the case of two variables.
10.10.3.
Apply Algorithm 10.10.2 to the polynomials
(a) f = X
2
+ 2XY
2
- XY, p
1
= 3X + Y - 1
(b) f = X
2
Y + 1, p
1
= X
2
+ X, p
2
= XY + X
Use the deglex order and assume that Y < X.
10.10.4.
If g(X
1
,X
2
,...,X
n
) Πk[X
1
,X
2
,...,X
n
] and a
1
, a
2
,..., a
n
Πk, show that
n
Â
(
)
=
(
)
(
)
g XXX h
,...,
XXXX
,
,...,
(
-
a
)
+
g
a
,
a
,...,
a
12
n
i
1 2
n
i
1
1 2
n
i
=
1
for h
i
(X
1
,X
2
,...,X
n
) Πk[X
1
,X
2
,...,X
n
].
10.10.5.
Let
3
2
2
2
f XYXYYY
P pXXpX
=
+
+
+
,
{
}
3
2
=
=
+
,
=
+
Yp
,
=
YY
-
.
1
2
3
Find a P-normal form for f with respect to the deglex order assuming that Y < X.
10.10.6.
Consider the polynomials
2
2
Œ
[
]
fXY X p XYY p Y X
=
-
,
=
-
,
=
-
R
XY
,
.
1
2
Let P = {p
1
,p
2
}. Show that
ææ
P
2
f
0
and
f
æ Ææ-
X
X
with respect to the deglex order assuming that Y < X. Since f = Yp
1
+ p
2
belongs to
the ideal <P> in
R
[X,Y], this shows that the mere fact that a polynomial belongs to
the ideal <P> does not guarantee that every one of its P-normal forms is zero.
10.10.7.
Use Theorem 10.10.12 to determine which of the following sets of polynomials P are
Gröbner bases for the ideal I, if any, with respect to the deglex order assuming that
Y < X:
P = { p
1
= XY - Y, p
2
= Y
2
(a)
- X }
P = { p
1
= X
2
+ X, p
2
= XY + Y, p
3
= Y
2
(b)
+ Y }
