Exercises Multiply in
Z
/
(
5
)
[
x
,
y
]
{\displaystyle {}\mathbb {Z} /(5)[x,y]}
x
4
+
2
x
2
y
2
−
x
y
3
+
2
y
3
and
x
4
y
+
4
x
2
y
+
3
x
y
2
−
x
2
y
2
+
2
y
2
.
{\displaystyle x^{4}+2x^{2}y^{2}-xy^{3}+2y^{3}{\text{ and }}x^{4}y+4x^{2}y+3xy^{2}-x^{2}y^{2}+2y^{2}.}
Multiply in
Q
[
x
,
y
,
z
]
{\displaystyle {}\mathbb {Q} [x,y,z]}
x
5
+
3
x
2
y
2
−
x
y
z
3
and
2
x
3
y
z
+
z
2
+
5
x
y
2
z
−
x
2
y
.
{\displaystyle x^{5}+3x^{2}y^{2}-xyz^{3}{\text{ and }}2x^{3}yz+z^{2}+5xy^{2}z-x^{2}y.}
Show that the
ideal
(
X
,
Y
)
{\displaystyle {}(X,Y)}
polynomial ring
K
[
X
,
Y
]
{\displaystyle {}K[X,Y]}
field
K
{\displaystyle {}K}
principal ideal .
Sketch in
R
2
{\displaystyle {}\mathbb {R} ^{2}}
x
2
−
y
2
−
1
=
0
{\displaystyle {}x^{2}-y^{2}-1=0}
x
2
+
x
y
+
y
2
=
0
{\displaystyle {}x^{2}+xy+y^{2}=0}
x
2
+
y
2
+
1
=
0
{\displaystyle {}x^{2}+y^{2}+1=0}
x
2
+
y
2
=
0
{\displaystyle {}x^{2}+y^{2}=0}
x
2
+
y
3
=
0
{\displaystyle {}x^{2}+y^{3}=0}
x
3
−
y
5
=
0
{\displaystyle {}x^{3}-y^{5}=0}
x
2
−
x
3
=
0
{\displaystyle {}x^{2}-x^{3}=0}
x
3
+
y
3
=
1
{\displaystyle {}x^{3}+y^{3}=1}
x
4
+
y
4
=
1
{\displaystyle {}x^{4}+y^{4}=1}
−
5
+
3
x
+
4
x
2
+
x
3
−
y
2
=
1
{\displaystyle {}-5+3x+4x^{2}+x^{3}-y^{2}=1}
Which of the quadrics on the right can be described
(in what sense)
with less than three variables?
Determine which quadric from
Example 43.9
can be described as a
graph ,
and which can be described as a
surface of revolution.
Theorem 43.10
.
Bring the real pure-quadratic polynomial
3
X
2
−
5
Y
2
+
8
X
Y
{\displaystyle 3X^{2}-5Y^{2}+8XY}
into
standard form.
Bring the real pure-quadratic polynomial
2
X
2
−
Y
2
+
3
Z
2
+
4
Y
Z
{\displaystyle 2X^{2}-Y^{2}+3Z^{2}+4YZ}
into
standard form.
Theorem 43.13
.
We are looking for the new basis, the transformation of the variables
(coordinate transformation),
and the simplified quadratic polynomial.
We consider the real quadratic polynomial
F
=
X
2
−
4
Y
2
+
6
X
Y
−
3
X
+
Y
+
2
{\displaystyle {}F=X^{2}-4Y^{2}+6XY-3X+Y+2\,}
with the pure-quadratic part
G
=
X
2
−
4
Y
2
+
6
X
Y
.
{\displaystyle {}G=X^{2}-4Y^{2}+6XY\,.}
a) Determine the
standard form
of
G
{\displaystyle {}G}
b) Determine an orthonormal basis such that
G
{\displaystyle {}G}
X
,
Y
{\displaystyle {}X,Y}
c) Express
F
{\displaystyle {}F}
d) Determine the
standard form
of
F
{\displaystyle {}F}
Bring the real quadratic polynomial
5
X
2
−
2
Y
2
−
6
X
Y
−
5
X
−
3
Y
−
7
{\displaystyle 5X^{2}-2Y^{2}-6XY-5X-3Y-7}
into
standard form .
Let
Q
1
,
Q
2
∈
R
2
{\displaystyle {}Q_{1},Q_{2}\in \mathbb {R} ^{2}}
c
>
0
{\displaystyle {}c>0}
E
=
{
P
∈
R
2
∣
d
(
P
,
Q
1
)
+
d
(
P
,
Q
2
)
=
c
}
.
{\displaystyle {}E={\left\{P\in \mathbb {R} ^{2}\mid d(P,Q_{1})+d(P,Q_{2})=c\right\}}\,.}
Show that
E
{\displaystyle {}E}
0
,
1
,
−
1
{\displaystyle {}0,1,-1}
Determine the
normalized standard form
of the real
quadric
7
x
2
−
11
y
2
+
15
x
y
.
{\displaystyle 7x^{2}-11y^{2}+15xy.}
Determine the
normalized standard form
of the real
quadric
3
x
2
+
2
y
2
+
2
x
y
−
2
y
z
.
{\displaystyle 3x^{2}+2y^{2}+2xy-2yz.}
Determine the
normalized standard form
of the real
quadric
x
2
+
y
2
−
2
z
2
−
4
x
y
+
6
x
y
−
2
y
z
.
{\displaystyle x^{2}+y^{2}-2z^{2}-4xy+6xy-2yz.}
Let
V
{\displaystyle {}V}
Minkowski space
of dimension
n
{\displaystyle {}n}
T
=
{
v
∈
V
∣
⟨
v
,
v
⟩
=
1
}
.
{\displaystyle {}T={\left\{v\in V\mid \left\langle v,v\right\rangle =1\right\}}\,.}
For what
n
{\displaystyle {}n}
T
{\displaystyle {}T}
n
{\displaystyle {}n}
Let
V
{\displaystyle {}V}
Minkowski space
of dimension
n
{\displaystyle {}n}
T
=
{
v
∈
V
∣
⟨
v
,
v
⟩
=
1
}
.
{\displaystyle {}T={\left\{v\in V\mid \left\langle v,v\right\rangle =1\right\}}\,.}
Let
w
{\displaystyle {}w}
observer vector
of an observer
B
{\displaystyle {}B}
V
B
{\displaystyle {}V_{B}}
T
∩
V
B
{\displaystyle {}T\cap V_{B}}
Let
K
{\displaystyle {}K}
field .
The
image
of the curve defined by
K
⟶
K
2
,
t
⟼
(
t
2
,
t
3
)
,
{\displaystyle K\longrightarrow K^{2},t\longmapsto \left(t^{2},\,t^{3}\right),}
is called semicubical parabola . Show that a point
(
x
,
y
)
∈
K
2
{\displaystyle {}(x,y)\in K^{2}}
x
3
=
y
2
{\displaystyle {}x^{3}=y^{2}}
Let
f
:
R
⟶
R
2
,
t
⟼
(
t
2
,
t
3
)
.
{\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} ^{2},t\longmapsto \left(t^{2},\,t^{3}\right).}
Determine the points
t
0
∈
R
{\displaystyle {}t_{0}\in \mathbb {R} }
f
(
t
)
=
(
t
2
,
t
3
)
{\displaystyle {}f(t)=\left(t^{2},\,t^{3}\right)}
(
1
,
0
)
{\displaystyle {}(1,0)}
We consider the curve
R
⟶
R
2
,
t
⟼
(
t
2
−
1
,
t
3
−
t
)
.
{\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ^{2},t\longmapsto (t^{2}-1,t^{3}-t).}
a) Show that the
image points
(
x
,
y
)
{\displaystyle {}(x,y)}
y
2
=
x
2
+
x
3
.
{\displaystyle {}y^{2}=x^{2}+x^{3}\,.}
(
x
,
y
)
∈
R
2
{\displaystyle {}(x,y)\in \mathbb {R} ^{2}}
y
2
=
x
2
+
x
3
{\displaystyle {}y^{2}=x^{2}+x^{3}}
t
1
{\displaystyle {}t_{1}}
t
2
{\displaystyle {}t_{2}}
Let
C
⊆
R
2
{\displaystyle {}C\subseteq \mathbb {R} ^{2}}
image
under the polynomial mapping
R
⟶
R
2
,
t
⟼
(
t
3
−
1
,
t
2
−
1
)
.
{\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ^{2},t\longmapsto \left(t^{3}-1,\,t^{2}-1\right).}
Determine a polynomial
F
≠
0
{\displaystyle {}F\neq 0}
C
{\displaystyle {}C}
F
{\displaystyle {}F}
Let
T
{\displaystyle {}T}
graph
of the standard parabola
y
=
x
2
,
{\displaystyle {}y=x^{2}\,,}
and let
M
⊆
R
3
{\displaystyle {}M\subseteq \mathbb {R} ^{3}}
T
{\displaystyle {}T}
x
{\displaystyle {}x}
a) Show that
M
{\displaystyle {}M}
quadric .
M
{\displaystyle {}M}
Hand-in-exercises How many monomials of
degree
d
{\displaystyle {}d}
Determine all solutions of the circle equation
x
2
+
y
2
=
1
{\displaystyle {}x^{2}+y^{2}=1\,}
for the
fields
K
=
Z
/
(
2
)
{\displaystyle {}K=\mathbb {Z} /(2)}
Z
/
(
3
)
{\displaystyle {}\mathbb {Z} /(3)}
Z
/
(
5
)
{\displaystyle {}\mathbb {Z} /(5)}
Z
/
(
7
)
{\displaystyle {}\mathbb {Z} /(7)}
We consider the real quadratic polynomial
F
=
3
X
2
−
5
Y
2
+
7
X
Y
+
4
X
−
2
Y
+
5
{\displaystyle {}F=3X^{2}-5Y^{2}+7XY+4X-2Y+5\,}
with the pure-quadratic part
G
=
3
X
2
−
5
Y
2
+
7
X
Y
.
{\displaystyle {}G=3X^{2}-5Y^{2}+7XY\,.}
a) Determine the
standard form
of
G
{\displaystyle {}G}
b) Determine an orthonormal basis such that
G
{\displaystyle {}G}
X
,
Y
{\displaystyle {}X,Y}
c) Express
F
{\displaystyle {}F}
d) Determine the
standard form
of
F
{\displaystyle {}F}
We consider the cone
K
=
{
(
x
,
y
,
z
)
∈
R
2
∣
x
2
+
y
2
=
z
2
}
⊆
R
3
,
{\displaystyle {}K={\left\{(x,y,z)\in \mathbb {R} ^{2}\mid x^{2}+y^{2}=z^{2}\right\}}\subseteq \mathbb {R} ^{3}\,,}
and let
E
⊆
R
3
{\displaystyle {}E\subseteq \mathbb {R} ^{3}}
K
∩
E
{\displaystyle {}K\cap E}
conic section .
a) Show that every conic section
K
∩
E
⊆
E
≅
R
2
{\displaystyle {}K\cap E\subseteq E\cong \mathbb {R} ^{2}\,}
is, in suitable coordinates
u
,
v
{\displaystyle {}u,v}
R
2
{\displaystyle {}\mathbb {R} ^{2}}
u
,
v
{\displaystyle {}u,v}
Example 43.8
can be realized as a conic section.
Determine the
normalized standard form
of the real
quadric
5
x
2
−
4
y
2
+
z
2
−
x
y
+
3
x
z
.
{\displaystyle 5x^{2}-4y^{2}+z^{2}-xy+3xz.}
![{\displaystyle {}\mathbb {Z} /(5)[x,y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e19b61b10b0498e3b079717f7dc95077268dec41)
![{\displaystyle {}\mathbb {Q} [x,y,z]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc077fc3a7098445bfa98091d13fa9f57cce2b0e)
![{\displaystyle {}K[X,Y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2979004ad27b9ba60c428e400cbf04d9f0957730)
