Cryptography Reference
In-Depth Information
Definition 5.2.6
Let
f
∈ k
[
x
0
,...,x
n
] be a homogeneous polynomial. A point
P
=
n
(
(
x
0
:
···
:
x
n
)
∈ P
k
)isa
zero
of
f
if
f
(
x
0
,...,x
n
)
=
0 for some (hence, every) point
(
x
0
,...,x
n
) in the equivalence class (
x
0
:
···
:
x
n
). We therefore write
f
(
P
)
=
0. Let
S
be
a set of polynomials and define
n
(
V
(
S
)
={
P
∈ P
k
):
P
is a zero of
f
(
x
) for all homogeneous
f
(
x
)
∈
S
}
.
n
(
A
projective algebraic set
is a set
X
=
V
(
S
)
⊆ P
k
)forsome
S
⊆ k
[
x
]. Such a set is
k
an algebraic extension of
also called a
k
-algebraic set. For
X
=
V
(
S
) and
k
define
k
)
n
(
k
):
f
(
P
)
X
(
={
P
∈ P
=
0 for all homogeneous
f
(
x
)
∈
S
}
.
=
=
Example 5.2.7
The hyperbola
y
1
/x
can be described as the affine algebraic set
X
V
(
xy
−
1)
⊂ A
2
over
R
. One can consider the corresponding projective algebraic set
V
(
xy
whose points consist of the points of
X
together with the points
(1 : 0 : 0) and (0 : 1 : 0). These two points correspond to the asymptotes
x
−
z
2
)
⊆ P
2
over
R
0
of the hyperbola and they essentially “tie together” the disconnected components of the
affine curve to make a single closed curve in projective space.
=
0 and
y
=
Describe the sets
V
(
x
2
y
2
z
2
)(
2
(
x
2
)(
Exercise 5.2.8
+
−
R
)
⊂ P
R
) and
V
(
yz
−
R
)
⊆
2
(
P
R
).
A set of homogeneous polynomials does not in general form an ideal as one cannot
simultaneously have closure under multiplication and addition. Hence, it is necessary to
introduce the following definition.
Definition 5.2.9
A
k
[
x
0
,...,x
n
]-ideal
I
⊆ k
[
x
0
,...,x
n
]isa
homogeneous ideal
if for
=
i
f
i
we have
f
i
∈
every
f
∈
I
with homogeneous decomposition
f
I
.
Exercise 5.2.10
Let
S
⊂ k
[
x
] be a set of homogeneous polynomials. Define (
S
)tobethe
={
j
=
1
f
j
(
x
)
s
j
(
x
):
n
k
[
x
]-ideal generated by
S
in the usual way, i.e. (
S
)
∈ N
,f
j
(
x
)
∈
k
. Prove that (
S
) is a homogeneous ideal. Prove that if
I
is a
homogeneous ideal then
I
[
x
0
,...,x
n
]
,s
j
(
x
)
∈
S
}
=
(
S
) for some set of homogeneous polynomials
S
.
⊆ P
n
(
k
Definition 5.2.11
For any set
X
) define
=
{
}
.
I
k
(
X
)
f
∈ k
[
x
0
,...,x
n
]:
f
is homogeneous and
f
(
P
)
=
0 for all
P
∈
X
We stress that
I
k
(
X
) is not the stated set of homogeneous polynomials but the ideal generated
by them. We write
I
(
X
)
=
I
(
X
).
k
n
is
defined over
An algebraic set
X
⊆ P
k
if
I
(
X
) can be generated by homogeneous
polynomials in
k
[
x
].
Proposition 5.2.12
Let
k
be a field:
n
(
1. If S
1
⊆
S
2
⊆ k
[
x
0
,...,x
n
]
then V
(
S
2
)
⊆
V
(
S
1
)
⊆ P
k
)
.
2. If fg is a homogeneous polynomial
then V
(
fg
)
=
V
(
f
)
∪
V
(
g
)
(recall
from
Lemma
A.5.4
that f and g are both homogeneous).
