Cryptography Reference
In-Depth Information
The
n
g
◦
ϕ
∈
I
k
(
X
) and so
g
=
g
◦
ϕ
∈
I
k
(
X
). It follows from part 6 of Lemma
5.2.20
that
f
∈
I
k
(
X
).
Theorem 5.2.26
Let f
(
x
0
,x
1
,x
2
)
∈ k
[
x
0
,x
1
,x
2
]
be a
k
-irreducible homogeneous polyno-
mial. Let
2
.
=
⊆ P
X
V
(
f
(
x
0
,x
1
,x
2
))
Then I
(
X
)
=
(
f
(
x
0
,x
1
,x
2
))
.
k
Proof
Let 0
2 be such that
f
(
x
0
,x
1
,x
2
) has a monomial that does not feature
x
i
(such an
i
must exist since
f
is irreducible). Without loss of generality, suppose
i
≤
i
≤
=
2.
ϕ
∗
(
f
)
Write
g
(
y
1
,y
2
)
=
=
f
(
y
1
,y
2
,
1). By part 6 of Lemma
5.2.20
the homogenisation of
g
is
f
.
Let
Y
2
=
X
∩ A
=
V
(
g
). No
te that
g
is
k
-irreducible (since
g
=
g
1
g
2
implies, by tak-
ing homogenisation,
f
=
g
1
g
2
). Let
h
∈
I
(
X
). Then
h
◦
ϕ
∈
I
(
Y
), and so, by Corol-
k
k
lary
5.1.20
,
h
◦
ϕ
∈
(
g
). In othe
r w
ords, there is some
h
1
(
y
1
,y
2
) such that
h
◦
ϕ
=
gh
1
.
Taking homogenisations gives
fh
1
|
h
and so
h
∈
(
f
).
∈ k
k
=
⊆
Corollar
y 5
.2.27
L
et f
(
x,y
)
[
x,y
]
be a
-irreducible polynomial and let X
V
(
f
)
A
2
. Then X
=
⊆ P
2
.
V
(
f
)
Exercise 5.2.28
Prove Corollary
5.2.27
.
Example 5.2.29
The projective closure of
V
(
y
2
x
3
n
is
V
(
y
2
z
x
3
=
+
Ax
+
B
)
⊆ A
=
+
Axz
2
Bz
3
).
+
2
2
Exercise 5.2.
30
Let
X
=
V
(
f
(
x
0
,x
1
))
⊆ A
and let
X
⊆ P
be the projective closure of
X
. Show that
X
−
X
is finite (in other words, there are only finitely many points at infinity).
A generalisation of projective space, called
weighted projective space
, is defined as
follows: for
i
0
,...,i
n
∈ N
···
denote by (
a
0
:
a
1
:
:
a
n
) the equivalence class of elements
k
n
+
1
in
under the equivalence relation
(
λ
i
0
a
0
,λ
i
1
a
1
,
,λ
i
n
a
n
)
(
a
0
,a
1
,
···
,a
n
)
≡
···
∈ k
∗
. The set of equivalence classes is denoted
for any
λ
P
(
i
0
,...,i
n
)(
k
). For example, it
makes sense to consider the curve
y
2
x
4
ax
2
z
2
z
4
as lying in
(1
,
2
,
1). We will not
discuss this topic further in the topic (we refer to Reid [
448
] for details), but it should be
noted that certain coordinate systems used for efficient elliptic curve cryptography naturally
live in weighted projective space.
=
+
+
P
5.3 Irreducibility
We have seen that
V
(
fg
) decomposes as
V
(
f
)
V
(
g
) and it is natural to consider
V
(
f
)
and
V
(
g
) as being 'components' of
V
(
fg
). It is easier to deal with algebraic sets that cannot
∪
