Cryptography Reference
In-Depth Information
Let
ϕ
−
1
∗
i
:
k
[
y
1
,...,y
n
]
→ k
[
x
0
,...,x
n
]bethe
homogenisation
x
deg(
f
i
f
(
x
0
/x
i
,...,x
i
−
1
/x
i
,x
i
+
1
/x
i
,...,x
n
/x
i
)
ϕ
−
1
i
(
f
)(
x
0
,...,x
n
)
=
where deg(
f
)
is the total degree.
We write
f
as an abbreviation for
ϕ
−
1
n
(
f
).
For notational simplicity we often consider
polynomials
f
(
x,y
); in this case, we define
f
z
deg(
f
)
f
(
x/z,y/z
).
=
We now state some elementary relations between projective algebraic sets
X
and their
affine parts
X
∩
U
i
.
Lemma 5.2.20
Let the notation be as above.
[
y
1
,...,y
n
]
and ϕ
−
1
∗
1. ϕ
i
:
k
[
x
0
,...,x
n
]
→ k
:
k
[
y
1
,...,y
n
]
→ k
[
x
0
,...,x
n
]
are
i
-algebra homomorphisms.
2. LetP
k
n
(
=
(
P
0
:
···
:
P
n
)
∈ P
k
)
withP
i
=
0
and letf
∈ k
[
x
0
,...,x
n
]
be homogeneous.
0
implies ϕ
i
(
f
)(
ϕ
−
i
(
P
))
Then f
(
P
)
=
=
0
.
[
x
0
,...,x
n
]
be homogeneous. Then ϕ
−
i
(
V
(
f
))
∈ k
=
V
(
ϕ
i
(
f
))
. In particular,
3. Let f
∩ A
n
=
◦
V
(
f
)
V
(
f
ϕ
)
.
I
k
(
X
)
implies ϕ
i
(
f
)
I
k
(
ϕ
−
1
4. Let X
⊆ P
n
(
k
)
. Then f
∈
∈
(
X
))
. In particular, f
∈
I
k
(
X
)
i
implies f
◦
ϕ
∈
I
k
(
X
∩ A
n
)
.
n
(
0
implies ϕ
−
1
∗
i
5. If P
∈ A
k
)
and f
∈ k
[
y
1
,
..
.,y
n
]
then f
(
P
)
=
(
f
)(
ϕ
i
(
P
))
=
0
.In
particular, f
(
P
)
=
0
implies f
(
ϕ
(
P
))
=
0
.
[
x
0
,...,x
n
]
then ϕ
−
1
∗
i
(
ϕ
i
(
f
))
6. For f
∈ k
|
f . Furthermo
re, if
f has a monomial that
does not include x
i
then ϕ
−
1
∗
i
(
ϕ
i
(
f
))
=
f (in particular, f
◦
ϕ
=
f).
Exercise 5.2.21
Prove Lemma
5.2.20
.
Definition
5.2.22
Let
I
⊆ k
[
y
1
,...,y
n
]
.
Define
the
homogenisation
I
to
be
the
k
[
x
0
,...,x
n
]-ideal generated by the set
{
f
(
x
0
,...,x
n
):
f
∈
I
}
.
Exercise 5.2.23
Let
I
⊆ k
[
x
1
,...,x
n
]. Show that
I
is a homogeneous ideal.
⊆ A
n
(
k
=
De
finitio
n 5.2.24
Let
X
). Define the
projective closure
of
X
to be
X
V
I
(
X
)
⊆ P
n
.
Lemma 5.2.25
Let the notation be as above.
n
.
T
hen ϕ
(
X
)
n
1. Let X
⊆ A
⊆
X and X
∩ A
=
X.
)
be non-empty. Then I
k
X
=
⊆ A
n
(
k
2. Let X
I
k
(
X
)
.
Proof
Part 1 follows directly from the definitions.
Part 2 is essentially that the homogenisation of a radi
ca
l ideal is a radical ideal, we
give a direct proof. Let
f
x
0
g
where
∈ k
[
x
0
,...,x
n
] be such that
f
(
X
)
=
0. Write
f
=
g
∈ k
[
x
0
,...,x
n
] has a monomial that does not include
x
0
.Bypart1,
g
is not constant.
