Cryptography Reference
In-Depth Information
Lemma A.5.4
Let R be an integral domain.
1. If F
(
x
)
∈
R
[
x
0
,...,x
n
]
is homogeneous and λ
∈
R then F
(
λx
0
,...,λx
n
)
=
λ
d
F
(
x
0
,...,x
n
)
.
2. If F
1
,F
2
∈
R
[
x
0
,...,x
n
]
are non-zero and homogeneous of degreesr ands respectively
then F
1
(
x
)
F
2
(
x
)
is homogeneous of degree r
+
s.
3. Let F
1
,F
2
∈
R
[
x
0
,...,x
n
]
be non-zero. If F
1
(
x
)
F
2
(
x
)
is homogeneous then F
1
(
x
)
and
F
2
(
x
)
are both homogeneous.
Proof
See Exercise 1-1 (page 6) of Fulton [
199
].
A.5.2 Resultants
Let
R
be a commutative integral domain. Let
F
(
x
)
F
n
x
n
F
n
−
1
x
n
−
1
=
+
+···+
F
0
and
G
m
x
m
G
n
−
1
x
n
−
1
G
(
x
)
G
0
be two polynomials over
R
with
F
0
,F
n
,G
0
,G
m
=
0. The polynomials
F,xF,...,x
m
−
1
F,G,xG,...,x
n
−
1
G
can be written as
n
=
+
+···+
+
m
lin-
m
variables 1
,x,...,x
n
+
m
−
1
and so the variable
x
may be
eliminated to compute the
resultant
(there should be no confusion between the use of the
symbol
R
for both the ring and the resultant)
ear combinations of the
n
+
F
0
F
1
···
F
n
00
···
0
0
F
0
···
F
n
−
1
F
n
0
···
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
00
···
0
F
0
F
1
···
F
n
R
(
F,G
)
=
R
x
(
F,G
)
=
det
.
G
0
···
G
m
0
··· ··· ···
0
0
G
0
···
G
m
0
··· ···
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
00
···
0
0
G
0
···
G
m
=
i
=
0
F
i
x
i
and
Theorem A.5.5
Let
k
be a field and F
(
x
)
,G
(
x
)
∈ k
[
x
]
. Write F
(
x
)
=
i
=
0
G
i
x
i
. Suppose
F
0
G
0
=
G
(
x
)
0
. Then R
(
F
(
x
)
,G
(
x
))
=
0
if and only if F
(
x
)
and
G
(
x
)
have a common root in
k
.
Proof
See Proposition IV.8.1 and Corollary IV.8.4 of [
329
].
Theorem
A.5.5
is generalised to polynomials in
R
[
x
] where
R
is a UFD in Lemma 2.6
on page 41 of Lorenzini [
355
]. Section IV.2.7 of [
355
] also describes the relation between
R
(
F,G
) and the norm of
G
(
α
) in the number ring generated by a root
α
of
F
(
x
).
If
F
(
x,y
)
,G
(
x,y
)
∈ Z
∈ Z
[
y
] for the resultant, which is a
polynomial in
y
, obtained by treating
F
and
G
as polynomials in
x
over the ring
R
[
x,y
] then write
R
x
(
F,G
)
= Z
[
y
].
If
F
and
G
have total degree
d
in
x
and
y
then the degree in
y
of
R
x
(
F,G
)is
O
(
d
2
).
A.6 Field extensions
General references for fields and their extensions are Chapter II of Artin [
14
], Chapter V
of Hungerford [
271
] or Chapter V of Lang [
329
].
