August 2011
Intermediate to advanced
332 pages
9h 10m
English
Content preview from Lattice Basis Reduction
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286 Lattice Basis Reduction
S(f, g) =
a
n
a
n−1
.
.
.
.
.
.
.
.
.
a
n
a
0
a
n−1
.
.
.
.
.
.
a
0
|
{z }
m columns
b
m
b
m−1
.
.
.
.
.
.
.
.
.
b
m
b
0
b
m−1
.
.
.
.
.
.
b
0
|
{z }
n columns
FIGURE 15.9
The Sylvester matrix S(f, g) of polynomials f, g ∈ Z[x]
Proof. Apply Hadamard’s inequality to the Sylvester matrix.
If g = f
′
in Lemma 15.30, then m = n−1 and |f
′
|
∞
≤ n|f |
∞
, and hence
|res(f, f
′
)| ≤ (n+1)
(n−1)/2
n
n/2
|f|
n−1
∞
|f
′
|
n
∞
≤ (n+1)
(n−1)/2
n
n/2
|f|
n−1
∞
n|f|
∞
n
= (n+1)
(n−1)/2
n
n/2
n
n
|f|
2n−1
∞
≤ (n+1)
n/2
(n+1)
n/2
(n+1)
n
|f|
2n−1
∞
= (n+1)
2n
|f|
2n−1
∞
.
Therefore
|disc(f)| ≤ (n+1)
2n
|f|
2n−1
∞
. (15.5)
Lemma 15.31. Let F
q
(q = p
n
) be a finite field, and let
f ∈ F
q
[x] be a
nonconstant polynomial. The following ...
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