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We saw in the first chapter that running the Euclidean
Algorithm in
[x] can lead to a significant expression swell.
Let us try to estimate how coefficients grow.
That is, we want to estimate the maximal space
needed to store one coefficient of an intermediate polynomial.
Let N be the number of bits of a machine word.
Definition 1
For an integer
a we define
(
a) the
length of
a as the minimum number of words to encode
a.
Hence for
a 0 we have
(a) = log_{2}(a)/N + 1 and (0) = 0 
(1) 
For a rational number
a =
b/
c with
b,
c and
gcd(
c,
d )= 1
we define
(
a), the
length of
a, as the maximum
among
(
b) and
(
c).
Definition 2
For a polynomial
a [
x] of degree
n 0
such that all coefficients have denominator
b and such that the
numerators
a_{n},
a_{n1},...,
a_{1},
a_{0} have gcd 1
then we define the
length of
a as
(a) = max((b),((a_{i}))) 
(2) 
Remark 1
A polynomial
a [x] of degree n 0
can be represented with
words.
Proposition 2
Let
a,
b be univariate polynomials over
in
x.
We assume that
a and
b are monic and that
deg(
a) = deg(
b) + 1 holds.
Let
q and
r be the quotient and the remainder of
a w.r.t.
b.
Then we have
Proof.
We define
a = x^{n} + a_{n1}x^{n1} + ^{ ... } + a_{0} and b = x^{n1} + b_{n2}x^{n2} + ^{ ... } + b_{0} 
(5) 
It is not hard to see that
q = x + a_{n1}  b_{n2} 
(6) 
Then
Thus
(bq) max((a),(b)) + 1 + (b) + (deg(q) + 1) 
(8) 
Recall that
Since
(
bq)
(
a) we obtain
(r) 

max((a),(b)) + 1 + (b) + (deg(q) + 1) + 1 


(a) + 2(b) + 3 

(10) 
Indeed
max(
(
a),
(
b))
(
a) +
(
b)
and
(deg(
q) + 1) =
(2)
1.
Remark 2
A similar result as Proposition 2
holds for monic polynomials
a, b [x] such that
deg(a) = deg(b) + 1.
See [vzGG99].
These results provide us with an exponential upper bound for the
coefficient of the gcd computed by the Euclidean Algorithm.
But in fact a polynomial bound can be established.
Moreover there is an efficient modular approach for computing gcds in
[x] and
[x]
as we shall see in the next sections.
Next: Relation between gcds in [x]
Up: Advanced Computer Algebra: The resultant
Previous: Advanced Computer Algebra: The resultant
Marc Moreno Maza
20040427