Journal of the Association for Computing Machinery, Vol. 18, No. 4, October 1971
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Euclid's Algorithm and Computation of Polynomial GCD's
497
(A3)
In Algorithms C and 5/[, it is assumed that the integerlength (Section
5.2) of an exact quotient of polynomials does not exceed the integerlength of the
dividend. In Algorithm C, it is further assumed that the integerlength of a sum or
difference of polynomials is the maximum of the integerlengths of the summands,
and that the integerlength of a product does not exceed the sum of the integer
lengths of the factors.
(A4)
In Algorithm M, it is assumed that the integerlengths of the given poly
nomials are small compared to the largest singleword integer fl so that the supply of
singleword primes will not be exhausted. Similarly it is assumed that the number of
terms in the GCD is small compared to ~, so that the required number of primes
can be bounded in a simple and realistic way.
After introducing some basic concepts in Section 5.2, we shall discuss integer
operations in Section 5.3, polynomial operations in Section 5.4, Algorithm C in
Section 5.5, Algorithm P in Section 5.6, and Algorithm M in Section 5.7. Finally we
compare Algorithms C and 5I in Section 5.8.
5.2
BASIC CONCEPTS, Let f and g be real functions defined on a set S. If there
is a positive real number c such that
If(x)l < clg(x)l
for all x ~ S, we write f ,~ g,
and say that f is
dominated
by g. If f ~ g and g ,~ f, we write f ~
g, and say that f
and g are
eodominant.
Finally, if f ~ g but f ~~ g, we write f < g, and say that. f is
strictly dominated
by g. Clearly codominance is an equivalence relation among the
real functions on S, while strict dominance defines a partial ordering among the
resulting codominance classes. In the author's opinion, this notation and terminology
(from [10]) are significantly superior to the traditional "littleoh" and "bigoh"
notation [11, p. 5].
Since the purpose of the analysis is to provide insight, not detail, we shall con
sider only a minimum number of independent parameters. Every polynomial F which
is considered in the analysis will be characterized either by its dimension vector
(1, d), defined below, or by its degree in the main variable together with a single
dimension vector for all its coefficients.
First we define the
length
of a nonzero integer to be the logarithm (to some fixed
base such as 2 or 10) of its magnitude. Next we define the
integerlength
of a nonzero
polynomial
F ~ Z[x~, ... , x~]
to be the maximum of the lengths of its nonzero
coefficients; this will be denoted by il(F). Finally, for nonzero
F C Z[Xl, ... , x,]
we define the
dimension vector
(l, d) by the relations 1 = il(F) and d = max (Oi (F)).
For integers it is clear that the length of a product is the sum of the lengths of the
factors. For polynomials, the integerlength of a product may be smaller or larger
than the sum of the integerlengths of the factors, because of the additive combina
tion of terms in polynomial multiplication. For example, if A (x) = x 
1, then
A (x)2
=
x 2 _
2x
+ 1, and il(A 2) = log 2 > 2 il (A) = 0. On the other hand, letting
B(x)
= x s+2S+3x
6+4x 5+5x 4+4x 3+3x 2+ 2x+
1, we have A (x )B (x )
=
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 Math, Coefficient, F~

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