THEOREM 1.
Let F~ and F: be given nonzero primitive polynomials in Z[x~ , . • • , x,] ,
let G =
gcd(F1,
Fo), and let d~ = O~(G), where O~ denotes the degree in X~. Also, let
( i)
Sj
(F1,
F2) :lenote the jth subresultant of F~ and F~ viewed as univariate polynomials
in xi with polynomial coe~cients, and let ~
be the (integer) content of ~(~) (F~, F~)
*Jdi
viewed as a multivariate polynomial with integer coe~cients.
(Here :l~ is the degree of
G in X~, and not, as in Section 3.5, the degree of the ith polynomial in a PRS. )
Finally, let
o" =
l~I ¢i.
(44)
i=1
Then every unlucky prime divides ¢.
PROOF.
For fixed p from Step (6) of Algorithm M, let P~
=
Fi
rood p and F~ =
F~ rood p. Also, let G = gcd (F1, F2) in
Z,[x~, ... ,
x,,], and let e~ =
di(G).
If p is unluckv, then there is some i with
di <
e~,
and therefore S (~) (P~,
d i
F~) ~ 0 rood p. Now let Cl and c~ denote the leading (polynomial) coefficients of
F~ and F:
respectively, relative to x~ If
p ~ oct,
then S (~) (F1, F~)
S (~) (F~
F:)
,
•
d i

di
,
rood p. Otherwise, suppose p divides the first k leading coefficients of F~ for some
c(i)(F~
F:)
~
~,~i)(Fi
F~)
rood
p.
In
either
k > 0. Then it can be shown that ~,d~
,
C: ~d~
case
t.Jdi~*(i)(Fi,
F~) = 0 mod p, p [
~(i)~di/F~
1,
F2),
p
i o'i,
and finally p [ a, as was to be
shown.
THEOREM 2.
Let u be the number of unlucky primes p > a, where a >_ 2 is a given
integer. Let
c = max [~[,
(45)
where ~ ranges over the coefficients of
F1
and F~ . Let
v
m = ½ max
(Oi(F~) + O~(F:)).
(46)
i~1
Let
v
t = max h,
(47)
i~1
where t~ is the maximum number of terms in any polynomial coe~cient of
F1 or
F2
viewed as univariate polynomials in x~. Finally, let
~7 = my
log,
(2mc2t 2).
(48)
Then u < ~.
PROOF.
Let P be the product of the unlucky primes p > o~. Since P I o by Theorem
m
/,~
2.2
\mv
1, wehavea ~ < P < a. Nowby (21),o~<
(2mc2t 2) ,soo_<
~2mc~)
.Itfollows
that u < log, a _< ~, as was to be shown.
THEOREM 3.
The probability that p is unlucky is at most v/p.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Euclid's Algorithm and Computation of Polynomial GCD's
493
PROOF.
For all sufficiently small problems, we have o~ < p for i = 1, ... ,
v,
and therefore p cannot be unlucky. However, in the general case, we may assume that
the quantities ~ mod p are independent random variables in Zp. Hence the proba
bility that p [ ~isp~
and the probability that p [cris 1 
(1 
p~)~ <
vp 1
where this last inequality follows by induction on v. This completes the proof.
4.5
ALGORITHSI P.
Let
Fi'
and
F2'
be given nonzero polynomials in
Zp[xi, .. • ,
x~], where p is a fixed, sufficiently large prime. [If p is too small, the elements of Zp
may be exhausted by Step (6) of the algorithm.] Algorithm P computes
G'
= ged (F~',
F/), H~' =
F(/G',
and H( =
F(/G'.
Since Zp is a field,
Fi'
and F2' are automatically
primitive, and
G'
is monie.
In the univariate ease, Algorithm P simply invokes Algorithm U, which is pre
sented in Section 4.7.
In the multivariate ease, we could single out a main variable, and then apply
Algorithm C. If so, we would still be faced with the problem of coefficient growth,
since the polynomial coefficients would grow in degree even though their integer
coefficients (in Zp) could not grow in size.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '13
 MRR
 Math, Coefficient, F~

Click to edit the document details