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.