502
w.s. BROWN
Mi,~ 12(d + 1)~',
M~ ~
1 ~ (d
+
1 )v,
MT,~l(d+
1) ~,
Ms~P(v,d)+
(d+ 1)~,~, (d+l)'+',
Mll ~
n(d
+
l)V,
M14~l 2(d+
1)~,
M15 ,~ 12 (c/ -t- 1 )~.
(92)
Total Time.
By assumption (A2) no unlucky primes will occur. Hence if F1
and F2 are relatively prime, only a single prime will be needed; otherwise the re
quired number is
< log~u
,
(93)
where ~
is the final value of u* in Step (13). Clearly u
need not exceed
2c2t,
where t is the number of terms in (~, and where c bounds the magnitudes of the
coefficients of
(~, flit,
and /t~. It follows immediately that ~ < 2 logs c -t- logs t
+ logs '2. But by assumption (A3), logs c < 21, and by assumption (A4), log~ t < 1.
Hence
< 4l
+ 2.
(94)
Summing (92) with Steps (6)- (12) weighted by (94), we obtain
M(v, l, d) £ 12(d +
1) ~ ÷
l(d +
1) ~+1.
(95)
When F~ and F2 are relatively prime (RP), it suffices to sum Steps (1)-(9) and
(15); thus we find the smaller bound
211(Re)(V, l, d) £ 12(d +
1) ~ +
(d + 1) ~'+~.
(96)
5.8
COMPARISON. From (80) and (95), we see that the bound on M is strictly
dominated by the bound on C. We shall now prove that M is strictly dominated by
C in the region v >_ 2.
Let F~ and F2 be random polynomials in v variables subject only to the constraints
imposed by the dimension vector (l, d), and let C_ be the maximum computing time
to obtain their pseudo-remainder F3. Then by (69), C_ dominates 12 (d +
1)2~-~.
On the other hand, by (95), M is strictly dominated by
M+ = 12(d +
1) "+~.
v 2
Since
C_/M+
= (d -I- 1)
, it follows that M < M+ < C_ ~ C in the region v _> 2.
This completes the proof.
6.
Summary and conclusions
In attempting to generalize Euclid's algorithm to the case of univariate or multi-
variate polynomials with integer or rational coefficients, one immediately encounters
the problem of coefficient growth. This is most serious when the growth is allowed to
go unchecked, as in the Euclidean PRS algorithm, or when the algorithm is recur-
sively applied to inflated polynomial coefficients, as in the multivariate case of the
primitive PRS algorithm.