(77)
Total Time.
Summing these bounds, we find
C(v, l, d) • d.C(v

1, 21, 2d) +
12(d +
1)4v22v3~,
(78)
for v > 0. Starting with the formula
C (0, l) £
12,
(79)
which follows from (63), we can now prove by induction on v that
C(v, l, d) £ 12(g ¢
1)4~22~23~.
(80)
5.6
ALGORITHM P.
In analyzing the computing time for Algorithm P, we shall
use the notation of Section 4.5. Let F~' and F2' be the given nonzero polynomials in
Zp[Xl,
• • • , x~], and let d bound the components of their degree vectors.
Let P (v, d) denote the maximum computing time for Algorithm P, and let Pi (v, d)
denote the time for the ith step. We shall omit the analyses of several steps which
obviously make no contribution to the final bound.
Step (1).
Since F~' and
F~'
each have at most (d + 1)~1 terms, and since the
time to compute a GCD in the coefficient domain
~ = Zp[x]
by Algorithm U is at
most d 2 [by (70)], we have
P~ .~ (d + 1) ~+~.
(81)
Step
(2).
By (67), the time required to divide Cl or c~ into a coefficient of Fi'
or F( is dominated by (d t 1 )5. Since there are at most 2 (d + 1 )~'~ such divisions,
we have
P2 £ (d
+ 1)~+~.
(82)
Let ~ denote either 0 or any coefficient of F~ or F2. Thus ~ C
Z~,[x~],
and 0~. (~) < d. Since the time to map ~ into ~ mod(x~  b) = ~ (b) ~ Zp, either by
division or by Horner's rule [1, p. 423], is dominated by d +
1, the time to map 0
and all of the coefficients of F1 and F2 into Zp is dominated by (d + 1 )". Since this
Journal of the Association for Computing Machinery, Vol. 18, No. 4, October 1971
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View Full DocumentEuclid's Algorithm and Computation of Polynomial GCD's
501
dominates the time required for the ensuing multiplications, we have
P7 ,~ (d + 1)v.
(83)
Step (8).
Here we invoke Algorithm P recursively, and then multiply the re
sulting GCD and cofactors by ~ and '
g , respectively. Thus
Ps(v,d)
~P(v
1,~/) + (d+ 1)v1.
(84)
Step (11).
Here each of the coefficients of G*, H**, and H2* must be extended
from degree n 
2 in x~ to degree n 
1. By (71), the required time for each suell
extension is codominant with n; hence
Pu ~ n (d + 1)~1.
(85)
Step (1~).
Recall that
G* = (~ = (O/g)G.
BythesamereasoningasinStep(1),
eont(G*) can be computed in time (d + l) v+'. By ~he same reasoning as in Step
(2), the ensuing divisions of this content into G*, and of g into H,* and H2*, can also
be performed in time (d + 1)v+l. Hence
P14 ,~ (d + 1)v+~.
(86)
Step (15).
By (66), each multiplication of a coefficient of G by c can be per
formed in time (d + 1)2. Hence the time to compute G' is bounded by (d + 1)~+~.
Since the same reasoning holds for H( and H(, we have
P15 ,~
(d + 1
)'+~.
(87)
Total Time.
By assumption (A2) no unlucky bvalues will occur. Hence if F,
and F2 are relatively prime, only one bvalue will be needed; otherwise the required
number is
~=o+1
= max (0~ (fi,), O~ (/Tz)) + 1
< 2 d + 1.
(88)
Summing the preceding bounds, with Steps (6)(12) weighted by (88), we obtain
P(v,d)~d.P(v
1, d) + (d+ 1)~+t,
(89)
for v > 1. Starting with the formula
P(1, d) ~ d ~,
(90)
which follows from (70), we can now prove by induction on v that
P(v, d) ~
(d +
1) ~+~.
(91)
5.7
ALGORITHM M.
In analyzing the computing time for Algorithm M, we
shall use the notation of Section 4.3. Let
F~'
and F'
2 be the given nonzero poly
nomials in
Z[xl, • • • , x~J,
and let (1, d) bound their dimension vectors.
Let
M(v, l, d)
denote the maximum computing time for Algorithm 5'I, and let
Mi(v, l, d) denote the time for the ith step. In view of the similarity of Algorithm
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