Euclid's Algorithm and Computation of Polynomial GCD's
dominates the time required for the ensuing multiplications, we have
P7 ,~ (d + 1)v.
Here we invoke Algorithm P recursively, and then multiply the re-
sulting GCD and cofactors by ~ and --'
g , respectively. Thus
1,~/) + (d+ 1)v-1.
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.
G* = (~ = (O/g)G.
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+~.
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
(d + 1
By assumption (A2) no unlucky b-values will occur. Hence if F,
and F2 are relatively prime, only one b-value will be needed; otherwise the required
= max (0~ (fi,), O~ (/Tz)) + 1
< 2 d + 1.
Summing the preceding bounds, with Steps (6)-(12) weighted by (88), we obtain
1, d) + (d+ 1)~+t,
for v > 1. Starting with the formula
P(1, d) ~ d ~,
which follows from (70), we can now prove by induction on v that
P(v, d) ~
In analyzing the computing time for Algorithm M, we
shall use the notation of Section 4.3. Let
2 be the given nonzero poly-
Z[xl, • • • , x~J,
and let (1, d) bound their dimension vectors.
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