2) d, and similarly [using assumption (A3)]
large random problems we find that d' > d,
1 ) (~ --l-- 2)~+I (d --k 1 )2,-2,
1) (d +
1)2~-2.
(69)
For polynomials in Zp[x], assumption (A4) permits us to restrict our attention to
single-word primes; hence all coefficient operations can be performed in some fixed
amount of time. For Algorithm U, let dl =
O(Fi)
and d2 = 0(F2), and suppose
dl >_ d2 > 0. Then, by essentially the same argument that led to (63),
T~ (U) ~ d, (< -
d3),
(70)
where da is the degree of the GCD. For the Chinese remainder algorithm, we again
use the notation of Section 4.8. Let d~ and d2 be the degrees of m~ and m2, respec-
tively. By (70), the time to compute c in Step (1) is codominant with
did2.
Since
Steps (2) and (3) can also be performed within this bound, we have
Tp (CRA) ~
&d2.
(71)
5.5
ALGORITmr C.
In analyzing the computing time for Algorithm C, we shall
use the notation of Section 2.4, and we shall assume that Step (4) is performed by
means of the subresultant PRS algorithm (Section 3.6). Let F~' and F2' be the given
nonzero polynomials in v variables, and let
(1, d)
bound their dimension vectors.
Let C(v, l, d) denote the maximum computing time for Algorithm C, and let
C~(v, l, 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 this step involves at most 2d +
1 GCD's of polynomial co-
efficients, we have
C~(v, l, d) ~ d.C(v -
1, l, d).
(72)
!
Step
(2).
By (67), the time required to divide Cl or c2 into a coefficient of F~
or F2' '
"
~
2
1)2~-2.
is dominated bv 1 (4 +
Since there are at most, 2 (d +
1 ) such divisions,
we have
C~ ,~, 12 (d +
1 )2~-1.
(73)
Step (4).
Clearly most of the work in this step is in the pseudo-divisions which
yield prem(Fi_2, Fi_~) for i = 3, • • • , k. As in Section 3.1, let dl denote the degree
of F~ in the main variable. Then by assumption (A1), we have d~ = d2 -t- 2 -
i,
for i = 3, • • • , k, and therefore k < d2 -4- 2. Furthermore, by (20) the degree of F~
in any auxiliary variable cammt exceed
d(d~ -4-
d2) _<
2d 2.
Similarly, by (22),
ignoring the logarithm terms in accordance with assumption (A3), the integer-
length of Fi cannot exceed
l(d~ -4- d2) < 2ld.
Hence we can bound the time for a
single pseudo-division by replacing l by
2hl, d~
by d, 5 by 1, and d by 2 d 2 in (69).
Multiplying the result by d, which bounds the number of pseudo-divisions, and
applying some obvious simplifying inequalities, we obtain
C4 ,~, l 2 (d +
1 )4~22~3~.
(74)
Step
(6).
To bound the time for computing
fk/O,
we can replace v by v -
1,
12 by l, d2 by d, 13 by
21d,
and da by 2d 2 in
(67); the result is dominated
Journal of the Association for Computing Machinery, Vol. 18, No. 4, October 1971