which differs very little from the corresponding primitive PRS (12), except at the
last step.
Although it is not known whether this behavior is typical, it is known that there
is a broad spectrum of possibilities. On the one hand, there are subresultant PRS's
in which all polynomials except the last are primitive. On the other hand [1, p. 377],
let F~, F2, •   , Fk be a normal subresultant PRS, and let G be a primitive poly
nomial with leading coefficient g. Then it is easy to show that the subresultant PRS
for
GFi
and
GF2
is
GF1, GF~ , g~+~GF3 , g~l+3GF4 , • • • , g6~+2k 5GFk ,
which diverges
linearly from the primitive PRS
GF~ , GF2 , • • • , GFk .
3.7
FURTHER IMPROVEMENTS.
Having used the subresultant PRS algorithm to
compute Fi =
Ti [see (16)] for i = 3, .. , j, it may happen that a divisor rj of
cont(Ti)
is available with little or no extra work. For example, we know that
0 = gcd (f~, f2) divides Ti for j
= 3, .. , k, and it occasionally happens that f~
divides T1 for some i < j.
When such a rj is available, we would like to incorporate it into/3j, so that Fj =
TJrj.
However, if we do so, we cannot necessarily complete the PRS by direct
application of (24). Fortunately, it is possible to modify (24) so as to complete the
PRS and
furthermore to guarantee that
Fi I Ti
for i = j +
1, •  • , k.
Let F1, F2, T3
....
, Tk be a subresultant PRS, and let F1, F~, F3,..., F~ be an
im
proved PRS
with Fi =
Ti/ri
for i = 3, • • • , k. Let
Ti = F1,
T2 = F2, and rl =
r2 = 1, and let t~ denote the leading coefficient of T~. Then
T~ = riF~,
(27)
where i =
1, • • • , k. Now from (9), (24), (25), and (27) it is easy to show that the
improved PRS is defined by
/~3/r3 =
( 1)~'+t,
~i/r, =
f,2¢~ i~rT~ 2',
where
Given F~,
i = 4, ...,k,
(28)
@3 ~
1,
(29)
~b~ =
(r,_2fi_2)~~b~~ '~,
i = 4, ...,
k.
• ",
F~_i,
we first compute prem(Fi_2,
Fi1) and divide by
([~i/rl)
from (28) to obtain the subresultant Ti =
riFi.
We then choose
ri,
and divide by
it to obtain Fi.
Clearly, there are many possible variations on this theme. To illustrate the theme,
let us return to the example (4), for which the primitive PRS is (12) and the sub
resultant PRS is (26). The reader should imagine the problem to be enough harder
that he could not easily discover the contents of the subresultants T~. Choosing
ri = fil, whenever
fi1 I T~,
and rl = 1 otherwise, the improved PRS is
Journal of the Association for Computing Machinery, Vol. 18, No. 4, October 1971
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488
w.s.
BROWN
1, 0, 1, 0, 3, 3, 8, 2, 5
3, 0, 5, 0, 4, 9, 21
5,0, 1,0,3
13, 25, 49
9326,  12300
260708.
(30)
By this simple strategy, we have succeeded in making F3 and F4 primitive, but we
have failed to remove the factor of 2 from Fs.
3.8
COMPARISON. Let us now compare the foregoing algorithms. It is clear that
the Euclidean PRS algorithm suffers so severely from coefficient growth that it does
not merit further consideration. We also exclude the reduced 1)RS algorithm be
cause it is never better than the subrcsultant 1)RS algorithm, and it can be extremely
inefficient in certain abnormal cases.
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 Spring '13
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 Math, Coefficient, F~

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