486
w.s.
BROWN
(2mlc2t 2 )m~.
(21)
This generalization of (18) follows easily from the results of [9]. Taking the loga-
rithm, we see that these integer coefficients are bounded in length by
m~[21 ~-
log (2m~) + 2 log t],
(22)
which is a generalization of (19).
In the case of a primitive PRS, (15) and (16) imply that F~ I Ti, so the bounds
(18) and (20) on the coefficients of T~ also apply to the coefficients of F~. When the
coefficients of F~ and T~ are polynomials, it may happen in rare cases (see Section
5.2) that one or more integer coefficients of polynomial coefficients of F~ are larger
than any integer coefficient of any polynomial coefficient of T~. Nevertheless, we
conjecture that no integer coefficient of any polynomial coefficient of F~ can ever
exceed the bound (21).
To justify the reduced PRS algorithm (Section 3.4), it is shown in [7] and [8]
that T~ I F~ for i = 3, • • • , k, and therefore all coefficients of F~ are in 9. In the case
of a normal reduced PRS, it is further shown that
Fi =
±Ti,
i = 3, ... , k,
(23)
and therefore the bounds (18)-(22) apply to the coefficients of F~ in this case as
well.
3.6
THE SUBRESVLTANT PRS ALGORITmL
If we could choose the ~ in such a
way as to satisfy (23) even in abnormal cases, then no coefficient GCD's would be
needed to compute the F~, and yet the bounds (18)-(22) would apply to their
coefficients. Fortunately this can be done. It is shown in [8] that F~ = T~ for i =
3, ... , ]c provided that we choose
~
=
(- 1)~+1,
(24)
~
=
--fi_2~b~ '-2,
i = 4, ...
, k,
where
(25)
(
4"
~i--3.Ll--~i--3
~
=
~--ji-2j
~i-1
,
i = 4, ...
,k.
At the present time it is not known whether or not these equations imply ~, ~ E ~"
In any event, 6~ and 3~ belong to the quotient field 5: of ~, and they yield the PRS,
F1, F2, T3, ... , Tk in 9[x].
It is possible to eliminate 6~ from (24), and write ~ explicitly as a product of
powers of f2, • • " ,
fi-2,
and (- 1). The resulting formula is given in [7].
In the case of a normal PRS, note that the subresultant PRS algorithm (24) and
the reduced PRS algorithm (13) agree up to signs as required by (23); in fact, the
signs also agree if ~1 is odd.
It is natural to wonder how close a subresultant PRS is likely to be to the cor-
responding primitive PRS. In the example (4), the subresultant PRS is
1, 0, 1, 0, --3, --3, 8, 2, --5
3, 0, 5, 0, --4, --9, 21
15, 0, --3, 0, 9
Journal of the Association for Computing Machinery, Vol. 18, No. 4, October 1971