Euclid's Algorithm and Computation of Polynomial GCD's
485
Notice that the coefficient sizes appear to be doubling at each step until the last.
It is easy to show that the small size of F6 results from the fact that ~4 < ~3 • If all of
the ~ were the same and greater than 1, then the growth would be uniformly ex-
ponential.
3.5
SUBRESULTANTS. Let F~, F2, • • • , Fk be any PRS in 9Ix]. The major result
of Collins [7] is that
F, ~ Sd~ (Fi, F2)
~
Sd~_~-l(F~,
F2),
i = 3, ..., k,
(15)
where ~ denotes similarity (Section 2.3) and where, for 0 _< j < d:,
Si(F~, F2)
is a
polynomial of degree at most j, each of whose coefficients is a determinant of order
d~ + d2 -
2j with coefficients of F~ and F~ as its elements. In particular,
So(F~,
F2)
is the classical
resultant
[5, Ch. 12] of
F1
and F2 ; in general, following Collins, we
shall call S~ (F~, F~) the jth
subresultant
of F~ and F2. The constants of similarity for
(15) may be expressed as products of powers of a3, - • • , ai, ~3, • • • , ~i,f2, • • • ,f~,
and (-1).
Brown and Traub [8] rederive these results in somewhat greater generality; their
treatment is brief and simple, and shows clearly how the subresultants arise.
We shall now establish bounds on the coefficients of the subresultants
Let
T~ = Sji_,_i(F1 ,
F:),
i = 3, "" , k.
(16)
1
ml = ~(dl + d2 -+-
2) -- di-1
= [dl+
d~ -- 2(4i-1 -
1)],
i = 3, ...,
k.
(17)
This is an approximate measure of degree loss. As we proceed through the PRS, it
increases monotonically from m3 > 1 to mk _< ½ (dl + d2). In a normal PRS,
mi
in-
creases by 1 when i increases by 1; in general, the increment is ~i-~. Since each
coefficient of Ti is a determinant of order 2mi with coefficients of F1 and F2 as its
elements, our bounds depend simply on m~.
If the coefficients of F1 and F2 are integers bounded in magnitude by c, then by
Hadamard's theorem [1, p. 375] the coefficients of T~ are bounded in magnitude by
(2m~J) m'.
(18)
Taking the logarithm (to the same base as the base of the number system), we see
that the coefficients of Ti are bounded in length by
mi[2l -4-
log (2m~)],
(19)
where 1 = log c bounds the lengths of the coefficients of Fa and F~. Although the