482
w.s. BROWN
Since the familiar process of polynomial division with remainder requires exact
divisibility in the coefficient domain, it is usually impossible to carry it out for non
zero F1, F2 C 9[x]. However, the process of
pseudodivision
[1, p. 369] always yields a
unique
pseudoquotient
Q = pquo(F1, F2) and
pseudoremainder
R = prem (F1, F2),
such that
f25+lF1 
QF2 = R
and 0 (R) < 0 (F2), where
f2
is the leading coefficient of
F~, and ~ = 0 (F1) 
0 (F2).
For nonzero F1, F2 C 9[x], we say that F1 is
similar
to F2 (F1 ~ F~) if there exist
al, a~ E 9 such that
alF1 = a2F2.
Here al and a2 are called
coe~cients of similarity.
For nonzero F1, F2 C 9Ix] with 0 (F1) > 0 (F2), let F1, F2, • • • , Fk be a sequence of
nonzero polynomials such that F~ ~
prem(F~_2, F~i) for i = 3, ...,
k, and
prem (Fk1,
Fk)
= 0. Such a sequence is called a
polynomial remainder sequence
(PRS). From the definitions, it follows that there exist nonzero a~, f~i C 9 and Q~ ~~
pquo(F1, F2) such that
~,F, = o~iFi_2


Q,F~_i,
O(F~) <
0(F~_i),
i = 3, ..., k.
(7)
Because of the uniqueness of pseudodivision, the PRS beginning with F~ and F
is unique up to similarity. Furthermore, it is easy to see that gcd(F1, F2) ~ gcd(F ,2
F3)
.....
gcd(Fk_~, Fk) ~ Fk. Thus, the construction of the PRS yields t2he
desired GCD to within similarity.
2.4
ALGORITHM C.
A polynomial F C 9Ix] will be called
primitive
if its nonzero
coefficients are relatively prime; in particular, all polynomials over a field are primi
tive. Since 9 is a unique factorization domain, it follows [6, pp, 7477] that 9Ix] is a
unique factorization domain whose units are the units of 9. Hence each polynomial
F C 9[x] has a unique representation of the form F = cont(F)pp(F), where eont(F)
is the (unit normal) GCD of the coefficients of F, and pp(F) is a primitive polyno
mial. We shall refer to cont(F) and pp(F) as the
content
and
primitive part,
re
spectively, of F.
Let
F~'
and
F2'
be given nonzero polynomials in 9[x] with 0 (Fi') >
0 (F21),
and
let G' be their GCD. Also, let el = cont(F~'), c: = cont(F'2), c = gcd(c~, c2),
F1 = pp(F1'), F2 = pp(F2'), and G = gcd(F1, F2). Now, if F~, F2, ..., F~is a
PRS, it is easy to show that G = pp(Fk) and
G' = cG.
Because of coefficient growth,
the coefficients of Fk are likely to be much larger than those of F~ and F2. However,
since G divides both F~ and F~, the coefficients of G are usually smaller than those
of F1 and F2. Thus, Fk is likely to have a very large content. Fortunately, most of
this unwanted content can be removed without computing any GCD's involving
coefficients of Fk.
Let fl
=
lc (F1), f: = lc (F2), fk = lc (Fk), g = lc (G), and ~ = gcd(fl,
f2).
Since
G divides both El and F2, it follows that g divides both fl and f2, and therefore g [ ~.
Let G =
(~/g)G.
Clearly, G has ~ as its leading coefficient, and G as its primitive
part, and it is easy to see that
G = OFk/f~.
In the case of the Euclidean PRS algorithm (Section 3.2), the reduced PRS al
gorithm (Section 3.4), and the subresultant PRS algorithm (Section 3.6), it can
be shown (see [7] and [8] ) that 0 divides a subresultant (Section 3.5) which in turn
divides Fk, and therefore 0 I f~. Hence, G =
F~/(f~/O).