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
[1, p. 369] always yields a
Q = pquo(F1, F2) and
R = prem (F1, F2),
QF2 = R
and 0 (R) < 0 (F2), where
is the leading coefficient of
F~, and ~ = 0 (F1) -
For nonzero F1, F2 C 9[x], we say that F1 is
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, .
= 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
i = 3, .
Because of the uniqueness of pseudo-division, 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
gcd(Fk_~, Fk) ~ Fk. Thus, the construction of the PRS yields t2he
desired GCD to within similarity.
A polynomial F C 9Ix] will be called
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, 74-77] 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
spectively, of 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 =
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  and  ) that 0 divides a subresultant (Section 3.5) which in turn
divides Fk, and therefore 0 I f~. Hence, G =
Thus, the (large) factor