Contents - Student subdomain for University of Bath

2.3. GREATEST COMMON DIVISORS 47
93060801700
1557792607653 x + 23315940650
173088067517
and, finally,
761030000733847895048691
.
86603128130467228900
Since this is a number, it follows that no polynomial can divide both A and B,
i.e. that gcd(A, B) = 1.
It is obvious that these calculations on polynomials with rational coefficients
require several g.c.d. calculations on integers, and that the integers in these
calculations are not always small.
We can eliminate these g.c.d. calculations by working all the time with
polynomials with integer coefficients, and this gives a generalisation of the a i of
algorithm 1, known as polynomial remainder sequences or p.r.s., by extending
the definition of division.
Definition 30 Instead of dividing f by g in K[x], we can multiply f by a
suitable power of the leading coefficient of g, so that the divisions stay in R.
The pseudo-remainder of dividing f by g, written prem(f, g), is the remainder
when one divides lc(g) deg(f)−deg(g)+1 f by g, conceptually in K[x], but in fact all
the calculations can be performed in R, i.e. all divisions are exact in R. This
is denoted 25 by prem(f, g).
In some applications (section 3.1.9) it is necessary to keep track of the signs:
we define a signed polynomial remainder sequence or s.p.r.s. of f 0 = f and
f 1 = g to have f i proportional by a positive constant to −rem(f i−2 , f i−1 ).
This gives us a pseudo-euclidean algorithm, analogous to algorithm 1 where we
replace rem by prem, and fix up the contents afterwards. In the above example,
we deduce the following sequence:
and
−15 x 4 + 381 x 2 − 261,
6771195 x 2 − 30375 x − 4839345,
500745295852028212500 x + 1129134141014747231250
7436622422540486538114177255855890572956445312500.
Again, this is a number, so gcd(A, B) = 1. We have eliminated the fractions,
but at a cost of even larger numbers. Can we do better?
2.3.2 Subresultant sequences
One option would be to make the a i primitive at each step, since we are going
to fix up the content terms later: giving the so-called primitive p.r.s. algorithm,
which in this case would give
−5x 4 + 127x 2 − 87; 5573x 2 − 25x − 3983; −4196868317x − 1861216034; 1
25 This definition agrees with Maple, but not with all software systems, which often use prem
to denote what Maple calls sprem, i.e. only raising lc(g) to the smallest power necessary.