.. , k,
(1)
and ak I ak1. From this it is easy to see that gcd (al, a~) =gcd (as, a3) .
....
gcd (ak1, ak) = ak, and therefore ak is the desired GCD.
This algorithm can be extended [1, p. 302] to yield integers u~ and v~ such that
ulal ~ via2 ~ al,
i = 1, .
.. , k.
(2)
When gcd (al, a~) = 1, it follows that
uka~
+ vka2 = 1, and therefore uk is an inverse
Journal of the Association for Computing Machinery, Vol. 18, No. 4, October 1971
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w.s.
BROWN
of al modulo a2, while vk is an inverse of a2 modulo al. If only uk is needed, as in
Step (1) of the Chinese remainder algorithm (Section 4.8), then one need not
compute v~, • • • , vk ; if al >>
a2,
the time saved may be substantial.
1.5
THE ALGORITHM FOR POLYNOMIALS. We shall consider two fundamentally
different generalizations of Euclid's algorithm (Section 1.4) to domains of poly
nomials.
In the classical algorithm (Section 2), we view a multivariate polynomial as a
univariate polynomial with polynomial coefficients, and we construct a sequence of
polynomials of successively smaller degree. Unfortunately, as the polynomials de
crease in degree, their coefficients (which may themselves be polynomials) tend to
grow, so the successive steps tend to become harder as the calculation progresses.
If the GCD's of these inflated coefficients are required, the problem is aggravated
especially in the multivariate case, where the grov¢th may be compounded through
several levels of recursion. If the coefficient domain is a field, this same remark ap
plies to any GCD's of numerators and denominators that are required to simplify
inflated coefficients. If coefficients in a field are not simplified, then the division steps
become harder faster, and the final result, although formally correct, may be prac
tically useless.
In the modular algorithm (Section 4) we first project the given polynomials into
one or more simpler domains in which images of the GCD can more easily be com
puted. The true GCD is then constructed from these images with the aid of the
Chinese remainder algorithm. Since the same method is used for the required GCD
computations in the image spaces, it is only necessary to apply Euclid's algorithm
to integers and to univariate polynomials with coefficients in a finite field.
1.6
RECENT HISTORY. During the past decade these algorithms have been
studied intensively by G. E. Collins, and (mostly in response to Collins' work) by
the author.
The first major advance was the discovery by Collins [7] of the subresultant PRS
algorithm (Section 3.6), which effectively controls coefficient growth without any
GCD computations in the coefficient domain or any subdomain thereof. Then, after
several years of improvement and consolidation, Collins and the author (working
independently but with some communication) discovered the essentials of the modu
lar algorithm (Section 4) which completely eliminates the problem of coefficient
growth by using modular arithmetic. With a very few hints from Collins and the
author, D. E. Knuth immediately grasped most of the key ideas and published a
sketch of a similar algorithm [1, pp. 393395].
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