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As discussed in §9, the principal computation which the
Gröbner Basis Algorithm performs is the computation of an
S-polynomial *s* from two polynomials (say
and ) and then the computation of this S-polynomial's normal
form (say *r*) using other polynomials
(say, ).
One of the side effects of this type of computation is
the fact that *s* lies in the ideal generated by
. If this ideal membership
relationship is
stored during the process of running the GBA and
one wants to find a smaller generating set for the output of
the GBA,
then it is helpful
to use these ideal membership relationships.

*Helton *

Wed Jul 3 10:27:42 PDT 1996