Often the Gröbner Basis algorithm gives a generating set for
a polynomial ideal which
is large enough
so that looking
for ``interesting'' polynomials
can be overwhelming (see
§);
therefore, it is necessary
to find a smaller generating set.
It is our belief and experience that most highly algebraic
mathematics theorems amount to giving a *small generating set*
for an ideal.
This is consistent with the esthetic that one wants simple hypotheses.
Also with respect to the use of the theorem for numerics, if
one tries to solve redundant equations, then small errors in data
and roundoff make the equations contradictory.
One reason that one does
not seek a *minimal* (in cardinality) generating set
for the ideal is that sometimes
one does not
want to eliminate certain equations involving knowns
(see
§).
Algorithms for finding small generating sets for ideals
is the topic of Part II in
§.

We remark to those already familiar with
Gröbner Basis that in
the case of commutative polynomial rings,
if *G* is a reduced Gröbner Basis, then none of the proper subsets
of *G* is a generating set for the ideal generated
by *G*.
In contrast, for the case of noncommutative
polynomial rings, a
reduced Gröbner Basis is not necessarily
a minimal generating set (see Example
12.1).
Thus, from a purely
mathematical point of view,
*one natural generalization
of a commutative reduced Gröbner Basis
is a minimal generating subset of a reduced Gröbner Basis
(even though this minimal generating subset is itself not a
Gröbner Basis).
*

*
Section
§
discusses the theory behind the algorithms
which we use
for finding small generating
subsets
of a given set.
*
These algorithms also pertain to finding minimal generating
sets although we do not typically employ them
for that purpose.

Wed Jul 3 10:27:42 PDT 1996