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.