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This shows that the algorithm produces the basis G = {p
1
,p
2
,XY - X,-2X}, which is
easily shown to satisfy the definition of a Gröbner basis. Of course, since there were
choices made along the way (we could have chosen different elements of B), we should
not expect to end up with a unique basis.
Because finding a Gröbner basis can be very time consuming, it is important that
one proceeds in as efficient manner as possible. The basic Buchberger algorithm is
not very efficient. For one thing, the set B may get large. We shall briefly mention two
ways to improve the algorithm. The first simplification of Gröbner bases rests on the
following lemma.
10.10.15. Lemma.
If G is a Gröbner basis for an ideal I and if p Œ G is a polynomial
with the property that lt(p) Œ<lt(G - {p})>, then G - {p} is also a Gröbner basis for I.
Proof.
Easy.
Definition.
A Gröbner basis G = {g
1
,g
2
,...,g
m
} is called
minimal
if
(1) lc(g
i
) = 1 for all i.
(2) For all i and j, i π j, lpp(g
i
) does not divide lpp(g
j
).
For example, given the Gröbner basis
{
}
2
2
G
=
X Y
+
X XY
,
+
1
,
XY
-
X
,
-
2
X
that we got in Example 10.10.14, we can normalize the fourth element and apply
Lemma 10.10.15 to the first three elements of G to show that {X} is also a Gröbner
basis that is in fact minimal.
Minimal Gröbner bases get rid of some extra elements but may not be unique.
For example, the ideals <X
2
+ aXY,XY> are minimal Gröbner bases for the ideal
<X
2
,XY> for any constant a Πk.
Definition.
A Gröbner basis G = {g
1
,g
2
,...,g
m
} is said to be
reduced
if
(1) lc(g
i
) = 1 for all i.
(2) For all i, g
i
is reduced with respect to G - {g
i
}. Equivalently, no nonzero term
in g
i
is divisible by any lpp(g
j
) for j π i.
10.10.16. Theorem.
If we fix a monomial order, then every nonzero ideal I in
k[X
1
,X
2
,...,X
n
] has a
unique
reduced Gröbner basis with respect to that order.
Proof.
See [CoLO97] or [AdaL94].
One immediate application of the uniqueness in Theorem 10.10.16 is the
following:
A test for when ideals are equal:
Simply compute the reduced Gröbner basis for both
and check if they are the same.
