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##

3.3 The Hilbert series of resolutions

The Hilbert function has become an important tool for
computations of Gröbner bases of quasihomogeneous ideals or modules
during recent years. For a detailed discussion of the usage of Hilbert
functions in that context I would like to refer to [GMRT].
Let the
-module *M* be given by generators
constituting a standard base of *M* with respect to the degree compatible
ordering <.

**Lemma 3..3**
Resolving the initial module

by the
STZS-method, one obtains the Hilbert series as well as the
numbers of generators and their degrees of the STZS-resolution of

*M*.

Proof: Of course, this resolution of leading terms can be continued
to a resolution of
(see [MM]). The Hilbert
functions are invariants of the initial modules (see [GMRT]
Theorem 1) and from the properties of the STZS-resolution follows
*Syz*_{i}(*In*(*M*))=*In*(*Syz*_{i}(*M*)).
Now, we assume to minimize the STZS-resolution *Syz*_{i}(*M*),
of *M*. This means,
to find constant entres in the syzygies of level *i*
and to cancel the corresponding element
.
The component
*i*_{l} in *Syz*_{i}(*M*) is set to zero by Gaussian eliminations and the
syzygy is taken out together with its component in *Syz*_{i+1}.
(compare R. La Scala [LS] Chapter 4)

Denote by *sf*_{i-1} the set of superfluous generators in
*Syz*_{i-1}(*M*).
By the descritpion above we get:

**Lemma 3..4**
The Hilbert function

*H*_{i} of the i-th module in the minimal resolution
is given by:

Remark: For computing the exact Hilbert function of the i-th module
in the minimal resolution it is not necessary to know the concrete
generators. It suffices to know the number and degree of a minimal set of
generators. However, one has to be
careful while counting explicit generators. They may be transformed
by the Gaussian elimination or by cancellation of components of syzygies
which are taken out into zero!

**Theorem 1**
By resolving the initial module (or ideal) of the input and detection
of direct reductions in
*Syz*_{i-1}(*M*) one obtains the Hilbert
function *H*_{i} of the module
*Syz*_{i}(*M*). The Hilbert functions
*H*_{i}*Syz*_{j} for *j*>*i* are upper bounds for the *H*_{j}.

Proof: By searching for direct reductions one obtains a minimal set of
generators for the actual module in the minimal resolution. Comparing them
with the generators in the STZS-resolution one finds all
superfluous elements in that resolution and can compute the Hilbert
functions of the minimal resolution according to the lemmas above.
The interpretation of Hilbert functions in the context of syzygies
differs from that in the Gröbner base algorithm. Once having detected
all necessary pairs (and checked this by the Hilbert function) of a certain
degree in the module
*Syz*_{i-1}(*M*), all remaining pairs of the same degree
will yield syzygies. It seems to be a good idea to wait with their
reduction until the next computation. Here one has a complete knowledge
of the Hilbert function and can decide how many pairs must be reduced
to obtain the correct Hilbert function in *Syz*_{i}(*M*) up to the current degree.
The rest of the pairs yields superfluous generators and their computation can
be skipped. Therefore, the usage of Hilbert functions suggests a computation
degree by degree over the whole resolvent.

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** Up:** 3. Improvements
** Previous:** 3.2 Direct minimizations
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