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 $k[\underline{x}]$-module M be given by generators $m_1,\ldots,m_n$ constituting a standard base of M with respect to the degree compatible ordering <.

Lemma 3..3   Resolving the initial module $In(M)=(lt(m_1),\ldots,lt(m_n))$ 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 $M=(m_1,\ldots,m_n)$ (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), $i=1,\ldots,p$ of M. This means, to find constant entres in the syzygies of level i and to cancel the corresponding element $m_{i_l}\in Syz_{i-1}$. 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:

$$H_i=H_{Syz_i}-\sum_{m\in sf_{i-1}}^r H_{k[\underline{x}]}T^{deg(m)}.$$

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_iSyz_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.