"Sparse structual Gr\"obner basis detection"
Karin Gatermann.
Konrad-Zuse-Zentrum Berlin
Takustr. 7
D-14195 Berlin
Germany
E-mail:gatermann@zib.de
The detection of a term order such that an existing set of polynomials
is a Gr\"obner basis is an important question since
there are several techniques known to convert a Gr\"obner basis
with respect to some term order to another Gr\"obner basis
with respect to another term order.
The well-known conversion techniques include FGLM-class algorithms
exploiting linear algebra technique as well as the Gr\"obner walk
by Collart, Kalbrener and Mall and its improvement, the fractal walk
by Amrhein and Gloor.
Also the Hilbert series driven version of the Buchberger algorithm
uses the knowledge of a Gr\"obner basis with respect
to some term order which is different from the order one is interested in.
Sturmfels and Wiegelmann gave a method in order to find a term order such that
the initial forms with respect to this term order form a Gr\"obner basis
by the 1. Buchberger criterion (Structual Gr\"obner basis detection).
That means that the leading monomials are coprime.
Their method is based on searching for a maximal matching in
a weighted bipartite graph and integer programming. In case there are more variables
than polynomials partitions of the variables are considered.
We analyse this approach for the case that the polynomials are sparse.
Then first the appearance of monomials is investigated in order to bound the
number of useful partitions. There are three informations necessary.
First the supports (set of variables) of all appearing monomials are extracted.
Then for each support
the polynomials which include monomials with this support are extracted.
The third information is a table with supports and the degrees of the polynomials
in these variable groups.
Based on the first two informations the partitions which makes sense for the
structual Gr\"obner basis detection problem are determined.
Secondly, the data of the
weighted bipartite graphs are easily obtained from the table.
This rearrangement of steps result in much fewer matching problems,
much easier available data during this process and also less
integer programming problems. Thus the complexity of the
sparse structual Gr\"obner basis detection is much better than without
exploiting sparsity. We will demonstrate this with examples.