As a first application we tried to specify a framework to unify the
different benchmark collections of systems of polynomials as, e.g.,
[1,2,4,8,11,12]. Each
such system of polynomials is defined through a finite basis in a
certain polynomial ring in a list of variables
over a base domain . It occurs that most examples may be reduced to
systems of polynomial with integer coefficients or with coefficients
in
where is a list of parameters. We
decided to focus on such systems and to define the corresponding table
` INTPS` accordingly.

A system of polynomials in ` INTPS` is defined through its basis,
list of variables, and list of parameters. The tags ` basis`, `
vars`, and ` parameters` correspond to these entries. They are the
most important tags: ` basis` and ` vars` of ` level==1`,
hence, mandatory; ` parameters` of ` level==2` since for there are no parameters.

For uniformity reasons and to ease comparison, we require of a
``valid'' INTPS record, that its basis polynomials are stored in
expanded form using the `+`

, `*`

, and `^`

operators,
and that the monomials of a polynomial and the polynomials of the
basis are ordered w.r.t. the degree reverse lexicographical
ordering. Based on SINGULAR, the (Perl) ` INTPS::validate`
routine defined in the ` INTPS` table module validates, and, if
requested, necessary, and possible, ``fixes'' these properties of an
INTPS record.

Further tags are defined to collect background information about the
different polynomial systems. Background information may be of
structural or relational type. Structural information about a
polynomial system concerns invariant properties of the basis and the
ideal generated by it, e.g., lists of the lengths and degrees of the
basis polynomials, the dimension or degree of the ideal, a prime or
primary decomposition of the ideal, or certain parameters of such a
description. Several optional tags, like ` llist`, ` dlist`,
` dim`, ` degree`, ` isoPrimes`, ` isoPrimeDims`, etc.,
and Perl routines are defined to collect or even generate such
information.

Relational information relates the polynomial systems to other tables.
This might be a bibliography reference of the origin of the example,
bibliography references of papers that considered the example, a
problem description of where the example came from or how it was
generated from certain parameters, etc. Since relational information
relates two tables we have to declare one of them as foreign and to
attach the information to the other table. For ` INTPS`, we define
optional tags ` BIB` containing a reference to the original
bibliography source described in the ` BIB` table and `
PROBLEMS` containing a reference to a problem description in the `
PROBLEMS` table. For the bibliography references to papers that
consider the given example we declare the ` INTPS` table as
foreign, i.e., we define a corresponding ` INTPS` tag in the `
BIB` table. The main reason for this decision is persistence in the
sense that we do not need to change an INTPS record each time a new
publication refers to it. For similar reasons, the bibliography
reference of the origin is attached to the ` INTPS` table, not to
` BIB`. Note that it is not always as easy as here to make such a
judicious decision.

For integrity reasons, we furthermore need to assure that there are no
``equal'' records in our collection of ` INTPS` records. The first
problem we face here, is to decide what we actually mean by
``equality'' of ` INTPS` records. Possible definitions range from
equality of the ideals generated by the basis polynomials up to string
equality of the ` basis` tag values. With benchmark computations in
mind, we decided on the following definition: Let
,
be -tuples of polynomials. Then we define to be equal
to iff there exist permutations
such
that

for all .

Having this definition at hand, we still need effective methods to
actually determine the equality of two ` INTPS` records: a
brute-force, trial-and-error method is certainly computationally
infeasible, since already by now we have ` INTPS` records with
polynomials in more than 40 variables. For this purpose, the first
author has developed and implemented within SINGULAR an
algorithm which uses structural information of the polynomials to
significantly cut-down the number of possible permutations. Tested
with random permutations on about 500 examples from our collection,
the implementation needs at most a minute or so to recover the input
permutations and hence, to decide the equality of ` INTPS` records
in the above sense. Details of the algorithm and its implementation
will be given in a forthcoming publication.