Polynomials receive our special attention. We formulate the Gröbner bases algorithm for ``sets of polynomials'', hence, the term ``polynomial'' must be kept rather general: it may be a polynomial in the strict mathematical sense (a ``pure polynomial'') but it may as well be a pure polynomial with some additional information attached, which allows further interpretation of the result. This additional information could for instance be the cofactors with respect to the initial polynomials. Computing a Gröbner basis for a set of ``polynomials with cofactors'' (see Section 3.13) yields a Gröbner basis plus a representation of each basis polynomial in terms of the initial polynomials.

Independent of this *structure* , we can choose
among different *representations for ``pure
polynomials''* , for instance the
distributive list representation (see Section 3.12) or the various
representations provided by SACLIB. Whatever structure is used for
``polynomials'', the representation of a ``pure polynomial'' will occur as a
virtual subdomain.

Thus, the domain ``polynomials'' as a virtual subdomain of ``sets of polynomials'' should have the 2-level structure as shown in Figure 5.

**Figure:** 2-level Structure of Polynomials

Thu Sep 3 14:50:07 MDT 1998