Chapter List

Introductory Remarks

When a polynomial system , has positive-dimensional zero manifolds, one way of representing these manifolds is by parametrization: with .

It is well known that such parametrizations do not always exist. If they exist, the are generally rational functions of t: there may also exist parametrizations by polynomials .

In the context of this book, we do not wish to enter into the algorithmic theory of these parametrizations. Instead, we attempt to extend the approach which we have used for 0-dimensional zero sets: We study how it may be possible to derive parametric representations of zero manifolds of the system P from representations of the multiplicative structure of the quotient ring . This approach has originated in the 1990s in collaboration with the group of Wu Wenda at the Mathematics Mechanization Research Center of the Chinese Academy of Sciences; some accounts of it have been presented at workshops and conferences and appeared in conference proceedings but not in a major journal. Describing this incomplete work here with the explicit agreement of my friend Wu may serve to expose it to a wider public; thus, the loose ends may be picked up by others and continued. In particular, it may be clarified whether our ideas can be turned into a general algorithmic theory for the computational determination of representations of zero manifolds of polynomial systems, or whether they only constitute interesting observations which work in sufficiently restricted situations.