Overview

We have previously emphasized that an indispensable prerequisite for the numerical treatment of an algebraic problem is its embedding into analysis. This is a straightforward matter for individual polynomials, univariate or multivariate: We can immediately see them as elements of a linear space, with the monomials of their support as basis elements; in this linear space, a topology is generated through the natural topology of the complex coefficients. Thus, as long as the structure of an algebraic task is determined by only one polynomial, its analytic embedding is so natural that we have often not bothered to display it explicitly. For univariate problems with several data polynomials (like g.c.d.), we have also been able to extend this approach.

For multivariate algebraic problems with several data polynomials, we have a different situation: The topology of the coefficients does not generally establish a suitable analytic embedding for the algebraic problem. In linear algebra, this is well known for singular or near-singular systems of linear polynomials. In polynomial algebra there are many more ways how an algebraic structure may change its character through (arbitrarily) small changes of its data. This is aggravated by the fact that many multivariate algebraic structures are commonly described in an overdetermined fashion: For example, the reduced Groebner basis of a polynomial ideal in s variables has, generally, more than s elements. This makes it impossible to define the "neighborhood of an ideal" naively by a neighborhood of its Groebner basis.

Fortunately,...