In particular, suppose I have an affine algebraic variety over $\mathbb{R}^n$ described by generators of a radical ideal $I$ and I want to find (perhaps not all of the) points in the variety. Several important questions come up in practice:

- are there versions of Buchberger's algorithm that work with inexact data? For instance, suppose that the coefficients of the polynomials generating $I$ are known only to floating point precision. Some CAS will try to find solutions assuming that these coefficients are
*exact*. Are there CAS that do something more intelligent (e.g., make certain guarantees given that the numerical coefficients are the truncation of exact coefficients)? - does a sparse system of polynomial equations yield a Gröbner basis with sparse elements? In other words, if each polynomial in the original system has a small number of non-zero coefficients relative to $n$, do the basis elements also have this property?
- what bounds are known for the size of a Gröbner basis in terms of size and sparsity of the original system?
- are there more appropriate algorithms (than Buchberger's) if we just want to find a
*single*point in the variety? (Suppose that any such point is sufficient.) More generally, which algorithms are better suited to address the kinds of issues mentioned above?