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:

 1. 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)?
 2. 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?
 3. what bounds are known for the size of a Gröbner basis in terms of size and sparsity of the original system?
 4. 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?