Numerical homotopy methods, pioneered largely by Sommese,
Verschelde, and Wampler, may be used to construct a numerical
irreducible decomposition of an algebraic set. In particular, these
methods will produce at least one approximation of a generic point on
each irreducible component of the solution set of a set of
polynomials. Applying a method such as LLL or PSLQ to certain
embeddings of one of these approximate generic points will yield all
integer relations (exact defining equations) for the irreducible
component on which the point lies. As a result, one may recover at
least some of the information contained in the symbolic decomposition
of an ideal without relying on symbolic methods such as Groebner basis
This is ongoing work with J. Hauenstein, T. McCoy, C.
Peterson, and A. Sommese.