Recovering exact results from inexact numerical data in algebraic geometry

Let {f1, f2,..., ft} ⊂ Q[z1,..., zN] be a set of homogeneous polyno-mials. Let Z denote the complex, projective, algebraic set determined by the homogeneous ideal I = (f1, f2,..., ft) ⊂ C[z1,..., zN]. Numer-ical continuation-based methods can be used to produce arbitrary pre-cision numerical approximations of generic points on each irreducible component of Z. Consider the prime decompositio