Recovering Exact Results from Inexact Numerical Data in Algebraic Geometry

Let {f1, f2, . . . , ft} ⊂ Q[z1, . . . , zN ] be a set of homogeneous polynomials. Let Z denote the complex, projective, algebraic set determined by the homogeneous ideal I = (f1, f2, . . . , ft) ⊂ C[z1, . . . , zN ]. Numerical continuation-based methods can be used to produce arbitrary precision numerical approximations of generic points on each irreducible component of Z. Consider the prime decomposition √ I = ⋂ i Pi over Q[z1, . . . , zN ]. In this article, it is shown that these approximated generic points may be used in an effective manner to extract exact elements Gi,j ∈ Z[z1, . . . , zN ] from each Pi. A collection of examples and applications serve to illustrate the approach.