Recovering Exact Results from Inexact Numerical Data in Algebraic Geometry
Daniel J. Bates, Jonathan D. Hauenstein, Timothy M. McCoy, Chris Peterson, Andrew John Sommese · Experimental Mathematics · 2013
Let be a set of homogeneous polynomials. Let Z denote the complex projective algebraic set determined by the zero locus of . Numerical-continuation-based methods can be used to produce arbitrary-precision numerical approximations of generic points on each irreducible component of Z. Consider the ideal and the prime decomposition over . This article illustrates how lattice-reduction algorithms may take as input numerically approximated generic points on Z and effectively extract exact elements for each Pi . A collection of examples serves to illustrate the approach and indicate some of the application areas for which this technique is valuable.