Recovering Exact Results from Inexact Numerical Data in Algebraic Geometry

Abstract

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 P_i . A collection of examples serves to illustrate the approach and indicate some of the application areas for which this technique is valuable.

2000 AMS Subject Classification:

• 65H10
• 68W30
• 14Q99

Keywords:

• witness set
• generic point
• homotopy continuation
• numerical algebraic geometry
• numerical irreducible decomposition
• primary decomposition
• algebraic set
• algebraic variety
• LLL

ACKNOWLEDGMENTS

We would like to thank several individuals and organizations for various forms of support during this project. The first author appreciates the useful comments and sound advice of H. Hong, E. Kaltofen, M. Singer, S. Sullivant, and A. Szänto received during a visit to North Carolina State University. The first, second, fourth, and fifth authors would like to express their gratitude to Doug Arnold and the organizers of the IMA (Institute for Mathematics and Its Applications) Thematic Year on Applications of Algebraic Geometry. The final draft of the paper was completed during the spring 2011 thematic semester of the Mittag-Leffler Institute on Algebraic Geometry with a View towards Applications. We would also like to express our appreciation to the referee, whose comments helped us to streamline the paper.

Daniel J. Bates received partial support from NSF grant DMS-0914674. Jonathan D. Hauenstein received partial support from NSF grant DMS-1114336. Chris Peterson received partial support from NSF grant DMS-0901770. Andrew J. Sommese received partial support from the Duncan Chair of the University of Notre Dame and NSF grant DMS-0712910.

Notes

¹Available online at http://www.nd.edu/~sommese/bertini.

²Macaulay2 is available at http://www.math.uiuc.edu/Macaulay2 ; Singular is available at http://www.singular.uni-kl.de ; CoCoA is available at http://cocoa.dima.unige.it .