Recovering Exact Results from Inexact Numerical Data in Algebraic Geometry

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.

[1]  E. Allgower,et al.  Introduction to Numerical Continuation Methods , 1987 .

[2]  Burton S. Kaliski,et al.  Polynomial Time , 2005, Encyclopedia of Cryptography and Security.

[3]  Jonathan D. Hauenstein,et al.  Adaptive Multiprecision Path Tracking , 2008, SIAM J. Numer. Anal..

[4]  Zhonggang Zeng,et al.  Computing the multiplicity structure in solving polynomial systems , 2005, ISSAC.

[5]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[6]  Andrew J. Sommese,et al.  The numerical solution of systems of polynomials - arising in engineering and science , 2005 .

[7]  David H. Bailey,et al.  Parallel integer relation detection: Techniques and applications , 2001, Math. Comput..

[8]  Chris Peterson,et al.  Computing intersection numbers of Chern classes , 2013, J. Symb. Comput..

[9]  Claus-Peter Schnorr,et al.  Lattice basis reduction: Improved practical algorithms and solving subset sum problems , 1991, FCT.

[10]  Claus-Peter Schnorr,et al.  Progress on LLL and Lattice Reduction , 2010, The LLL Algorithm.

[11]  Jan Verschelde,et al.  Using Monodromy to Decompose Solution Sets of Polynomial Systems into Irreducible Components , 2001 .

[12]  Tien Yien Li,et al.  Numerical solution of multivariate polynomial systems by homotopy continuation methods , 1997, Acta Numerica.

[13]  Andrew J. Sommese,et al.  Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components , 2000, SIAM J. Numer. Anal..

[14]  Andrew J. Sommese,et al.  Symmetric Functions Applied to Decomposing Solution Sets of Polynomial Systems , 2002, SIAM J. Numer. Anal..

[15]  Franklin T. Luk,et al.  An improved LLL algorithm , 2008 .

[16]  Jonathan D. Hauenstein,et al.  A Numerical Local Dimension Test for Points on the Solution Set of a System of Polynomial Equations , 2009, SIAM J. Numer. Anal..

[17]  Hans Schönemann,et al.  SINGULAR: a computer algebra system for polynomial computations , 2001, ACCA.

[18]  Damien Stehlé,et al.  An LLL Algorithm with Quadratic Complexity , 2009, SIAM J. Comput..

[19]  Charles W. Wampler,et al.  Interactions of Classical and Numerical Algebraic Geometry , 2009 .

[20]  D. Eisenbud,et al.  Direct methods for primary decomposition , 1992 .

[21]  Chris Peterson,et al.  A numerical-symbolic algorithm for computing the multiplicity of a component of an algebraic set , 2006, J. Complex..

[22]  Chris Peterson,et al.  NUMERICAL COMPUTATION OF THE DIMENSIONS OF THE COHOMOLOGY OF TWISTS OF IDEAL SHEAVES , 2008 .

[23]  David Mumford,et al.  What Can Be Computed in Algebraic Geometry , 1993, alg-geom/9304003.

[24]  Henri Cohen,et al.  A course in computational algebraic number theory , 1993, Graduate texts in mathematics.

[25]  H. Hironaka.  Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: II , 1964 .

[26]  Paul Kutler,et al.  A Polynomial Time, Numerically Stable Integer Relation Algorithm , 1998 .

[27]  Daniele Micciancio,et al.  Faster exponential time algorithms for the shortest vector problem , 2010, SODA '10.

[28]  David A. Cox,et al.  Ideals, Varieties, and Algorithms , 1997 .

[29]  Tien-Yien Li.  Numerical Solution of Polynomial Systems by Homotopy Continuation Methods , 2003 .

[30]  Susanne Wetzel,et al.  Heuristics on lattice basis reduction in practice , 2002, JEAL.

[31]  Jeffrey C. Lagarias,et al.  Polynomial Time Algorithms for Finding Integer Relations Among Real Numbers , 1989, STACS.

[32]  Anton Leykin.  Numerical primary decomposition , 2008, ISSAC '08.

[33]  Johan P. Hansen,et al.  INTERSECTION THEORY , 2011 .

[34]  S. L. Kleiman.  INTERSECTION THEORY (Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge. Band 2) , 1985 .