117
Views
17
CrossRef citations to date
0
Altmetric
Original Articles

Recovering Exact Results from Inexact Numerical Data in Algebraic Geometry

, , , &
Pages 38-50 | Published online: 18 Mar 2013

REFERENCES

  • Allgower , [Allgower and Georg 03] E. and Georg , K. 2003 . Introduction to Numerical Continuation Methods , Classics in Applied Mathematics 45. SIAM Press .
  • Backes , [Backes and Wetzel 02] W. and Wetzel , S. 2002 . Heuristics on Lattice Basis Reduction in Practice . J. Exp. Algorithmics , 7 : 1 – 21 .
  • Bailey , [Bailey and Broadhurst 01] D. and Broadhurst , D. 2001 . Parallel Integer Relation Detection: Techniques and Applications . Mathematics of Computation , 70 : 1719 – 1736 .
  • Bates , [Bates et al. 06] D. J. , Peterson , C. and Sommese , A. J. 2006 . A Numerical-Symbolic Algorithm for Computing the Multiplicity of a Component of an Algebraic Set . Journal of Complexity , 22 ( 4 ) : 475 – 489 .
  • Bates , [Bates et al. 08] D. J. , Hauenstein , J. D. , Sommese , A. J. and Wampler , C. W. 2008 . Adaptive Multiprecision Path Tracking . SIAM J. Numer. Anal. , 46 ( 2 ) : 722 – 746 .
  • Bates , [Bates et al. 09] D. J. , Hauenstein , J. D. , Peterson , C. and Sommese , A. J. 2009 . A Numerical Local Dimensions Test for Points on the Solution Set of a System of Polynomial Equations . SIAM J. Numer. Anal. , 47 ( 5 ) : 3608 – 3623 .
  • Bates , [Bates et al. 12] D. , Eklund , D. and Peterson , C. 2012 . Computing Intersection Numbers of Chern Classes . To appear in J. Symb Comp. ,
  • Bayer , [Bayer and Mumford 92] D. and Mumford , D. 1992 . “ What Can Be Computed in Algebraic Geometry ” . In Computational Algebraic Geometry and Commutative Algebra , 1 – 48 . University Press .
  • Cohen , [Cohen 93] H. 1993 . A Course in Computational Algebraic Number Theory , Graduate Texts in Mathematics 138. Springer .
  • Cox , [Cox et al. 07] D. , Little , J. and O'shea , D. 2007 . Ideals, Varieties, and Algorithms , third ed., Undergraduate Texts in Mathematics. Springer .
  • Dayton , [Dayton and Zeng 05] B. H. and Zeng , Z. 2005 . “ Computing the Multiplicity Structure in Solving Polynomial Systems ” . In Proceedings of ISSAC’05 , 116 – 123 . ACM .
  • Eisenbud , [Eisenbud et al. 92] D. , Huneke , C. and Vasconcelos , W. 1992 . Direct Methods for Primary Decomposition . Inventiones Mathematicae , 110 : 207 – 235 .
  • Ferguson , [Ferguson and Bailey 91] H. and Bailey , D. 1991 . A Polynomial Time, Numerically Stable Integer Relation Algorithm , Technical report .
  • Fulton , [Fulton 98] W. 1998 . Intersection Theory , second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, no. 2. Springer .
  • Gathen , [von zur Gathen and Gerhard 03] J. von zur and Gerhard , J. 2003 . Modern Computer Algebra , second edition. Cambridge University Press .
  • Hastad , [Hastad et al. 89] J. , Just , B. , Lagarias , J. C. and Schnorr , C. P. 1989 . Polynomial Time Algorithms for Finding Integer Relations among Real Numbers . SIAM Journal of Computing , 18 : 859 – 881 .
  • Hauenstein , [Hauenstein et al. 09] J. , Migliore , J. , Peterson , C. and Sommese , A. 2009 . “ Numerical Computation of the Dimension of the Cohomology of Twists of Ideal Sheaves ” . In Interactions of Classical and Numerical Algebraic Geometry (Notre Dame, IN 2008) , 235 – 242 . AMS . Contemporary Mathematics 496
  • Hironaka , [Hironaka 64] H. 1964 . Resolution of Singularities of an Algebraic Variety over a Field of Characteristic Zero . Annals of Mathematics , 79 ( 1 ) : 109 – 203 .
  • Lenstra , [Lenstra et al.] 82 A. , Lenstra , H. and Lovász , L. 1982 . Factoring Polynomials with Rational Coefficients . Mathematische Annalen , 261 : 515 – 534 .
  • Leykin , [Leykin 08] A. 2008 . “ Numerical Primary Decomposition ” . In Proceedings of ISSAC’08 , 165 – 172 . ACM .
  • Li , [Li 03] T. Y. 2003 . “ Numerical Solution of Polynomial Systems by Homotopy Continuation Methods ” . In Handbook of Numerical Analysis, Volume XI, Special Volume: Foundations of Computational Mathematics , Edited by: Cucker , Leykin 08 F. 209 – 304 . North-Holland .
  • Luk , [Luk and Tracy 08] F. and Tracy , D. 2008 . An Improved LLL Algorithm . Linear Algebra Appl. , 428 ( 2-3 ) : 441 – 452 .
  • Micciancio , [Micciancio and Voulgaris 09] D. and Voulgaris , P. 2009 . Faster Exponential Time Algorithms for the Shortest Vector Problem . Electronic Colloquium on Computational Complexity , : 1 – 19 . Report No. 65
  • Nguyen Micciancio and Voulgaris 09 , [Nguyen and Stehlé 09] P. Q. and Stehlé , D. 2009 . An LLL Algorithm with Quadratic Complexity . SIAM J. Computing , 39 : 874 – 903 .
  • Schnorr , [Schnorr 09] C. 2009 . “ Progress on LLL and Lattice Reduction ” . In Proceedings of LLL+25 , 1 – 24 . Springer-Verlag .
  • Schnorr , [Schnorr and Euchner 94] C. and Euchner , M. 1994 . Lattice Basis Reduction: Improved Practical Algorithms and Solving Subset Sum Problems . Mathemtical Programming , 66 ( 1-3 ) : 81 – 199 .
  • Sommese , [Sommese and Wampler 05] A. J. and Wampler , C. W. 2005 . The Numerical Solution to Systems of Polynomials Arising in Engineering and Science , World Scientific .
  • Sommese , [Sommese et al. 01a] A. J. , Verschelde , J. and Wampler , C. W. 2001 . Numerical Decomposition of the Solution Sets of Polynomials into Irreducible Components . SIAM J. Numer. Anal. , 38 : 2022 – 2046 .
  • Sommese , [Sommese et al. 01b] A. J. , Verschelde , J. and Wampler , C. W. 2001 . “ Using Monodromy to Decompose Solution Sets of Polynomial Systems into Irreducible Components ” . In Proceedings of the 2001 NATO Advanced Research Conference, Eilat, Israel, on Applications of Algebraic Geometry to Coding Theory, Physics, and Computation , Edited by: Ciliberto , C. , Hirzebruch , F. , Miranda , R. and Teicher , M. 297 – 315 . Springer .
  • Sommese , [Sommese et al. 02] A. J. , Verschelde , J. and Wampler , C. W. 2002 . Symmetric Functions Applied to Decomposing Polynomial Systems . SIAM J. Numer. Anal. , 40 : 2026 – 2046 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.