Advertisement

Application of a Numerical Version of Terr Acini’s Lemma for Secants and Joins

  • Daniel J. Bates
  • Chris Peterson
  • Andrew J. Sommese
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 146)

Abstract

This paper illustrates how methods such as homotopy continuation and monodromy, when combined with a numerical version of Terracini’s lemma, can be used to produce a high probability algorithm for computing the dimensions of secant and join varieties. The use of numerical methods allows applications to problems that are difficult to handle by purely symbolic algorithms.

Key words

Generic point witness point homotopy continuation irreducible components numerical algebraic geometry monodromy polynomial system secant join 

AMS(MOS) subject classifications

Primary 65H10 65H20 68W30 14Q99 14M99 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    B. ÅDLANDSVIK. Joins and higher secant varieties. Math. Scand. 61 (1987), no. 2, 213–222.zbMATHMathSciNetGoogle Scholar
  2. [2]
    H. ABO, G. Ottaviani, AND C. PETERSON. Induction for secant varieties of Segre varieties. Preprint: math.AG/0607191.Google Scholar
  3. [3]
    E. ALLGOWER AND K. GEORG. Introduction to numerical continuation methods. Classics in Applied Mathematics 45, SIAM Press, Philadelphia, 2003.Google Scholar
  4. [4]
    D.J. BATES, J.D. HAUENSTEIN, A.J SOMMESE, and C.W. WAMPLER. Bertini: Software for Numerical Algebraic Geometry. Available at www.nd.edu/~sommese/bertini.Google Scholar
  5. [5]
    M.V. CATALISANO, A.V. GERAMITA, AND A. GLMIGLIANO. Ranks of tensors, secant varieties of Segre varieties and fat points. Linear Algebra Appl. 355 (2002), 263–285.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    C. CLLIBERTO. Geometric aspects of polynomial interpolation in more variables and of Waring’s problem. European Congress of Mathematics, Vol. I (Barcelona, 2000), 289–316, Progr. Math., 201, Birkhauser, Basel, 2001.Google Scholar
  7. [7]
    CoCoATeam, CoCoA: a system for doing Computations in Commutative Algebra. Available at www.cocoa.dima.unige.it.Google Scholar
  8. [8]
    J. DRAISMA. A tropical approach to secant dimensions, math.AG/0605345.Google Scholar
  9. [9]
    H.R.P. FERGUSON AND D.H. BAILEY. A Polynomial Time, Numerically Stable Integer Relation Algorithm. RNR Techn. Rept. RNR-91-032, Jul. 14, 1992.Google Scholar
  10. [10]
    H.R.P. FERGUSON, D.H. BAILEY, AND S. ARNO. Analysis of PSLQ, An Integer Relation Finding Algorithm. Math. Comput. 68, 351–369, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    D. GRAYSON AND M. STILLMAN. MACAULAY 2: a software system for research in algebraic geometry. Available at www.math.uiuc.edu/Macaulay2.Google Scholar
  12. [12]
    G.M. GREUEL, G. PFISTER, AND H. SCHONEMANN. SINGULAR 3.0: A Computer Algebra System for Polynomial Computations. Available at www.s ingular.uni-kl.de.Google Scholar
  13. [13]
    J.M. LANDSBERG. The border rank of the multiplication of two by two matrices is seven., J. Amer. Math. Soc. 19 (2006), 447–159.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    T.Y. LI AND Z. ZENG. A rank-revealing method with updating, downdating and applications. SIAM J. Matrix Anal. Appl. 26 (2005), 918–946.Google Scholar
  15. [15]
    B. MCGILLIVRAY. A probabilistic algorithm for the secant defect of Grassmann varieties. Linear Algebra and its Applications 418 (2006), 708–718.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16] A.J. SOMMESE AND J. VERSCHELDE. Numerical Homotopies to compute generic points on positive dimensional Algebraic Sets. Journal of Complexity 16(3): 572–602 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    A.J. SOMMESE, J. VERSCHELDE, AND C.W. WAMPLER. Numerical decomposition of the solution sets of polynomials into irreducible components. SIAM J. Numer. Anal. 38 (2001), 2022–2046.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    A.J. SOMMESE, J. VERSCHELDE, AND C.W. WAMPLER. A method for tracking singular paths with application to the numerical irreducible decomposition. Algebraic Geometry, a Volume in Memory of Paolo Francia. Ed. by M.C. Beltrametti, F. Catanese, C. Ciliberto, A. Lanteri, and C. Pedrini. De Gruyter, Berlin, 2002, 329–345.Google Scholar
  19. [19]
    A.J. SOMMESE, J. VERSCHELDE, AND C.W. WAMPLER. Symmetric functions applied to decomposing solution sets of polynomial systems. SIAM Journal on Numerical Analysis 40 (2002), 2026–2046.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    A.J. SOMMESE AND C.W. WAMPLER. The Numerical Solution to Systems of Polynomials Arising in Engineering and Science. World Scientific, Singapore, 2005.CrossRefGoogle Scholar
  21. [21]
    H. STETTER. Numerical Polynomial Algebra. SIAM Press, Philadelphia, 2004.zbMATHGoogle Scholar
  22. [22]
    G.W. STEWART. Matrix Algorithms 1: Basic Decompositions. SIAM Press, Philadelphia, 1998.zbMATHGoogle Scholar
  23. [23]
    B. STURMFELS AND S. SULLIVANT. Combinatorial secant varieties. Pure Appl. Math. Q. 2 (2006), 867–891zbMATHMathSciNetGoogle Scholar
  24. [24]
    A. TERRACINI. Sulla rappresentazione delle forme quaternarie mediante somme di potenze di forme lineari. Atti R. Accad. delle Scienze di Torino, Vol. 51, 1915–16.Google Scholar
  25. [25]
    L. TREFBTHEN AND D. BAU. Numerical Linear Algebra. SIAM Press, Philadelphia, 1997.Google Scholar
  26. [26] G.H. GOLUB AND C.F. VAN LOAN. Matrix computations, 3rd edition. Johns Hopkins University Press, Baltimore, MD, 1996.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Daniel J. Bates
    • 1
  • Chris Peterson
    • 2
  • Andrew J. Sommese
    • 3
  1. 1.Institute for Mathematics and Its Applications (IMA)University of MinnesotaMinneapolis
  2. 2.Department of MathematicsColorado State UniversityFort Collins
  3. 3.Department of MathematicsUniversity of Notre DameNotre Dame

Personalised recommendations