Advertisement

Foundations of Computational Mathematics

, Volume 18, Issue 4, pp 867–890 | Cite as

Numerical Computation of Galois Groups

  • Jonathan D. Hauenstein
  • Jose Israel Rodriguez
  • Frank Sottile
Article

Abstract

The Galois/monodromy group of a family of geometric problems or equations is a subtle invariant that encodes the structure of the solutions. We give numerical methods to compute the Galois group and study it when it is not the full symmetric group. One algorithm computes generators, while the other studies its structure as a permutation group. We illustrate these algorithms with examples using a Macaulay2 package we are developing that relies upon Bertini to perform monodromy computations.

Keywords

Galois group Monodromy Fiber product Homotopy continuation Numerical algebraic geometry Polynomial system 

Mathematics Subject Classification

65H10 65H20 14Q15 

References

  1. 1.
    H. Alt, Über die erzeugung gegebener ebener kurven mit hilfe des gelenkviereckes, Zeitschrift für Angewandte Mathematik und Mechanik, 3 (1923), pp. 13–19.CrossRefzbMATHGoogle Scholar
  2. 2.
    B. Anderson and U. Helmke, Counting critical formations on a line, SIAM Journal on Control and Optimization, 52 (2014), pp. 219–242.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    D. Bates, E. Gross, A. Leykin, and J. Rodriguez, Bertini for Macaulay2. arXiv:1603.05908, 2013.
  4. 4.
    D. Bates, J. Hauenstein, A. Sommese, and C. Wampler, Bertini: Software for numerical algebraic geometry. Available at http://bertini.nd.edu.
  5. 5.
    D. Bates, J. Hauenstein, A. Sommese, and C. Wampler, Adaptive multiprecision path tracking, SIAM J. Numer. Anal., 46 (2008), pp. 722–746.Google Scholar
  6. 6.
    D. Bates, J. Hauenstein, A. Sommese, and C. Wampler, Numerically solving polynomial systems with Bertini, vol. 25 of Software, Environments, and Tools, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013.Google Scholar
  7. 7.
    D. Brake, J. Hauenstein, A. Murray, D. Myszka, and C. Wampler, The complete solution of Alt-Burmester synthesis problems for four-bar linkages, Journal of Mechanisms and Robotics, 8 (2016), p. 041018.CrossRefGoogle Scholar
  8. 8.
    C. Brooks, A. Martín del Campo, and F. Sottile, Galois groups of Schubert problems of lines are at least alternating, Trans. Amer. Math. Soc., 367 (2015), pp. 4183–4206.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    L. Burmester, Lehrbuch der Kinematic, Verlag Von Arthur Felix, Leipzig, Germany, 1886.zbMATHGoogle Scholar
  10. 10.
    P. Cameron, Permutation groups, vol. 45 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 1999.Google Scholar
  11. 11.
    A. Dimca, Singularities and topology of hypersurfaces, Universitext, Springer-Verlag, New York, 1992.CrossRefzbMATHGoogle Scholar
  12. 12.
    J. Draisma and J. I. Rodriguez, Maximum likelihood duality for determinantal varieties, International Mathematics Research Notices, 2014(2014), pp. 5648–5666.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    A. Galligo and A. Poteaux, Computing monodromy via continuation methods on random Riemann surfaces, Theoret. Comput. Sci., 412 (2011), pp. 1492–1507.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    D. Grayson and M. Stillman, Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/.
  15. 15.
    P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978 original.Google Scholar
  16. 16.
    J. Harris, Galois groups of enumerative problems, Duke Math. J., 46 (1979), pp. 685–724.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    J. Hauenstein, I. Haywood, and A. Liddell, Jr., An a posteriori certification algorithm for Newton homotopies, in ISSAC 2014—Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2014, pp. 248–255.Google Scholar
  18. 18.
    J. Hauenstein, J. Rodriguez, and B. Sturmfels, Maximum likelihood for matrices with rank constraints, Journal of Algebraic Statistics, 5 (2014), pp. 18–38.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    J. Hauenstein and A. Sommese, Witness sets of projections, Appl. Math. Comput., 217 (2010), pp. 3349–3354.MathSciNetzbMATHGoogle Scholar
  20. 20.
    N. Hein, F. Sottile, and I. Zelenko, A congruence modulo four for real Schubert calculus with isotropic flags, 2016. Canadian Mathematical Bulletin, to appear.Google Scholar
  21. 21.
    N. Hein, F. Sottile, and I. Zelenko, A congruence modulo four in real Schubert calculus, J. Reine Angew. Math., 714 (2016), pp. 151–174.Google Scholar
  22. 22.
    C. Hermite, Sur les fonctions algébriques, CR Acad. Sci.(Paris), 32 (1851), pp. 458–461.Google Scholar
  23. 23.
    D. Higman, Intersection matrices for finite permutation groups, J. Algebra, 6 (1967), pp. 22–42.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    S. Hoşten, A. Khetan, and B. Sturmfels, Solving the likelihood equations, Found. Comput. Math., 5 (2005), pp. 389–407.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    J. Huh and B. Sturmfels, Likelihood geometry, in Combinatorial algebraic geometry, vol. 2108 of Lecture Notes in Math., Springer, 2014, pp. 63–117.Google Scholar
  26. 26.
    C. Jordan, Traité des Substitutions, Gauthier-Villars, Paris, 1870.zbMATHGoogle Scholar
  27. 27.
    A. Leykin and F. Sottile, Galois groups of Schubert problems via homotopy computation, Math. Comp., 78 (2009), pp. 1749–1765.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    A. Martín del Campo and F. Sottile, Experimentation in the Schubert calculus, in Schubert Calculus—Osaka 2012, Mathematical Society of Japan, Tokyo, 2016, pp. 295–336.Google Scholar
  29. 29.
    D. Molzahn, M. Niemerg, D. Mehta, and J. Hauenstein, Investigating the maximum number of real solutions to the power flow equations: Analysis of lossless four-bus systems. arXiv:1603.05908, 2016.
  30. 30.
    A. Poteaux, Computing monodromy groups defined by plane algebraic curves, in SNC’07, ACM, New York, 2007, pp. 36–45. Google Scholar
  31. 31.
    N. Rennert and A. Valibouze, Calcul de résolvantes avec les modules de Cauchy, Experiment. Math., 8 (1999), pp. 351–366.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    J. Rodriguez and X. Tang, Data-discriminants of likelihood equations, in Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation, ISSAC ’15, New York, NY, USA, 2015, ACM, pp. 307–314.Google Scholar
  33. 33.
    J. Ruffo, Y. Sivan, E. Soprunova, and F. Sottile, Experimentation and conjectures in the real Schubert calculus for flag manifolds, Experiment. Math., 15 (2006), pp. 199–221. MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    L. Scott, Representations in characteristic \(p\), in The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), vol. 37 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, R.I., 1980, pp. 319–331. Google Scholar
  35. 35.
    A. Sommese, J. Verschelde, and C. Wampler, Introduction to numerical algebraic geometry, in Solving polynomial equations, vol. 14 of Algorithms Comput. Math., Springer, Berlin, 2005, pp. 301–335. Google Scholar
  36. 36.
    A. Sommese and C. Wampler, Numerical algebraic geometry, in The mathematics of numerical analysis (Park City, UT, 1995), vol. 32 of Lectures in Appl. Math., Amer. Math. Soc., Providence, RI, 1996, pp. 749–763. Google Scholar
  37. 37.
    A. Sommese and C. Wampler, The numerical solution of systems of polynomials, World Scientific Publishing Co. Pte.Ltd., Hackensack, NJ, 2005.Google Scholar
  38. 38.
    A. Sommese and C. Wampler, Exceptional sets and fiber products, Found. Comput. Math., 8 (2008), pp. 171–196. Google Scholar
  39. 39.
    F. Sottile and J. White, Double transitivity of Galois groups in Schubert calculus of Grassmannians, Algebr. Geom., 2 (2015), pp. 422–445. MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Y. Tong, D. Myszka, and A. Murray, Four-bar linkage synthesis for a combination of motion and path-point generation, Proceedings of the ASME International Design Engineering Technical Conferences, DETC2013-12969 (2013). Google Scholar
  41. 41.
    R. Vakil, Schubert induction, Ann. of Math. (2), 164 (2006), pp. 489–512. Google Scholar
  42. 42.
    O. Zariski, A theorem on the Poincaré group of an algebraic hypersurface, Annals of Mathematics, 38 (1937), pp. 131–141. MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© SFoCM 2017

Authors and Affiliations

  • Jonathan D. Hauenstein
    • 1
  • Jose Israel Rodriguez
    • 2
  • Frank Sottile
    • 3
  1. 1.Department of Applied and Computational Mathematics and StatisticsUniversity of Notre DameNotre DameUSA
  2. 2.Department of StatisticsUniversity of ChicagoChicagoUSA
  3. 3.Department of MathematicsTexas A & M UniversityCollege StationUSA

Personalised recommendations