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Research of Hauenstein supported in part by NSF Grant ACI-1460032, Sloan Research Fellowship BR2014-110 TR14, and Army Young Investigator Program (YIP) W911NF-15-1-0219.
Research of Rodriguez supported in part by NSF Grant DMS-1402545.
Research of Sottile supported in part by NSF Grant DMS-1501370.
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.
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D. Bates, E. Gross, A. Leykin, and J. Rodriguez, Bertini for Macaulay2. arXiv:1603.05908, 2013.
D. Bates, J. Hauenstein, A. Sommese, and C. Wampler, Bertini: Software for numerical algebraic geometry. Available at http://bertini.nd.edu.
D. Bates, J. Hauenstein, A. Sommese, and C. Wampler, Adaptive multiprecision path tracking, SIAM J. Numer. Anal., 46 (2008), pp. 722–746.
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.
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. CrossRef
L. Burmester, Lehrbuch der Kinematic, Verlag Von Arthur Felix, Leipzig, Germany, 1886. MATH
P. Cameron, Permutation groups, vol. 45 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 1999.
D. Grayson and M. Stillman, Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/.
P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978 original.
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.
N. Hein, F. Sottile, and I. Zelenko, A congruence modulo four for real Schubert calculus with isotropic flags, 2016. Canadian Mathematical Bulletin, to appear.
N. Hein, F. Sottile, and I. Zelenko, A congruence modulo four in real Schubert calculus, J. Reine Angew. Math., 714 (2016), pp. 151–174.
C. Hermite, Sur les fonctions algébriques, CR Acad. Sci.(Paris), 32 (1851), pp. 458–461.
J. Huh and B. Sturmfels, Likelihood geometry, in Combinatorial algebraic geometry, vol. 2108 of Lecture Notes in Math., Springer, 2014, pp. 63–117.
C. Jordan, Traité des Substitutions, Gauthier-Villars, Paris, 1870. MATH
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.
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.
A. Poteaux, Computing monodromy groups defined by plane algebraic curves, in SNC’07, ACM, New York, 2007, pp. 36–45.
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.
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.
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.
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.
A. Sommese and C. Wampler, The numerical solution of systems of polynomials, World Scientific Publishing Co. Pte.Ltd., Hackensack, NJ, 2005.
A. Sommese and C. Wampler, Exceptional sets and fiber products, Found. Comput. Math., 8 (2008), pp. 171–196.
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).
R. Vakil, Schubert induction, Ann. of Math. (2), 164 (2006), pp. 489–512.
- Numerical Computation of Galois Groups
Jonathan D. Hauenstein
Jose Israel Rodriguez
- Publication date
- Springer US
- Foundations of Computational Mathematics
The Journal of the Society for the Foundations of Computational Mathematics
Print ISSN: 1615-3375
Electronic ISSN: 1615-3383