Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

Book Review

The AMS does not provide abstracts of book reviews. You may download the entire review from the links below.


Full text of review: PDF   This review is available free of charge.
Book Information:

Authors: Daniel J. Bates, Jonathan D. Hauenstein, Andrew J. Sommese and Charles W. Wampler
Title: Numerically solving polynomial systems with Bertini
Additional book information: Software, Environments, and Tools, Vol. 25, SIAM, Philadelphia, PA, 2013, xii+352 pp., ISBN 978-1-611972-69-6, US.00 $95.00

References [Enhancements On Off] (What's this?)

  • [1] E. Allgower, K. Georg, Introduction to numerical continuation methods, Classics in Applied Mathematics, 45, SIAM, Philadelphia, 2003. (Reprint of the 1990 Springer-Verlag edition).
  • [2] H. Alt, Über die Erzeugung gegebener ebener Kurven mit Hilfe des Gelenkviereckes, Zeitschrift fur Angewandte Math. Mech., 3, 13-19 (1923).
  • [3] Daniel J. Bates, Wolfram Decker, Jonathan D. Hauenstein, Chris Peterson, Gerhard Pfister, Frank-Olaf Schreyer, Andrew J. Sommese, and Charles W. Wampler, Comparison of probabilistic algorithms for analyzing the components of an affine algebraic variety, Appl. Math. Comput. 231 (2014), 619-633. MR 3174059, https://doi.org/10.1016/j.amc.2013.12.165
  • [4] Daniel J. Bates, Jonathan D. Hauenstein, Andrew J. Sommese, and Charles W. Wampler, Numerically solving polynomial systems with Bertini, Software, Environments, and Tools, vol. 25, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. MR 3155500
  • [5] Daniel J. Bates, Jonathan D. Hauenstein, Andrew J. Sommese, and Charles W. Wampler, Adaptive multiprecision path tracking, SIAM J. Numer. Anal. 46 (2008), no. 2, 722-746. MR 2383209 (2008m:65147), https://doi.org/10.1137/060658862
  • [6] Daniel J. Bates, Chris Peterson, Andrew J. Sommese, and Charles W. Wampler, Numerical computation of the genus of an irreducible curve within an algebraic set, J. Pure Appl. Algebra 215 (2011), no. 8, 1844-1851. MR 2776427 (2012f:65079), https://doi.org/10.1016/j.jpaa.2010.10.016
  • [7] D. N. Bernstein, The number of roots of a system of equations, Funkcional. Anal. i Priložen. 9 (1975), no. 3, 1-4 (Russian). MR 0435072 (55 #8034)
  • [8] B. Buchberger, A theoretical basis for the reduction of polynomials to canonical forms, ACM SIGSAM Bull. 10 (1976), no. 3, 19-29. MR 0463136 (57 #3097)
  • [9] David Cox, John Little, and Donal O'Shea, Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. MR 1189133 (93j:13031)
  • [10] D. F. Davidenko, On a new method of numerical solution of systems of nonlinear equations, Doklady Akad. Nauk SSSR (N.S.) 88 (1953), 601-602 (Russian). MR 0054339 (14,906f)
  • [11] D. F. Davidenko, On approximate solution of systems of nonlinear equations, Ukrain. Mat. Žurnal 5 (1953), 196-206 (Russian). MR 0057029 (15,164i)
  • [12] W. Gröbner, Über die Eliminationstheorie, Monatsh. Math. 54 (1950), 71-78 (German). MR 0034750 (11,638c)
  • [13] E. Gross, B. Davis, K. Ho, D. J. Bates, H. Harrington, Numerical algebraic geometry for model selection, preprint, arXiv 1507.04331
  • [14] E. Gross, H. Harrington, Z. Rosen, B. Sturmfels, Algebraic systems biology: a case study for the WNT pathway, preprint, arXiv 1502.03188
  • [15] Wenrui Hao, Jonathan D. Hauenstein, Chi-Wang Shu, Andrew J. Sommese, Zhiliang Xu, and Yong-Tao Zhang, A homotopy method based on WENO schemes for solving steady state problems of hyperbolic conservation laws, J. Comput. Phys. 250 (2013), 332-346. MR 3079538, https://doi.org/10.1016/j.jcp.2013.05.008
  • [16] J. D. Hauenstein, Y. H. He, D. Mehta, Numerical analyses on moduli space of vacua, Journal of High Energy Physics, 9, 27 pages, (2013).
  • [17] J. D. Hauenstein, L. Oeding, G. Ottaviani, A. J. Sommese, Homotopy techniques for tensor decomposition and perfect identifiability, preprint, arXiv 1501.00090
  • [18] Jonathan Hauenstein, Jose Israel Rodriguez, and Bernd Sturmfels, Maximum likelihood for matrices with rank constraints, J. Algebr. Stat. 5 (2014), no. 1, 18-38. MR 3279952
  • [19] Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109-203; ibid. (2) 79 (1964), 205-326. MR 0199184 (33 #7333)
  • [20] Birkett Huber and Bernd Sturmfels, A polyhedral method for solving sparse polynomial systems, Math. Comp. 64 (1995), no. 212, 1541-1555. MR 1297471 (95m:65100), https://doi.org/10.2307/2153370
  • [21] Ernst W. Mayr and Albert R. Meyer, The complexity of the word problems for commutative semigroups and polynomial ideals, Adv. in Math. 46 (1982), no. 3, 305-329. MR 683204 (84g:20099), https://doi.org/10.1016/0001-8708(82)90048-2
  • [22] Alexander Morgan and Andrew Sommese, A homotopy for solving general polynomial systems that respects $ m$-homogeneous structures, Appl. Math. Comput. 24 (1987), no. 2, 101-113. MR 914806 (88j:65110), https://doi.org/10.1016/0096-3003(87)90063-4
  • [23] A. P. Morgan, A. J. Sommese, C. W. Wampler, Complete solution of the nine-point path synthesis problem for four-bar linkages, J. Mech. Des., 114, 153-159 (1992).
  • [24] T. Y. Li, Numerical solution of multivariate polynomial systems by homotopy continuation methods, Acta Numer., 1997, Acta Numer., vol. 6, Cambridge Univ. Press, Cambridge, 1997, pp. 399-436. MR 1489259 (2000i:65084), https://doi.org/10.1017/S0962492900002749
  • [25] J. I. Rodriguez, B. Wang, The maximum likelihood degree of rank 2 matrices via Euler characteristic, preprint, arXiv:1505.06536v1 (2015).
  • [26] David Rupprecht, Semi-numerical absolute factorization of polynomials with integer coefficients, J. Symbolic Comput. 37 (2004), no. 5, 557-574. MR 2094614 (2005m:13036), https://doi.org/10.1016/S0747-7171(02)00011-1
  • [27] Andrew J. Sommese, Jan Verschelde, and Charles W. Wampler, Numerical decomposition of the solution sets of polynomial systems into irreducible components, SIAM J. Numer. Anal. 38 (2001), no. 6, 2022-2046. MR 1856241 (2002g:65064), https://doi.org/10.1137/S0036142900372549
  • [28] Andrew J. Sommese, Jan Verschelde, and Charles W. Wampler, Symmetric functions applied to decomposing solution sets of polynomial systems, SIAM J. Numer. Anal. 40 (2002), no. 6, 2026-2046 (2003). MR 1974173 (2004m:65069), https://doi.org/10.1137/S0036142901397101
  • [29] A. J. Sommese, J. Verschelde, and C. W. Wampler, Using monodromy to decompose solution sets of polynomial systems into irreducible components, Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001) NATO Sci. Ser. II Math. Phys. Chem., vol. 36, Kluwer Acad. Publ., Dordrecht, 2001, pp. 297-315. MR 1866906 (2002k:65087)
  • [30] Andrew J. Sommese, Jan Verschelde, and Charles W. Wampler, Symmetric functions applied to decomposing solution sets of polynomial systems, SIAM J. Numer. Anal. 40 (2002), no. 6, 2026-2046 (2003). MR 1974173 (2004m:65069), https://doi.org/10.1137/S0036142901397101
  • [31] Andrew J. Sommese, Jan Verschelde, and Charles W. Wampler, Homotopies for intersecting solution components of polynomial systems, SIAM J. Numer. Anal. 42 (2004), no. 4, 1552-1571. MR 2114290 (2005m:65111), https://doi.org/10.1137/S0036142903430463
  • [32] Andrew J. Sommese and Charles W. Wampler, Numerical algebraic geometry, The mathematics of numerical analysis (Park City, UT, 1995) Lectures in Appl. Math., vol. 32, Amer. Math. Soc., Providence, RI, 1996, pp. 749-763. MR 1421365 (98d:14079)
  • [33] A. J. Sommese, C. W. Wampler, The numerical solution of systems arising in engineering and science, World Scientific, Singapore, 2005.
  • [34] Jan Verschelde and Ann Haegemans, The $ GBQ$-algorithm for constructing start systems of homotopies for polynomial systems, SIAM J. Numer. Anal. 30 (1993), no. 2, 583-594. MR 1211406 (94b:65075), https://doi.org/10.1137/0730028
  • [35] Charles W. Wampler and Andrew J. Sommese, Numerical algebraic geometry and algebraic kinematics, Acta Numer. 20 (2011), 469-567. MR 2805156 (2012h:70010), https://doi.org/10.1017/S0962492911000067

Review Information:

Reviewer: Henry Schenck
Affiliation: University of Illinois at Urbana–Champaign
Journal: Bull. Amer. Math. Soc. 53 (2016), 179-186
MSC (2000): Primary 13P15; Secondary 65D99, 68W30
DOI: https://doi.org/10.1090/bull/1520
Published electronically: August 28, 2015
Review copyright: © Copyright 2015 American Mathematical Society
American Mathematical Society