Bulletin of the American Mathematical Society

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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

E. Allgower, K. Georg, Introduction to numerical continuation methods, Classics in Applied Mathematics, 45, SIAM, Philadelphia, 2003. (Reprint of the 1990 Springer-Verlag edition).

H. Alt, Über die Erzeugung gegebener ebener Kurven mit Hilfe des Gelenkviereckes, Zeitschrift fur Angewandte Math. Mech., 3, 13–19 (1923).

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, DOI 10.1016/j.amc.2013.12.165

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

Daniel J. Bates, Jonathan D. Hauenstein, Andrew J. Sommese, and Charles W. Wampler II, Adaptive multiprecision path tracking, SIAM J. Numer. Anal. 46 (2008), no. 2, 722–746. MR 2383209, DOI 10.1137/060658862

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, DOI 10.1016/j.jpaa.2010.10.016

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

B. Buchberger, A theoretical basis for the reduction of polynomials to canonical forms, ACM SIGSAM Bull. 10 (1976), no. 3, 19–29. MR 463136, DOI 10.3982/te103bm

David Cox, John Little, and Donal O’Shea, Ideals, varieties, and algorithms, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. An introduction to computational algebraic geometry and commutative algebra. MR 1189133, DOI 10.1007/978-1-4757-2181-2

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

D. F. Davidenko, On approximate solution of systems of nonlinear equations, Ukrain. Mat. Žurnal 5 (1953), 196–206 (Russian). MR 0057029

W. Gröbner, Über die Eliminationstheorie, Monatsh. Math. 54 (1950), 71–78 (German). MR 34750, DOI 10.1007/BF01304105

E. Gross, B. Davis, K. Ho, D. J. Bates, H. Harrington, Numerical algebraic geometry for model selection, preprint, arXiv 1507.04331

E. Gross, H. Harrington, Z. Rosen, B. Sturmfels, Algebraic systems biology: a case study for the WNT pathway, preprint, arXiv 1502.03188

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, DOI 10.1016/j.jcp.2013.05.008

J. D. Hauenstein, Y. H. He, D. Mehta, Numerical analyses on moduli space of vacua, Journal of High Energy Physics, 9, 27 pages, (2013).

J. D. Hauenstein, L. Oeding, G. Ottaviani, A. J. Sommese, Homotopy techniques for tensor decomposition and perfect identifiability, preprint, arXiv 1501.00090

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, DOI 10.18409/jas.v5i1.23

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, DOI 10.2307/1970547

Birkett Huber and Bernd Sturmfels, A polyhedral method for solving sparse polynomial systems, Math. Comp. 64 (1995), no. 212, 1541–1555. MR 1297471, DOI 10.1090/S0025-5718-1995-1297471-4

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, DOI 10.1016/0001-8708(82)90048-2

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, DOI 10.1016/0096-3003(87)90063-4

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).

T. Y. Li, Numerical solution of multivariate polynomial systems by homotopy continuation methods, Acta numerica, 1997, Acta Numer., vol. 6, Cambridge Univ. Press, Cambridge, 1997, pp. 399–436. MR 1489259, DOI 10.1017/S0962492900002749

J. I. Rodriguez, B. Wang, The maximum likelihood degree of rank 2 matrices via Euler characteristic, preprint, arXiv:1505.06536v1 (2015).

David Rupprecht, Semi-numerical absolute factorization of polynomials with integer coefficients, J. Symbolic Comput. 37 (2004), no. 5, 557–574. MR 2094614, DOI 10.1016/S0747-7171(02)00011-1

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, DOI 10.1137/S0036142900372549

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, DOI 10.1137/S0036142901397101

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

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, DOI 10.1137/S0036142901397101

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, DOI 10.1137/S0036142903430463

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

A. J. Sommese, C. W. Wampler, The numerical solution of systems arising in engineering and science, World Scientific, Singapore, 2005.

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, DOI 10.1137/0730028

Charles W. Wampler and Andrew J. Sommese, Numerical algebraic geometry and algebraic kinematics, Acta Numer. 20 (2011), 469–567. MR 2805156, DOI 10.1017/S0962492911000067

• E. Allgower, K. Georg, Introduction to numerical continuation methods, Classics in Applied Mathematics, 45, SIAM, Philadelphia, 2003. (Reprint of the 1990 Springer-Verlag edition).
• H. Alt, Über die Erzeugung gegebener ebener Kurven mit Hilfe des Gelenkviereckes, Zeitschrift fur Angewandte Math. Mech., 3, 13–19 (1923).
• 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, DOI 10.1016/j.amc.2013.12.165
• 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
• 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), DOI 10.1137/060658862
• 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), DOI 10.1016/j.jpaa.2010.10.016
• 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)
• 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)
• 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), DOI 10.1007/978-1-4757-2181-2
• 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)
• D. F. Davidenko, On approximate solution of systems of nonlinear equations, Ukrain. Mat. Žurnal 5 (1953), 196–206 (Russian). MR 0057029 (15,164i)
• W. Gröbner, Über die Eliminationstheorie, Monatsh. Math. 54 (1950), 71–78 (German). MR 0034750 (11,638c)
• E. Gross, B. Davis, K. Ho, D. J. Bates, H. Harrington, Numerical algebraic geometry for model selection, preprint, arXiv 1507.04331
• E. Gross, H. Harrington, Z. Rosen, B. Sturmfels, Algebraic systems biology: a case study for the WNT pathway, preprint, arXiv 1502.03188
• 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, DOI 10.1016/j.jcp.2013.05.008
• J. D. Hauenstein, Y. H. He, D. Mehta, Numerical analyses on moduli space of vacua, Journal of High Energy Physics, 9, 27 pages, (2013).
• J. D. Hauenstein, L. Oeding, G. Ottaviani, A. J. Sommese, Homotopy techniques for tensor decomposition and perfect identifiability, preprint, arXiv 1501.00090
• 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
• 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)
• 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), DOI 10.2307/2153370
• 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), DOI 10.1016/0001-8708(82)90048-2
• 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), DOI 10.1016/0096-3003(87)90063-4
• 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).
• 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), DOI 10.1017/S0962492900002749
• J. I. Rodriguez, B. Wang, The maximum likelihood degree of rank 2 matrices via Euler characteristic, preprint, arXiv:1505.06536v1 (2015).
• David Rupprecht, Semi-numerical absolute factorization of polynomials with integer coefficients, J. Symbolic Comput. 37 (2004), no. 5, 557–574. MR 2094614 (2005m:13036), DOI 10.1016/S0747-7171(02)00011-1
• 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), DOI 10.1137/S0036142900372549
• 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), DOI 10.1137/S0036142901397101
• 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)
• 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), DOI 10.1137/S0036142901397101
• 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), DOI 10.1137/S0036142903430463
• 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)
• A. J. Sommese, C. W. Wampler, The numerical solution of systems arising in engineering and science, World Scientific, Singapore, 2005.
• 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), DOI 10.1137/0730028
• Charles W. Wampler and Andrew J. Sommese, Numerical algebraic geometry and algebraic kinematics, Acta Numer. 20 (2011), 469–567. MR 2805156 (2012h:70010), DOI 10.1017/S0962492911000067