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Numerical-symbolic exact irreducible decomposition of cyclic-12

Part of: Nonlinear algebraic or transcendental equations Algorithms - Computer Science Computational aspects and applications Computational aspects in algebraic geometry

Published online by Cambridge University Press:  01 August 2011

Rostam Sabeti
Rostam Sabeti*
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA (email: sabetiro@math.msu.edu)

Abstract

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In 1992, Göran Björck and Ralf Fröberg completely characterized the solution set of cyclic-8. In 2001, Jean-Charles Faugère determined the solution set of cyclic-9, by computer algebra methods and Gröbner basis computation. In this paper, a new theory in matrix analysis of rank-deficient matrices together with algorithms in numerical algebraic geometry enables us to present a symbolic-numerical algorithm to derive exactly the defining polynomials of all prime ideals of positive dimension in primary decomposition of cyclic-12. Empirical evidence together with rigorous proof establishes the fact that the positive-dimensional solution variety of cyclic-12 just consists of 72 quadrics of dimension one.

MSC classification

Secondary: 65H10: Systems of equations 68W30: Symbolic computation and algebraic computation 13P05: Polynomials, factorization 14Q99: None of the above, but in this section
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 14 , 2011 , pp. 155 - 172
Copyright
Copyright © London Mathematical Society 2011

References

[1]Backelin, J. and Fröberg, R., How to prove that there are 924 cyclic-7 roots?, Proc. ISSAC’91 (ed. Watt, S. M.; ACM, New York, NY, 1991) 103–111.Google Scholar
[2]Bates, D. J., Hauenstein, J. D., Peterson, C. and Sommese, A. J., ‘A numerical local dimension test for points on the solution set of a system of polynomial equations’, SIAM J. Numer. Anal. 47 (2009) no. 5, 3608–3623.CrossRefGoogle Scholar
[3]Bates, D. J., Hauenstein, J. D., Sommese, A. J. and Wampler, C. W., ‘Software for numerical algebraic geometry: a paradigm and progress toward its implementation’, Software for algebraic geometry, The IMA Volumes in Mathematics and its Applications 148 (Springer, New York, 2008) 1–14.Google Scholar
[4]Bates, D. J., Peterson, C. and Sommese, A. J., ‘A numerical-symbolic algorithm for computing the multiplicity of a component of an algebraic set’, J. Complexity 22 (2006) 475–489.CrossRefGoogle Scholar
[5]Bates, D. J., Peterson, C. and Sommese, A. J., ‘Applications of a numerical version of Terracini’s lemma for secants and joins’, Algorithms in algebraic geometry, The IMA Volumes in Mathematics and its Applications 146 (Springer, New York, 2008) 1–14.Google Scholar
[6]Bates, D. J., Sommese, A. J., Hauenstein, J. D. and Wampler, C. W., ‘Adaptive multiprecision path tracking’, SIAM J. Numer. Anal. 46 (2008) 722–746.CrossRefGoogle Scholar
[7]Bini, D. and Morrain, B., ‘Polynomial test suite’, INRIA Sophia Antipolis-Méditerranée, France, available at http://www-sop.inria.fr/saga/POL/BASE/2.multipol/cyclic.html.Google Scholar
[8]Björck, G., ‘Functions of modulus 1 on ℤn whose Fourier transforms have constant modulus, and cyclic n-roots’, Recent advances in Fourier analysis and its applications (eds Byrnes, J. S. and Byrnes, J. F.; Kluwer Academic, Dordrecht, 1990) 131–140.CrossRefGoogle Scholar
[9]Björck, G. and Fröberg, R., ‘A faster way to count the solutions of inhomogeneous systems of algebraic equations, with applications to cyclic n-roots’, J. Symbolic Comput. 12 (1991) 329–336.CrossRefGoogle Scholar
[10]Björck, G. and Fröberg, R., ‘Methods to ‘divide out’ certain solutions from systems of algebraic equations, applied to find all cyclic-8 roots’, Analysis, algebra, and computers in mathematical research, Lecture Notes in Pure and Applied Mathematics 156 (Marcel Dekker, New York, 1992) 57–70.Google Scholar
[11]Cox, D., Little, J. and O’Shea, D., Ideals, varieties and algorithms: an introduction to computational algebraic geometry and commutative algebra, 2nd edn Undergraduate Texts in Mathematics (Springer, New York, 1997).Google Scholar
[12]Faugère, J. C., ‘Finding all the solutions of cyclic-9 using Gröbner basis techniques’, Computer mathematics (Matsuyama, 2001), Lecture Notes Series on Computing 9 (World Scientific, River Edge, NJ, 2001) 1–12.Google Scholar
[13]Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics 52 (Springer, Berlin–New York, 1977).CrossRefGoogle Scholar
[14]Kuo, Y.-C. and Li, T. Y., ‘Determine whether a numerical solution of a polynomial system is isolated’, J. Math. Anal. Appl. 338 (2008) no. 2, 840–851.CrossRefGoogle Scholar
[15]Lee, T., Li, T. Y. and Tsai, C., ‘HOM4PS-2.0: a software package for solving polynomial systems by the polyhedral homotopy continuation method’, Computing 83 (2008) no. 2, 109–133.CrossRefGoogle Scholar
[16]Lu, Y., Bates, D. J., Sommese, A. J. and Wampler, C. W., ‘Finding all real points of a complex curve’, Algebra, geometry, and their interactions, Contemporary Mathematics 448 (American Mathematical Society, Providence, RI, 2007) 183–205.CrossRefGoogle Scholar
[17]Sommese, A. J. and Verschelde, J., ‘Numerical homotopies to compute points on positive dimensional algebraic sets’, J. Complexity 16 (2000) no. 3, 572–602.CrossRefGoogle Scholar
[18]Sommese, A. J., Verschelde, J. and Wampler, C. W., ‘Numerical decomposition of the solution sets of polynomial systems into irreducible components’, SIAM J. Numer. Anal. 38 (2001) no. 6, 2022–2046.CrossRefGoogle Scholar
[19]Sommese, A. J., Verschelde, J. and Wampler, C. W., ‘Numerical irreducible decomposition using projections from points on the components’, Symbolic computation: solving equations in algebra, geometry, and engineering, Contemporary Mathematics 286 (American Mathematical Society, Providence, RI, 2001) 37–51.CrossRefGoogle Scholar
[20]Sommese, A. J., Verschelde, J. and Wampler, C. W., ‘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 Science Series II: Mathematics, Physics and Chemistry 36 (Springer, Dordrecht, 2001) 297–315.CrossRefGoogle Scholar
[21]Sommese, A. J., Verschelde, J. and Wampler, C. W., ‘Symmetric functions applied to decomposing solution sets of polynomial systems’, SIAM J. Numer. Anal. 40 (2002) no. 6, 2026–2046.CrossRefGoogle Scholar
[22]Sommese, A. J., Verschelde, J. and Wampler, C. W., ‘Homotopies for intersecting solution components of polynomial systems’, SIAM J. Numer. Anal. 42 (2004) 1552–1571.CrossRefGoogle Scholar
[23]Sommese, A. J., Verschelde, J. and Wampler, C. W., ‘Numerical factorization of multivariate complex polynomials’, Theoret. Comput. Sci. 315 (2004) 651–669.CrossRefGoogle Scholar
[24]Sommese, A. J., Verschelde, J. and Wampler, C. W., ‘An intrinsic homotopy for intersecting algebraic varieties’, J. Complexity 21 (2005) 593–608.CrossRefGoogle Scholar
[25]Sommese, A. J. and Wampler, C. W., ‘Numerical algebraic geometry’, Mathematics of numerical analysis, Lectures in Applied Mathematics 32 (eds Renegar, J., Shub, M. and Smale, S.; American Mathematical Society, Providence, RI, 1996) 749–763.Google Scholar
[26]Sommese, A. J. and Wampler, C. W., The numerical solution of systems of polynomials, arising in engineering and science (World Scientific, Hackensack, NJ, 2005).CrossRefGoogle Scholar
[27]Verschelde, J., ‘Polyhedral methods in numerical algebraic geometry’, Interaction of classical and algebraic geometry, Contemporary Mathematics 496 (eds Bates, D., Besana, G., DiRocco, S. and Wampler, C.; American Mathematical Society, Providence, RI, 2009) 243–263.Google Scholar