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Multiple zeros of nonlinear systems. (English) Zbl 1242.65102
The authors study the multiplicity for general nonlinear systems at an isolated zero. It formulates the multiplicity, presents a method for computing the multiplicity structure, proposes an algorithm for the accurate computation of multiple zeros, and introduces a basic algebraic theory of multiplicity. The results give a consistent bridge between numerical analysis and algebraic geometry. Numerical results are shown to accurately compute the multiplicity and the multiple zeros for perturbed nonlinear systems.

MSC:
65H10 Numerical computation of solutions to systems of equations
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[1] Dan Bates, Chris Peterson, and Andrew J. Sommese, A numerical-symbolic algorithm for computing the multiplicity of a component of an algebraic set, J. Complexity 22 (2006), no. 4, 475 – 489. · Zbl 1100.65046 · doi:10.1016/j.jco.2006.04.003 · doi.org
[2] Daniel J. Bates, Jonathan D. Hauenstein, Chris Peterson, and Andrew J. Sommese, A numerical local dimensions test for points on the solution set of a system of polynomial equations, SIAM J. Numer. Anal. 47 (2009), no. 5, 3608 – 3623. · Zbl 1211.14066 · doi:10.1137/08073264X · doi.org
[3] Shui Nee Chow and Jack K. Hale, Methods of bifurcation theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 251, Springer-Verlag, New York-Berlin, 1982. · Zbl 0487.47039
[4] David A. Cox, John Little, and Donal O’Shea, Using algebraic geometry, 2nd ed., Graduate Texts in Mathematics, vol. 185, Springer, New York, 2005. · Zbl 1079.13017
[5] Barry H. Dayton and Zhonggang Zeng, Computing the multiplicity structure in solving polynomial systems, ISSAC’05, ACM, New York, 2005, pp. 116 – 123. · Zbl 1360.65151 · doi:10.1145/1073884.1073902 · doi.org
[6] James W. Demmel, Applied numerical linear algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997. · Zbl 0879.65017
[7] Jacques Emsalem, Géométrie des points épais, Bull. Soc. Math. France 106 (1978), no. 4, 399 – 416 (French, with English summary). · Zbl 0396.13017
[8] William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. · Zbl 0541.14005
[9] Gene H. Golub and Charles F. Van Loan, Matrix computations, 3rd ed., Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 1996. · Zbl 0865.65009
[10] Wolfgang Gröbner, Algebraische Geometrie. 2. Teil: Arithmetische Theorie der Polynomringe, Bibliographisches Institut, Mannheim-Vienna-Zurich, 1970 (German). B. I. Hochschultaschenbücher, 737/737a*. · Zbl 0206.23901
[11] Gert-Martin Greuel and Gerhard Pfister, A Singular introduction to commutative algebra, Second, extended edition, Springer, Berlin, 2008. With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann; With 1 CD-ROM (Windows, Macintosh and UNIX). · Zbl 1133.13001
[12] G.-M. GREUEL, G. PFISTER, AND H. SCHÖNEMANN, Singular 3.0. A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, Univ. of Kaiserslautern, 2005. · Zbl 1344.13002
[13] Hidetsune Kobayashi, Hideo Suzuki, and Yoshihiko Sakai, Numerical calculation of the multiplicity of a solution to algebraic equations, Math. Comp. 67 (1998), no. 221, 257 – 270. · Zbl 0999.65038
[14] Martin Kreuzer and Lorenzo Robbiano, Computational commutative algebra. 2, Springer-Verlag, Berlin, 2005. · Zbl 1090.13021
[15] Y. C. Kuo and T. Y. Li, Determining dimension of the solution component that contains a computed zero of a polynomial system, J. Math. Anal. Appl. 338 (2008), no. 2, 840 – 851. · Zbl 1133.65028 · doi:10.1016/j.jmaa.2007.05.049 · doi.org
[16] E. Lasker, Zur Theorie der moduln und Ideale, Math. Ann. 60 (1905), no. 1, 20 – 116 (German). · JFM 36.0292.01 · doi:10.1007/BF01447495 · doi.org
[17] Anton Leykin, Jan Verschelde, and Ailing Zhao, Newton’s method with deflation for isolated singularities of polynomial systems, Theoret. Comput. Sci. 359 (2006), no. 1-3, 111 – 122. · Zbl 1106.65046 · doi:10.1016/j.tcs.2006.02.018 · doi.org
[18] Anton Leykin, Jan Verschelde, and Ailing Zhao, Higher-order deflation for polynomial systems with isolated singular solutions, Algorithms in algebraic geometry, IMA Vol. Math. Appl., vol. 146, Springer, New York, 2008, pp. 79 – 97. · Zbl 1138.65038 · doi:10.1007/978-0-387-75155-9_5 · doi.org
[19] Bang-He Li, A method to solve algebraic equations up to multiplicities via Ritt-Wu’s characteristic sets, Acta Anal. Funct. Appl. 5 (2003), no. 2, 97 – 109 (English, with English and Chinese summaries). · Zbl 1030.65055
[20] T. Y. Li and Zhonggang Zeng, A rank-revealing method with updating, downdating, and applications, SIAM J. Matrix Anal. Appl. 26 (2005), no. 4, 918 – 946. · Zbl 1114.15004 · doi:10.1137/S0895479803435282 · doi.org
[21] F. S. Macaulay, The algebraic theory of modular systems, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1994. Revised reprint of the 1916 original; With an introduction by Paul Roberts. · Zbl 0802.13001
[22] M. G. Marinari, H. M. Möller, and T. Mora, On multiplicities in polynomial system solving, Trans. Amer. Math. Soc. 348 (1996), no. 8, 3283 – 3321. · Zbl 0910.13009
[23] Teo Mora, Solving polynomial equation systems. II, Encyclopedia of Mathematics and its Applications, vol. 99, Cambridge University Press, Cambridge, 2005. Macaulay’s paradigm and Gröbner technology. · Zbl 1161.13306
[24] B. Mourrain, Isolated points, duality and residues, J. Pure Appl. Algebra 117/118 (1997), 469 – 493. Algorithms for algebra (Eindhoven, 1996). · Zbl 0896.13020 · doi:10.1016/S0022-4049(97)00023-6 · doi.org
[25] Takeo Ojika, Modified deflation algorithm for the solution of singular problems. I. A system of nonlinear algebraic equations, J. Math. Anal. Appl. 123 (1987), no. 1, 199 – 221. · Zbl 0625.65043 · doi:10.1016/0022-247X(87)90304-0 · doi.org
[26] Andrew J. Sommese and Jan Verschelde, Numerical homotopies to compute generic points on positive dimensional algebraic sets, J. Complexity 16 (2000), no. 3, 572 – 602. Complexity theory, real machines, and homotopy (Oxford, 1999). · Zbl 0982.65070 · doi:10.1006/jcom.2000.0554 · doi.org
[27] Richard P. Stanley, Hilbert functions of graded algebras, Advances in Math. 28 (1978), no. 1, 57 – 83. · Zbl 0384.13012 · doi:10.1016/0001-8708(78)90045-2 · doi.org
[28] Hans J. Stetter and Günther H. Thallinger, Singular systems of polynomials, Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation (Rostock), ACM, New York, 1998, pp. 9 – 16. · Zbl 0924.65042 · doi:10.1145/281508.281525 · doi.org
[29] Hans J. Stetter, Numerical polynomial algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2004. · Zbl 1058.65054
[30] Joseph L. Taylor, Several complex variables with connections to algebraic geometry and Lie groups, Graduate Studies in Mathematics, vol. 46, American Mathematical Society, Providence, RI, 2002. · Zbl 1002.32001
[31] G. H. THALLINGER, Analysis of Zero Clusters in Multivariate Polynomial Systems, Diploma Thesis, Tech. Univ. Vienna, 1996.
[32] Xiaoli Wu and Lihong Zhi, Computing the multiplicity structure from geometric involutive form, ISSAC 2008, ACM, New York, 2008, pp. 325 – 332. · Zbl 1236.68302 · doi:10.1145/1390768.1390812 · doi.org
[33] Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-London-New York, 1960. · Zbl 0322.13001
[34] Zhonggang Zeng, Computing multiple roots of inexact polynomials, Math. Comp. 74 (2005), no. 250, 869 – 903. · Zbl 1079.12007
[35] -, ApaTools: A Maple and Matlab toolbox for approximate polynomial algebra, in Software for Algebraic Geometry, IMA Volume 148, M. Stillman, N. Takayama, and J. Verschelde, eds., Springer, 2008, pp.149-167. · Zbl 1148.68581
[36] -, The closedness subspace method for computing the multiplicity structure of a polynomial system. to appear: Interactions between Classical and Numerical Algebraic Geometry, Contemporary Mathematics series, American Mathematical Society, 2009. · Zbl 1181.65074
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