Computing the multiplicity structure in solving polynomial systems. (English) Zbl 1360.65151
Kauers, Manuel (ed.), Proceedings of the 2005 international symposium on symbolic and algebraic computation, ISSAC’05, Beijing, China, July 24–27, 2005. New York, NY: ACM Press (ISBN 1-59593-095-7). 116-123 (2005).
MSC:
| 65H10 | Numerical computation of solutions to systems of equations |
| 68W30 | Symbolic computation and algebraic computation |
References:
| [1] | R. M. Corless, P. M. Gianni, and B. M. Trager, A reordered Schur factorization method for zero-dimensional systems with multiple roots. Proc. ISSAC ’97, AMS Press, pp 133-140. 10.1145/258726.258767 |
| [2] | D. Cox, J. Little, and D. O’Shea, Using Algebraic Geometry, Springer Verlag, 1998. · Zbl 0920.13026 |
| [3] | D. W. Decker, H. B. Keller, and C. T. Kelly, Convergence rate for Newton’s method at singular points, SIAM J. Numer. Anal., 20 (1983), pp.296-314. · Zbl 0571.65046 |
| [4] | J. C. Faugère, A new efficient algorithm for computing Göbner bases, Journal of Pure and Applied Algebra, 139 (1998), pp. 61-88. · Zbl 0930.68174 |
| [5] | W. Fulton, Intersection Theory, Springer Verlag, Berlin, 1984. · Zbl 0541.14005 |
| [6] | G.-M. Greuel and G. Pfister, A Singular Introduction to Commutative Algebra, Springer Verlag, 2002. · Zbl 1023.13001 |
| [7] | G.-M. Greuel, G. Pfister, and H. Schönemann, Singular 2.0. A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern (2001). http://www.singular.uni-kl.de. |
| [8] | H. Kobayashi, H. Suzuki, and Y. Sakai, Numerical calculation of the multiplicity of a solution to algebraic equations, Math. Comp., 67 (1998), pp. 257-270. 10.1090/S0025-5718-98-00906-5 · Zbl 0999.65038 |
| [9] | A. Leykin, J. Verschelde, and A. Zhao, Newton’s method with deflation for isolated singularities of polynomial systems. Preprint, (2004). |
| [10] | T. Y. Li and Z. Zeng, A rank-revealing method with updating, downdating and applications. to appear: SIAM J. Matrix Anal. Appl. 10.1137/S0895479803435282 |
| [11] | F. S. Macaulay, The Algebraic Theory of Modular Systems, Cambridge Univ. Press, 1916. · JFM 46.0167.01 |
| [12] | D. Manocha and J. Demmel, Algorithms for intersecting parametric and algebraic curves II: multiple intersections, Graphical Models and Image Processing, 57 (1995), pp. 81-100. 10.1006/gmip.1995.1010 |
| [13] | M. G. Marinari, T. Mora, and H. M. Möller, On multiplicities in polynomial system solving, Trans. AMS, 348 (1996), pp. 3283-3321. · Zbl 0910.13009 |
| [14] | H. M. Möller and H. J. Stetter, Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems, Numer. Math., 70 (1995), pp. 311-329. 10.1007/s002110050122 · Zbl 0851.65029 |
| [15] | T. Mora, Solving Polynomial Equation Systems II. manuscript. |
| [16] | S. Moritsugu and K. Kuriyama, On multiple zeros of systems of algebraic equations. Proc. of ISSAC ’99, ACM Press, pp 23-30, 1999, 1999. 10.1145/309831.309846 |
| [17] | T. Ojika, Modified deflation algorithm for the solution of singular problems, J. Math. Anal. Appl., 123 (1987), pp.199-221. · Zbl 0625.65043 |
| [18] | A. J. Sommese, J. Verschelde, and C. W. Wampler, Numerical decomposition of the solution sets of polynomial systems into irreducible components, SIAM J. Numer. Anal., 38 (2001), pp. 2022-2046. 10.1137/S0036142900372549 · Zbl 1002.65060 |
| [19] | R. P. Stanley, Hilbert function of graded algebras, Advances in Math., 28 (1973), pp. 57-83. · Zbl 0384.13012 |
| [20] | H. J. Stetter, Analysis of zero clusters in multivariate polynomial systems. Proc. of ISSAC ’96, ACM Press, pp 127-136, 1996. 10.1145/236869.236919 · Zbl 0914.65053 |
| [21] | _____, Matrix eigenproblems are at the heart of polynomial system solving, SIGSAM Bull., (1996), pp. 22-25. 10.1145/242961.242966 · Zbl 1097.65530 |
| [22] | _____, Numerical Polynomial Algebra, SIAM, Philadelphia, 2004. |
| [23] | B. Sturmfels, Solving Systems of polynomial Equations, Number 97 in CBMS Regional Conference Series in Mathematics, AMS, 2002. · Zbl 1101.13040 |
| [24] | G. H. Thallinger, Analysis of zero clusters in multivariate polynomial systems. Diploma Thesis, Tech. Univ. Vienna, 1996. |
| [25] | Z. Zeng and B. H. Dayton, The approximate GCD of inexact polynomials, II: a multivariate algorithm. Proc. of ISSAC ’04, ACM Press, pp 320-327, 2004. 10.1145/1005285.1005331 · Zbl 1134.13313 |
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