Homotopies for intersecting solution components of polynomial systems

SIAM Journal on Numerical Analysis

Volumn 42, Issue 4, 2004, Pages 1552-1571

  Sommese, Andrew J ^a   Verschelde, Jan ^b   Wampler, Charles W ^c  

^a UNIVERSITY OF NOTRE DAME   (United States)

^b UNIVERSITY OF ILLINOIS AT CHICAGO   (United States)

^c GENERAL MOTORS CORPORATION   (United States)

Author keywords

Components of solutions; Embedding; Generic points; Homotopy continuation; Irre ducible components; Numerical algebraic geometry; Polynomial system

Indexed keywords

COMPONENTS OF SOLUTIONS; EMBEDDING; GENERIC POINTS; HOMOTOPY CONTINUATION; IRRE-DUCIBLE COMPONENTS; NUMERICAL ALGEBRAIC GEOMETRY; POLYNOMIAL SYSTEM;

ALGEBRA; NUMERICAL ANALYSIS; SET THEORY;

POLYNOMIALS;

EID: 21844480967     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142903430463     Document Type: Article

References (19)

1. E.L. ALLGOWER AND K. GEORG, Introduction to Numerical Continuation Methods, Classics Appl. Math. 45, SIAM, Philadelphia, 2003.
2. E.L. ALLGOWER AND K. GEORG, Numerical path following, in Techniques of Scientific Computing (Part 2), Handb. Numer. Anal. 5, P.G. Ciarlet and J.L. Lions, eds., North-Holland, Amsterdam, 1997, pp. 3-207.
3. M. BELTRAMETTI AND A.J. SOMMESE, The Adjunction Theory of Complex Projective Varieties, Expo. Math. 16, Walter De Gruyter, Berlin, 1995.
4. D. EISENBUD, Commutative Algebra with a View Toward Algebraic Geometry, Grad. Texts in Math. 150, Springer-Verlag, New York, 1995.
5. W. FULTON, Intersection Theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete 3, Springer-Verlag, Berlin, 1998.
6. M. GRIFFIS AND J. DUFFY, Method and Apparatus for Controlling Geometrically Simple Parallel Mechanisms with Distinctive Connections, U.S. Patent 5,179,525, 1993.
7. M.L. HUSTY AND A. KARGER, Self-motions of Griffis-Duffy type parallel manipulators, in Proceedings of the IEEE International Conference on Robotics and Automation, San Francisco, 2000, pp. 7-12.
8. T.Y. Li, Numerical solution of multivariate polynomial systems by homotopy continuation methods, Acta Numer., 6 (1997), pp. 399-436.
9. A. MORGAN, Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems, Prentice-Hall, Englewood Cliffs, NJ, 1987.
10. A. MORGAN AND A.J. SOMMESE, Coefficient-parameter polynomial continuation, Appl. Math. Comput., 29 (1989), pp. 123-160;
11. Errata, Appl. Math. Comput., 51 (1992), p. 207.
12. D. MUMFORD, Algebraic Geometry I: Complex Projective Varieties, Grundlehren Math. Wiss. 221, Springer-Verlag, Berlin, 1976.
13. A.J. SOMMESE AND J. VERSCHELDE, Numerical homotopies to compute generic points on positive dimensional algebraic sets, J. Complexity, 16 (2000), pp. 572-602.
14. 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.
15. A.J. SOMMESE, J. VERSCHELDE, AND C.W. WAMPLER, Using monodromy to decompose solution sets of polynomial systems into irreducible components, in Application of Algebraic Geometry to Coding Theory, Physics and Computation, C. Ciliberto, F. Hirzebruch, R. Miranda, and M. Teicher, eds., Kluwer Academic Publishers, Norwell, MA, pp. 297-315.
16. A.J. SOMMESE, J. VERSCHELDE, AND C.W. WAMPLER, Symmetric functions applied to decomposing solution sets of polynomial systems, SIAM J. Numer. Anal., 40 (2002), pp. 2026-2046.
17. A.J. SOMMESE, J. VERSCHELDE, AND C.W. WAMPLER, A method for tracking singular paths with application to the numerical irreducible decomposition, in Algebraic Geometry, a Volume in Memory of Paolo Francia, M.C. Beltrametti, F. Catanese, C. Ciliberto, A. Lanteri, and C. Pedrini, eds., Walter de Gruyter, Berlin, 2002, pp. 329-345.
18. A.J. SOMMESE AND C.W. WAMPLER, Numerical algebraic geometry, in The Mathematics of Numerical Analysis, Lectures in Appl. Math. 32, J. Renegar, M. Shub, and S. Smale, eds., Springer-Verlag, New York, 1996, pp. 749-763.
19. J. VERSCHELDE, Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation, ACM Trans. Math. Software, 25 (1999), pp. 251-276.