Using Monodromy to Decompose Solution Sets of Polynomial Systems into Irreducible Components

  • A. J. Sommese
  • J. Verschelde
  • C. W. Wampler
Part of the NATO Science Series book series (NAII, volume 36)


To decompose solution sets of polynomial systems into irreducible components, homotopy continuation methods generate the action of a natural monodromy group which partially classifies generic points onto their respective irreducible components. As illustrated by the performance on several test examples, this new method achieves a great increase in speed and accuracy, as well as improved numerical conditioning of the multivariate interpolation problem.

Key words

components of solutions embedding generic points homotopy continuation irreducible components monodromy group numerical algebraic geometry polynomial system primary decomposition 

Mathematics Subject Classification (2000)

Primary 65H10 Secondary 13P05 14Q99 68W30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Björck, G. (1985) Functions of modulus one on Z pb whose Fourier transforms have constant modulus, in Proceedings of the Alfred Haar Memorial Conference, Budapest, volume 49 of Colloquia Mathematica Societatis Jânos Bolyai, pp. 193–197.Google Scholar
  2. Björck, G. (1989) Functions of modulus one on Z n whose Fourier transforms have constant modulus, and “cyclic n-roots”, in J.S. Byrnes and J.F. Byrnes (eds.), Recent Advances in Fourier Analysis and its Applications, volume 315 of NATO Adv. Sci. Inst. Ser C: Math. Phys. Sci., Kluwer, pp. 131–140.Google Scholar
  3. Björck, G., and Fröberg, R. (1991) A faster way to count the solutions of inhomogeneous systems of algebraic equations, with applications to cyclic n-roots, J. Symbolic Computation 12(3), 329–336.zbMATHCrossRefGoogle Scholar
  4. Björck, G., and Fröberg, R. (1994) Methods to “divide out” certain solutions from systems of algebraic equations, applied to find all cyclic 8-roots, in M. Gyllenberg and L.E. Persson (eds.), Analysis, Algebra and Computers in Math, research, volume 564 of Lecture Notes in Applied Mathematics, Marcel Dekker, pp. 57–70.Google Scholar
  5. Davenport, J. (1987) Looking at a set of equations, Technical report 87-06, Bath Computer Science.Google Scholar
  6. Diaconis, P., Eisenbud, D., and Sturmfels, B. (1998) Lattice Walks and Primary Decomposition, in B.E. Sagan and R.P. Stanley (eds.), Mathematical Essays in Honor of Gian-Carlo Rota, volume 161 of Progress in Mathematics, Birkhäuser, pp. 173–193.Google Scholar
  7. Dietmaier, P. (1998) The Stewart-Gough platform of general geometry can have 40 real postures, in J. Lenarcic and M.L. Husty (eds.), Advances in Robot Kinematics: Analysis and Control, Kluwer Academic Publishers, Dordrecht, pp. 1–10.Google Scholar
  8. Emiris, I.Z. (1994) Sparse Elimination and Applications in Kinematics, PhD thesis, Computer Science Division, Dept. of Electrical Engineering and Computer Science, University of California, Berkeley.Google Scholar
  9. Emiris, I.Z., and Canny, J.F. (1995) Efficient incremental algorithms for the sparse resultant and the mixed volume, J. Symbolic Computation 20(2), 117–149. Software available at Scholar
  10. Faugère, J.C. (1999) A new efficient algorithm for computing Gröbner bases (F 4), Journal of Pure and Applied Algebra 139 (1–3), 61–88. Proceedings of MEGA′98, 22–27 June 1998, Saint-Malo, France.MathSciNetzbMATHCrossRefGoogle Scholar
  11. Haagerup, U. (1996) Orthogonal maximal abelian *-algebras of the n × n matrices and cyclic n-roots. in Operator Algebras and Quantum Field Theory, International Press, Cambridge, MA, pp. 296–322.Google Scholar
  12. Hosten, S., and Shapiro, J. (2000) Primary Decomposition of Lattice Basis Ideals, Journal of Symbolic Computation 29(4&5), 625–639.MathSciNetzbMATHCrossRefGoogle Scholar
  13. Husty, M.L. (1996) An algorithm for solving the direct kinematics of general Stewart-Gough Platforms, Mech. Mach. Theory 31(4), 365–380.CrossRefGoogle Scholar
  14. Husty, M.L., and Karger, A. (2000) Self-motions of GriflRs-Duffy type parallel manipulators, Proc. 2000 IEEE Int. Conf. Robotics and Automation, CDROM, 24-28 April 2000, San Francisco, CA.Google Scholar
  15. Isaacson, E., and Keller, H.B. (1994) Analysis of Numerical Methods, Dover Publications.Google Scholar
  16. Li, T.Y., and Li, X. (2001) Finding mixed cells in the mixed volume computation, Found. Comput. Math. 1(2), 161–181. Software available at http://www.math.msu.edurii.edurii.MathSciNetzbMATHCrossRefGoogle Scholar
  17. Möller, H.M. (1998) Gröbner bases and numerical analysis, in B. Buchberger and F. Winkler (eds.), Gröbner Bases and Applications, volume 251 of London Mathematical Lecture Note Series, Cambridge University Press, pp. 159–178.Google Scholar
  18. Raghavan, M. (1993) The Stewart platform of general geometry has 40 configurations, ASME J. Mech. Design 115, 277–282.CrossRefGoogle Scholar
  19. Sommese, A.J., and Verscheide, J. (2000) Numerical homotopies to compute generic points on positive dimensional algebraic sets, Journal of Complexity 16(3), 572–602.MathSciNetzbMATHCrossRefGoogle Scholar
  20. Sommese, A.J., Verscheide, J., and Wampler, C.W. (2001a) Numerical decomposition of the solution sets of polynomial systems into irreducible components, S/AM J. Nutner. Anal. 38(6), 2022–2046.zbMATHCrossRefGoogle Scholar
  21. Sommese, A.J., Verscheide, J., and Wampler, C.W. (2001b) Numerical irreducible decomposition using projections from points on the components, accepted by Contemporary Mathematics, available at http://www.nd.edursommese/sommese and http://www.math.uic.edurjan.edurjan.
  22. Sommese, A.J., and Wampler, C.W. (1995) Numerical algebraic geometry, in J. Renegar, M. Shub and S. Smale (eds.), The Mathematics of Numerical Analysis, Proceedings of the AMS-SIAM Summer Seminar in Applied Mathematics, July 1 7-August 11, 1995, Park City, Utah, volume 32 of Lectures in Applied Mathematics, pp. 749–763.Google Scholar
  23. Verscheide, J. (1999) Algorithm 795: PHCpack: A general-purpose solver for polynomial systerns by homotopy continuation, ACM Transactions on Mathematical Software 25(2), 251–276. Software available at Scholar
  24. Wampler, C.W. (1996) Forward displacement analysis of general six-in-parallel SPS (Stewart) platform manipulators using soma coordinates, Mech. Mach. Theory 31(3), 331–337.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • A. J. Sommese
    • 1
    • 4
  • J. Verschelde
    • 2
    • 5
  • C. W. Wampler
    • 3
  1. 1.University of Notre DameUSA
  2. 2.University of Illinois at ChicagoUSA
  3. 3.General Motors Research LaboratoriesWarrenUSA
  4. 4.Department of MathematicsNotre DameUSA
  5. 5.Department of MathematicsChicagoUSA

Personalised recommendations