Regular and Chaotic Dynamics

, Volume 12, Issue 6, pp 732–735 | Cite as

Stanley decomposition of the joint covariants of three quadratics



The Stanley decomposition of the joint covariants of three quadratics is computed using a new transvectant algorithm and computer algebra. This is sufficient to compute the general form of the normal form with respect to a nilpotent with three 3-dimensional irreducible blocks.

Key words

joint covariant quadratic nilpotent normal form 

MSC2000 numbers

13A50 34E10 


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  1. 1.
    Cushman R. and Sanders, J.A., A Survey of Invariant Theory Applied to Normal Forms of Vectorfields with Nilpotent Linear Part, in Invariant Theory and Tableaux (Minneapolis, MN, 1988), vol. 19 of IMA Vol. Math. Appl., Springer: New York, 1990, pp. 82–106.Google Scholar
  2. 2.
    Murdock, J., On the Structure of Nilpotent Normal Form Modules. J. Differential Equations, 2002, vol. 180, no. 1, pp. 198–237.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Apel, J., On a Conjecture of R.P. Stanley. I. Monomial Ideals. J. Algebraic Combin., 2003, vol. 17, no. 1, pp. 39–56.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Apel, J., On a Conjecture of R.P.Stanley. II. Quotients Modulo Monomial Ideals. J. Algebraic Combin., 2003, vol. 17, no. 1, pp. 57–74.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Murdock J. and Sanders, J.A., A New Transvectant Algorithm for Nilpotent Normal Forms, J. Differential Equations, 2007, vol. 238, pp. 234–256.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Sanders, J.A., Verhulst, F., and Murdock, J., Averaging Methods in Nonlinear Dynamical Systems, 2nd ed., vol. 59 of Applied Mathematical Sciences, New York: Springer, 2007.MATHGoogle Scholar
  7. 7.
    Olver, P.J., Classical Invariant Theory, vol. 44 of London Mathematical Society Student Texts, Cambridge: Cambridge University Press, 1999.MATHGoogle Scholar
  8. 8.
    Olver, P.J., Joint Invariant Signatures, Found. Comput. Math., 2001, vol. 1, no. 1, pp. 3–67.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cushman, R., Sanders, J.A., and White, N., Normal Form for the (2; n)-nilpotent Vector Field, Using Invariant Theory. Phys. D, 1988, vol. 30, no. 3, pp. 399–412.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Grace, J.H. and Young, M.A., The Algebra of Invariants, Cambridge: Cambridge University Press, 1903.MATHGoogle Scholar
  11. 11.
    Sturmfels, B., Gröbner Bases and Stanley Decompositions of Determinantal Rings. Math. Z., 1990, vol. 205, no. 1, pp. 137–144.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Develin, M. and Sturmfels, B., Tropical Convexity. Doc. Math., 2004, vol. 9, pp. 1–27 (electronic).MATHMathSciNetGoogle Scholar
  13. 13.
    Chuanming Zong, What is Known about Unit Cubes, Bull. Amer. Math. Soc. (N.S.), 2005, vol. 42, no. 2, pp. 181–211 (electronic).MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Vermaseren, J.A.M., New Features of FORM. Technical report, Nikhef, Amsterdam, 2000. Math-ph/0010025.Google Scholar
  15. 15.
    Cushman R. and Sanders, J.A., Nilpotent Normal Forms and Representation Theory of sl(2, R), in Multiparameter Bifurcation Theory (Arcata, Calif., 1985), vol. 56 of Contemp. Math., Amer. Math. Soc., Providence, RI, 1986, pp. 31–51.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2007

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of SciencesVrije UniversiteitAmsterdamThe Netherlands

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