Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Semi-Invariants, equivariants and algorithms

  • Published:
Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

The results from Invariant Theory and the results for semi-invariants and equivariants are summarized in a suitable way for combining with Gröbner basis computation. An algorithm for the determination of fundamental equivariants using projections and a Poincaré series is described. Secondly, an algorithm is given for the representation of an equivariant in terms of the fundamental equivariants. Several ways for the exact determination of zeros of equivariant systems are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Barany, E., Dellnitz, M., Golubitsky, M.: Detecting the Symmetry of Attractors. Physica D67, 66–87 (1993)

    Google Scholar 

  2. Becker, Th., Weispenning, V.: Gröbner Bases. A Computational Approach to Commutative Algebra. In: Cooperation with H. Kredel. Berlin, Heidelberg, New York: Springer 1993

    Google Scholar 

  3. Chevalley, C.: Invariants of finite groups generated by reflections. Am. J. Math.77, 778–782 (1955)

    Google Scholar 

  4. Cox, D., Little, J., O'Shea, D.: Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra. Berlin, Heidelberg, New York: Springer 1992

    Google Scholar 

  5. Dieudorme, J., Carrel, J. B.: Invariant Theory, Old and New. New York: Academic Press 1971

    Google Scholar 

  6. Fässler, A., Stiefel, E.: Group Theoretical Methods and Their Applications. Boston Birkhäuser 1992

  7. Field, M. J., Richardson, R. W.: Symmetry Breaking and the Maximal Isotropy Subgroup Conjecture for Reflection Groups. In: Truesdel, C. (ed) Archive for Rational Mechanics and Analysis Vol. 105. Berlin, Heidelberg, New York: Springer 1989

    Google Scholar 

  8. Gatermann, K.: Symbolic solution of polynomial equation systems with symmetry. In: Watanabe, Sh.: Nagata, M. (eds) Proceedings of ISSAC-90 (Tokyo, Japan, August 20–24, 1990), pp 112–119. New York: ACM 1990

    Google Scholar 

  9. Gaterman, K.: Werner, B.: Secondary Hopf Bifurcation Caused by Steady-state Steady-state Mode interaction. In: Chaddam, J., Golubitsky, M., Langford, W., Wetton, B. (eds) Pattern Formations: Symmetry Methods and Applications, Fields Institute Com. Series, 1994

  10. Gatermann, K.: A remark on the detection of symmetry of attractors. In: Chossat, P. (ed) Dynamics, Bifurcation and Symmetry. Kluwer Academic Publishers, Dordrecht, 1994

    Google Scholar 

  11. Golubitsky, M., Stewart, I., Schaeffer, D. G.: Singularities and Groups in Bifurcation Theory. Vol. II, Berlin, Heidelberg, New York: Springer 1988

    Google Scholar 

  12. Hearn, A. C.: REDUCE User's Manual, Version 3.5. The RAND Corp., Santa Monica, USA 1993

  13. Hochster, M., Eagon, J. A.: Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci. Am. J. Math.93, 1020–1058 (1971)

    Google Scholar 

  14. Hochster, M., Roberts, J.: Rings of invariants of reductive groups acting on regular rings are Cohen-Macauley. Adv. Math.13, 115–175 (1974)

    Google Scholar 

  15. Jaric, M. V., Michel, L., Sharp, R. T.: Zeros of covariant vector fields for the point groups: invariant formulation. J. Phys.45, 1–27 (1984)

    Google Scholar 

  16. Kemper, G. The Invar Package for Calculating Rings of Invariants. Maple Share library, 1993

  17. McShane, J. M., Grove, L. C.: Polynomial Invariants of Finite Groups. In Algebras, Groups and Geometries10, 1–12 (1993)

    Google Scholar 

  18. Melenk, H., Möller, H. M., Neun, W.: GROEBNER A Package for Calculating Groebner Bases. Available with REDUCE 1992

  19. Melenk, H., Möller, H. M., Neun, W.: Symbolic solution of large stationary chemical kinetics problems. Impact. Comput. Sci. Eng.1, 138–167 (1989)

    Google Scholar 

  20. Molien, T.: Über die Invarianten der linearen Substitutionsgruppe. Königl. Preuss. Akad. Wiss,pp. 1152–1156, 1897

  21. Noether, E.: Der Endlichkeitssatz der Invarianten endlicher Gruppen. Math. Ann.77, 89–92 (1916)

    Google Scholar 

  22. Sattinger, D. H.: Group Theoretic Methods in Bifurcation Theory. Lecture Notes in Mathematics vol. 762. Berlin, Heidelberg, New York: Springer 1979

    Google Scholar 

  23. Schwarz, G. W.: Lifting Smooth Homotopies of Orbit Spaces. In Institut des Hautes Ètudes Scientifiques, Publications Mathématiques 51, 1980

  24. Serre, J. P.: Linear Representations of Finite Groups. Berlin, Heidelberg, New York: Springer 1977

    Google Scholar 

  25. Sloane, N. J. A.: Error-correcting codes and invariant theory: new applications of a nineteenthcentury technique. Am. Math. Monthly84, 82–107 (1977)

    Google Scholar 

  26. Stanley, R. P.: Invariants of finite groups and their applications to combinatorics. Bulletin Am. Math. Soc.1, 475–511 (1979)

    Google Scholar 

  27. Sturmfels, B.: Algorithms in Invariant Theory. Wien: Springer 1993

    Google Scholar 

  28. Verscheide, J., Gatermann, K.: Symmetric Newton Polytopes for Solving Sparse Polynomial Systems. Konard-Zuse-Zentrum, Preprint SC 94-3, 1994. Accepted for publication by Advances in Applied Mathematics

  29. Workfold, P. A.: Zeros of equivariant vector fields: Algorithms for an invariant approach. To appear in J. Symbolic Computation 1994

Download references

Author information

Authors and Affiliations

  1. Konrad-Zuse-Zentrum für Informationstechnik Berlin, Heilbronner Str. 10, D-10711, Berlin, Wilmersdorf Germany

    Karin Gatermann

Authors
  1. Karin Gatermann
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gatermann, K. Semi-Invariants, equivariants and algorithms. AAECC 7, 105–124 (1996). https://doi.org/10.1007/BF01191379

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01191379

Keywords

Search

Navigation