Russian Mathematics

, Volume 63, Issue 1, pp 76–78 | Cite as

On Classification of Polynomial Hamiltonians With Nondegenerate Linearly Stable Singular Point

  • P. V. BibikovEmail author
Brief communications


We study the problem of classification of polynomial Hamiltonians with a non-degenerate linearly stable singular point on the two-dimensional complex plane with respect to the action of the group of polynomial symplectic automorphisms. To each Hamiltonian one can associate the set of polynomials in three variables which are the components of the Birkhoff normal form of the Hamiltonian and a finite group which is the Galois group of a finite-dimensional extension of the fields generated by the polynomials. Using these objects we provide an equivalence criterion for two polynomial Hamiltonians.

Key words

Hamiltonian symplectomorphism polynomial automorphism Birkhoff normal form Galois group 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



Supported by the Russian Foundation for Basic Research (Grant mol_a_dk No. 16-31-60018).


  1. 1.
    Alekseevskii, D. V., Vinogradov, A. M., and Lychagin, V. V. “Basic Ideas and Concepts of Differential Geometry”, Itogi nauki i tekhn. Ser. Sovrem. Probl. Mat. Fundam. Napravleniya 28 (VINITI, Moscow, 1988), pp. 5–289.Google Scholar
  2. 2.
    Kruglikov, B. and Lychagin, V. “Global Lie–Tresse Theorem”, Select. Math., New Ser., DOI 10.1007/s00029-015-0220-z (2016).Google Scholar
  3. 3.
    Birkhoff, G. D. Dynamical Systems, Amer. Math. Soc. Colloqium Publ. IX, Amer. Math. Soc. (1927).Google Scholar
  4. 4.
    Arzhantsev, I., Flener, H., Kaliman, S., Kutzshebauch, F., and Zaidenberg, M. “Infinite Transivity in Affine Varieties”, arxiv:1210.6937.Google Scholar

Copyright information

© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.Institute of Control Sciences of Russian Academy of SceincesMoscowRussia

Personalised recommendations