Skip to main content
SpringerLink
Log in

Algebraic nonsolvability of the problem of existence of holomorphic first integrals

  • Published:
Journal of Dynamical and Control Systems Aims and scope Submit manuscript

Abstract

We prove that the existence of a holomorphic first integral of an analytic differential equation is an algebraically nonsolvable problem. Moreover, we prove a perturbation lemma showing that a nilpotent singularity of a differential equation in the plane cannont have a holomorphic first integral independently of the prolongation of anynth-order jet. We give an application to the case of nilpotent centers.

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

Access this article

Log in via an institution

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

References

  1. D. Cerveau and R. Moussu, Groupes d'automorphismes de (ℂ) et équations différentiellesy dy+...=0.Bull. Soc. Math. de France 1/6 (1988), 459–488.

    Google Scholar 

  2. J. F. Mattei and R. Moussu, Holonomic et intégrales premières.Ann. Sci. Ecolè Norm. Sup. 16 (1980), 469–523.

    Google Scholar 

  3. C. Camacho and P. Sad, Invariant varieties through singularities of holomorphic vector fields.Ann. Math. 115 (1982), 579–595.

    Google Scholar 

  4. A. Lins Neto, Construction of singularities of holomorphic vector fields and foliations in dimension two.J. Differ. Geom. 26 (1987), 1–31.

    Google Scholar 

  5. P. M. Elizarov and Yu. S. Il'yashenko et al., Finitely generated groups of germs of one-dimensional conformal mappings and invariants for complex singular points of analytic foliations of the complex plane. InAdv. Sov. Math. (Ed. Yu. S. Il'yashenko)14, 1993.

  6. C. Camacho, A. Lins Neto, and P. Sad, Topological invariants and equidesingularization for homolomorphic vector fields.J. Differ. Geom. 20 (1982), 143–174.

    Google Scholar 

  7. G. Sansone and R. Conti, Equazioni differenziali non lineari.Cremonese, Roma, 1956. (English translation:Pergamon Press, Oxford, 1964.)

    Google Scholar 

  8. F. Takens, Singularities of vector fields.Publ. Math. I.H.E.S. 43 (1974), 47–100.

    Google Scholar 

  9. R. Moussu, Symétrie et forme normale des centres et foyers dégénérés.Ergod. Theor. of Dynam. Syst. 2 (1982), 241–251.

    Google Scholar 

Download references

Author information

Author notes

  1. M. Villarini

    Present address: Dipartimento di Matematica, “Ulisse Dini”-Università di Firenze, Viale Morgagni 67/A, 50134, Firenze, Itally

Authors and Affiliations

    Authors
    1. M. Villarini
      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

    Villarini, M. Algebraic nonsolvability of the problem of existence of holomorphic first integrals. Journal of Dynamical and Control Systems 1, 403–425 (1995). https://doi.org/10.1007/BF02269377

    Download citation

    • Received:

    • Issue Date:

    • DOI: https://doi.org/10.1007/BF02269377

    1991 Mathematics Subject Classification

    Search

    Navigation