A remark on first integrals of vector fields

• Autores: Andre Belotto da Silva, Martin Klimeš, Julio C. Rebelo, Helena Reis
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 68, Nº. 1, 2024, págs. 103-109
• Idioma: inglés
• DOI: 10.5565/PUBLMAT6812405
• Enlaces
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• Resumen

  • We provide examples of vector fields on (C3 , 0) admitting a formal first integral but no holomorphic first integral. These examples are related to a question raised by D. Cerveau and motivated by the celebrated theorems of Malgrange and Mattei–Moussu

• Referencias bibliográficas
  • W. Balser, Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations, Universitext, Springer-Verlag, New York,...
  • W. Balser, W. B. Jurkat, and D. A. Lutz, Birkhoff invariants and Stokes’ multipliers for meromorphic linear differential equations, J. Math....
  • A. Belotto da Silva, M. Klimes, J. C. Rebelo, and H. Reis, The global dynamics of Airy equation and of Painlev´e’s equations P-I and P-II,...
  • D. Cerveau and J.-F. Mattei, Formes int´egrables holomorphes singuli`eres, With an English summary, Ast´erisque 97, Soci´et´e Math´ematique...
  • G. Gorni and G. Zampieri, Analytic-non-integrability of an integrable analytic Hamiltonian system, Differential Geom. Appl. 22(3) (2005),...
  • Y. Ilyashenko and S. Yakovenko, Lectures on Analytic Differential Equations, Grad. Stud. Math. 86, American Mathematical Society, Providence,...
  • M. Klughertz, Existence d’une int´egrale premi`ere m´eromorphe pour des germes de feuilletages `a feuilles ferm´ees du plan complexe,...
  • B. Malgrange, Frobenius avec singularit´es. I. Codimension un, Inst. Hautes Etudes Sci. Publ. Math. 46 (1976), 163–173.
  • B. Malgrange, Frobenius avec singularit´es. II. Le cas g´en´eral, Invent. Math. 39(1) (1977), 67–89. DOI: 10.1007/BF01695953
  • J.-F. Mattei and R. Moussu, Holonomie et int´egrales premi`eres, Ann. Sci. Ecole Norm. Sup. (4) 13(4) (1980), 469–523. DOI: 10.24033/asens.1393.
  • S. Pinheiro and H. Reis, Topological aspects of completely integrable foliations, J. Lond. Math. Soc. (2) 89(2) (2014), 415–433. DOI: 10.1112/jlms/jdt069
  • L. Stolovitch, Progress in normal form theory, Nonlinearity 22(7) (2009), R77–R99. DOI: 10. 1088/0951-7715/22/7/R01
  • W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Pure and Applied Mathematics XIV, Interscience Publishers John Wiley...