Remarks on vector fields with simply connected trajectories and their associated derivations

  1. Bustinduy, Alvaro 1
  2. Giraldo, Luis
  1. 1 Universidad Nebrija
    info

    Universidad Nebrija

    Madrid, España

    ROR https://ror.org/03tzyrt94

Revista:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas

ISSN: 1578-7303,  1579-1505

Año de publicación: 2019

Volumen: 113

Número: 4

Páginas: 4119-4126

Tipo: Artículo

DOI: 10.1007/S13398-019-00670-Z GOOGLE SCHOLAR

Otras publicaciones en: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas

Resumen

Let X be a polynomial vector field on C2 with at most isolated zeros and whose trajectories are all simply connected. Let us suppose that there is a polynomial P∈ C[x, y] such that (i) dP(X) = 1 or (ii) dP(X) = a· P, with a∈ C∗. In (Bustinduy and Giraldo, in Adv Math 285:1339–1357, 2015; Bustinduy and Giraldo, in J Differ Equ 264:3933–3939, 2018) the authors determined X and P, up to an algebraic change of coordinates, when P∈ C[x, y] is primitive. In this note, we extend these results for an arbitrary P. Finally, as an application, we show that if a polynomial vector field X on C2 with at most isolated zeros has all its trajectories simply connected and there exist P∈ C[x, y] and n∈ N+ such that Xn(P) = 0 and Xn - 1(P) ≠ 0 or Xn + 1(P) = a· Xn(P) with a∈ C∗, X is complete and present some questions on the study of derivations whose image is a Mathieu subspace.

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Referencias bibliográficas

  • Brunella, M.: Complete vector fields on the complex plane. Topology 43(2), 433–445 (2004)
  • Bustinduy, A., Giraldo, L.: Vector fields with simply connected trajectories transverse to a polynomial. Adv. Math. 285, 1339–1357 (2015)
  • Bustinduy, A., Giraldo, L.: On vector fields with simply connected trajectories and one invariant line. J. Differ. Equ. 264, 3933–3939 (2018)
  • Bustinduy, A., Giraldo, L., Muciño-Raymundo, J.: Vector fields from locally invertible polynomial maps in $${\mathbb{C}}^n$$ C n . Colloq. Math. 140, 205–220 (2015)
  • Cerveau, D.: Dérivations surjectives de l’anneau $${\mathbb{C}}[x, y]$$ C [ x , y ] . J. Algebra 195, 320–335 (1997)
  • Cerveau, D.: Quelques problèmes en géométrie feuilletée pour les 60 années de l’IMPA. Bull. Braz. Math. Soc. New Ser. 44, 653–679 (2013)
  • Christopher, C., Llibre, J., Pereira, J.V.: Multiplicity of invariant algebraic curves in polynomial vector fields. Pac. J. Math. 229(1), 63–117 (2007)
  • Coomes, B., Zurkowski, V.: Linearization of polynomial flows and spectra of derivations. J. Dyn. Differ. Equ. 1, 29–66 (1991)
  • Corrêa Jr., M.: An improvement to Laguntinskii–Pereira integrability theorem. Math. Res. Lett 18, 645–661 (2011)
  • Corrêa, M., Corrêa Jr., M., Maza, L.G., Soares, M.G.: Hypersurfaces invariant by Pfaff systems. Commun. Contemp. Math. 17(16), 1450051 (2015)
  • Freudenburg, G.: Algebraic Theory of Locally Nilpotent Derivations. Encyclopaedia of Mathematical Sciences, vol. 136. Springer, Berlin (2006)
  • Gurjar, R.V., Masuda, K, Miyanishi, M.: Surjective derivations in small dimensions. J. Ramanujan Math. Soc. 28A(Spec. Iss.), 221–246 (2013)
  • Rentschler, R.: Opérations du groupe additif sur le plan affine. C. R. Acad. Sci. Paris 267, 384–387 (1968)
  • Sun, X.: Images of derivations of polynomial algebras with divergence zero. J. Algebra 492, 414–418 (2017)
  • Suzuki, M.: Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l’espace $${\mathbb{C}}^{2}$$ C 2 . J. Math. Soc. Jpn. 26, 241–257 (1974)
  • Suzuki, M.: Sur les opérations holomorphes du groupe additif complexe sur l’espace de deux variables complexes. Ann. Sci. École Norm. Sup. 10(4), 517–546 (1977)
  • van den Essen, A., Wright, D., Zhao, W.: Images of locally finite derivations of polynomial algebras in two variables. J. Pure Appl. Algebra 215, 2130–2134 (2011)
  • Zhao, W.: Generalizations of the image conjecture and the Mathieu conjecture. J. Pure Appl. Algebra 214, 1200–1216 (2010)