Advertisement

A Family of Foliations with One Singularity

  • S. C. CoutinhoEmail author
  • Filipe Ramos Ferreira
Article
  • 5 Downloads

Abstract

For every integer \(k \ge 3\) we describe a new family of foliations of degree k with one singularity. We show that a very generic member of this family has trivial isotropy group and a line as its unique Darboux polynomial.

Keywords

Holomorphic foliation Algebraic solution Singularity 

Mathematics Subject Classification

Primary 17B35 16S32 Secondary 37F75 

Notes

Acknowledgements

During the preparation of the paper the first author was partially supported by CNPq Grant 304543/2017-9 and the second author by a Grant PIBIC(CNPq). We also benefited from the access to on-line journals provided by CAPES. We would also like to thank the referee for numerous suggestions that greatly improved this paper.

References

  1. Alcántara, C.R.: Foliations on \(\mathbb{CP}^2\) of degree 2 with degenerate singularities. Bull. Braz. Math. Soc. (N.S.) 44(3), 421–454 (2013)MathSciNetCrossRefGoogle Scholar
  2. Alcántara, C.R.: Foliations on \(\mathbb{CP}^2\) of degree \(d\) with a singular point with Milnor number \(d^2+d+1\). Rev. Mat. Complut. 31(1), 187–199 (2018)MathSciNetCrossRefGoogle Scholar
  3. Cerveau, D., Déserti, J., Garba Belko, D., Meziani, R.: Géométrie classique de certains feuilletages de degré deux. Bull. Braz. Math. Soc. (N.S.) 41(2), 161–198 (2010)MathSciNetCrossRefGoogle Scholar
  4. Coutinho, S.C.: \(d\)-simple rings and simple \({\cal{D}}\)-modules. Math. Proc. Camb. Philos. Soc. 125(3), 405–415 (1999)MathSciNetCrossRefGoogle Scholar
  5. Coutinho, S.C., Menasché Schechter, L.: Algebraic solutions of holomorphic foliations: an algorithmic approach. J. Symb. Comput. 41, 603–618 (2006)MathSciNetCrossRefGoogle Scholar
  6. Daly, T.: Axiom: The Thirty Year Horizon, volume 1: Tutorial. Lulu Press, Morrisville (2005)Google Scholar
  7. Darboux, G.: Mémoire sur les équations différentielles algébriques du I\(^{\rm o}\) ordre et du premier degré. Bull. des Sci. Math. (Mélanges), 60–96, 123–144, 151–200 (1878)Google Scholar
  8. Doering, A.M.De S., Lequain, Y., Ripoll, C.: Differential simplicity and cyclic maximal ideals of the Weyl algebra \(A_2(K)\). Glasg. Math. J. 48(2), 269–274 (2006)MathSciNetCrossRefGoogle Scholar
  9. Goodearl, K.R., Warfield Jr., R.B.: An Introduction to Noncommutative Noetherian Rings, London Mathematical Society Student Texts, vol. 61, 2nd edn. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  10. Ince, E.L.: Ordinary Differential Equations. Dover, New York (1956). first edition published in 1926Google Scholar
  11. Jacobi, C.: De integratione aequationes differentiallis \((a+a^{\prime }x+a^{\prime \prime }y)(xdy-ydx)-(b+b^{\prime }x+b^{\prime \prime }y)dy+(c+c^{\prime }x+c^{\prime \prime }y)dx)=0\). J. für die reine und angewandte Mathematik 24, 1–4 (1842)Google Scholar
  12. Jordan, D.A.: Ore extensions and Poisson algebras. Glasg. Math. J. 56(2), 355–368 (2014)MathSciNetCrossRefGoogle Scholar
  13. Jouanolou, J.P.: Equations de Pfaff algébriques, Lect. Notes in Math., vol. 708. Springer, Heidelberg (1979)CrossRefGoogle Scholar
  14. Man, Y.-K., MacCallum, M.A.H.: A rational approach to the Prelle–Singer algorithm. J. Symb. Comput. 24, 31–43 (1997)MathSciNetCrossRefGoogle Scholar
  15. Zariski, O., Samuel, P.: Commutative Algebra, vol. I. Springer, New York (1975). (Reprint of the 1958 edition, Graduate Texts in Mathematics, vol. 28)Google Scholar

Copyright information

© Sociedade Brasileira de Matemática 2019

Authors and Affiliations

  1. 1.Departamento de Ciência da Computação, Instituto de MatemáticaUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil

Personalised recommendations