Vector fields, invariant varieties and linear systems

Jorge Vitório Pereira[1]

  • [1] IMPA, Estrada Dona Castorina, 110, Jardim Botanico, 22460-320 Rio de Janeiro (Brésil)

Annales de l’institut Fourier (2001)

  • Volume: 51, Issue: 5, page 1385-1405
  • ISSN: 0373-0956

Abstract

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We investigate the interplay between invariant varieties of vector fields and the inflection locus of linear systems with respect to the vector field. Among the consequences of such investigation we obtain a computational criterion for the existence of rational first integrals of a given degree, bounds for the number of first integrals on families of vector fields, and a generalization of Darboux's criteria. We also provide a new proof of Gomez--Mont's result on foliations with all leaves algebraic.

How to cite

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Pereira, Jorge Vitório. "Vector fields, invariant varieties and linear systems." Annales de l’institut Fourier 51.5 (2001): 1385-1405. <http://eudml.org/doc/115951>.

@article{Pereira2001,
abstract = {We investigate the interplay between invariant varieties of vector fields and the inflection locus of linear systems with respect to the vector field. Among the consequences of such investigation we obtain a computational criterion for the existence of rational first integrals of a given degree, bounds for the number of first integrals on families of vector fields, and a generalization of Darboux's criteria. We also provide a new proof of Gomez--Mont's result on foliations with all leaves algebraic.},
affiliation = {IMPA, Estrada Dona Castorina, 110, Jardim Botanico, 22460-320 Rio de Janeiro (Brésil)},
author = {Pereira, Jorge Vitório},
journal = {Annales de l’institut Fourier},
keywords = {holomorphic vector fields; linear systems; inflection points},
language = {eng},
number = {5},
pages = {1385-1405},
publisher = {Association des Annales de l'Institut Fourier},
title = {Vector fields, invariant varieties and linear systems},
url = {http://eudml.org/doc/115951},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Pereira, Jorge Vitório
TI - Vector fields, invariant varieties and linear systems
JO - Annales de l’institut Fourier
PY - 2001
PB - Association des Annales de l'Institut Fourier
VL - 51
IS - 5
SP - 1385
EP - 1405
AB - We investigate the interplay between invariant varieties of vector fields and the inflection locus of linear systems with respect to the vector field. Among the consequences of such investigation we obtain a computational criterion for the existence of rational first integrals of a given degree, bounds for the number of first integrals on families of vector fields, and a generalization of Darboux's criteria. We also provide a new proof of Gomez--Mont's result on foliations with all leaves algebraic.
LA - eng
KW - holomorphic vector fields; linear systems; inflection points
UR - http://eudml.org/doc/115951
ER -

References

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  2. J. Artés, B. Grünbaum, J. Llibre, On the number of invariant straight lines for polynomial differential systems, Pacific Journal of Mathematics 184 (1998) Zbl0930.34022MR1628583
  3. A. Cayley, On the sextactic points of a plane curve, 341 (1865), Trans. Royal Soc. London CLV 
  4. F. Cukierman, Determinant of complexes and higher Hessians, Mathematische Annalen 307 (1997) Zbl0870.14022MR1428872
  5. G. Fischer, Complex Analytic Geometry, 538 (1976), Springer Zbl0343.32002MR430286
  6. X. Gomez-Mont, Integrals for holomorphic foliations with singularities having all leaves compact, Ann. Inst. Fourier, Grenoble 39 (1989) Zbl0667.58052MR1017286
  7. J. P. Joaunolou, Équations de Pfaff algébriques, 708 (1979), Springer Zbl0477.58002MR537038
  8. A. Lins Neto, Some Examples for Poincaré and Painlevé Problems, (2000) Zbl1130.34301MR1914932
  9. A. Lins Neto, B. Scárdua, Folheações Algébricas Complexas, IMPA, 21 Colóquio Brasileiro de Matemática, Rio de Janeiro (1997) 
  10. J.-M. Lion, Un critère de Darboux d'existence d'intégrale première pour les 1-formes différentielles analytiques, Boletim da Sociedade Brasileira de Matemática 31 (2000) Zbl0964.37028MR1785265
  11. J. LLibre, J. V. Pereira, Multiplicity of Algebraic Invariant Curves and Darboux Integrability, (2000) 

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