Rational first integrals for polynomial vector fields on algebraic hypersurfaces of ℝ n+1

Jaume Llibre, Yudy Bolaños

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2 Citations (Scopus)

Abstract

Using sophisticated techniques of Algebraic Geometry, Jouanolou in 1979 showed that if the number of invariant algebraic hypersurfaces of a polynomial vector field in ℝ n of degree m is at least {n+m-1 \choose n}+n, then the vector field has a rational first integral. Llibre and Zhang used only Linear Algebra to provide a shorter and easier proof of the result given by Jouanolou. We use ideas of Llibre and Zhang to extend the Jouanolou result to polynomial vector fields defined on algebraic regular hypersurfaces of ℝ n+1 , this extended result completes the standard results of the Darboux theory of integrability for polynomial vector fields on regular algebraic hypersurfaces of ℝ n+1 . © 2012 World Scientific Publishing Company.
Original languageEnglish
Article number1250270
JournalInternational Journal of Bifurcation and Chaos
Volume22
Issue number11
DOIs
Publication statusPublished - 1 Jan 2012

Keywords

  • algebraic hypersurfaces
  • Darboux theory of integrability
  • polynomial vector fields
  • rational first integrals

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