Rational first integrals in the Darboux theory of integrability in Cn

Jaume Llibre; Xiang Zhang

doi:https://doi.org/10.1016/j.bulsci.2007.12.001

Rational first integrals in the Darboux theory of integrability in Cⁿ

Jaume Llibre, Xiang Zhang

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

31 Citations (Scopus)

Abstract

In 1979 Jouanolou showed that if the number of invariant algebraic hypersurfaces of a polynomial vector field in Rn or Cn of degree d is at least ((d + n - 1; n)) + n, then the vector field has a rational first integral. His proof used sophisticated tools of algebraic geometry. We provide an easy and elementary proof of Jouanolou's result using linear algebra. © 2007 Elsevier Masson SAS. All rights reserved.

Original language	English
Pages (from-to)	189-195
Journal	Bulletin des Sciences Mathematiques
Volume	134
DOIs	https://doi.org/10.1016/j.bulsci.2007.12.001
Publication status	Published - 1 Mar 2010

Keywords

Darboux integrability
Rational first integral

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abstract = "In 1979 Jouanolou showed that if the number of invariant algebraic hypersurfaces of a polynomial vector field in Rn or Cn of degree d is at least ((d + n - 1; n)) + n, then the vector field has a rational first integral. His proof used sophisticated tools of algebraic geometry. We provide an easy and elementary proof of Jouanolou's result using linear algebra. {\textcopyright} 2007 Elsevier Masson SAS. All rights reserved.",

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AU - Zhang, Xiang

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N2 - In 1979 Jouanolou showed that if the number of invariant algebraic hypersurfaces of a polynomial vector field in Rn or Cn of degree d is at least ((d + n - 1; n)) + n, then the vector field has a rational first integral. His proof used sophisticated tools of algebraic geometry. We provide an easy and elementary proof of Jouanolou's result using linear algebra. © 2007 Elsevier Masson SAS. All rights reserved.

AB - In 1979 Jouanolou showed that if the number of invariant algebraic hypersurfaces of a polynomial vector field in Rn or Cn of degree d is at least ((d + n - 1; n)) + n, then the vector field has a rational first integral. His proof used sophisticated tools of algebraic geometry. We provide an easy and elementary proof of Jouanolou's result using linear algebra. © 2007 Elsevier Masson SAS. All rights reserved.

KW - Darboux integrability

KW - Rational first integral

UR - https://ddd.uab.cat/record/226098

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DO - https://doi.org/10.1016/j.bulsci.2007.12.001

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EP - 195

JO - Bulletin des Sciences Mathematiques

JF - Bulletin des Sciences Mathematiques

SN - 0007-4497