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

32 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

Access to Document

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

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

Cite this

@article{bb5a5af12068471fbbd386967634c961,

title = "Rational first integrals in the Darboux theory of integrability in Cn",

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.",

keywords = "Darboux integrability, Rational first integral",

author = "Jaume Llibre and Xiang Zhang",

year = "2010",

month = mar,

day = "1",

doi = "https://doi.org/10.1016/j.bulsci.2007.12.001",

language = "English",

volume = "134",

pages = "189--195",

journal = "Bulletin des Sciences Mathematiques",

issn = "0007-4497",

publisher = "Elsevier Masson SAS",

}

TY - JOUR

T1 - Rational first integrals in the Darboux theory of integrability in Cn

AU - Llibre, Jaume

AU - Zhang, Xiang

PY - 2010/3/1

Y1 - 2010/3/1

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

U2 - https://doi.org/10.1016/j.bulsci.2007.12.001

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

M3 - Article

VL - 134

SP - 189

EP - 195

JO - Bulletin des Sciences Mathematiques

JF - Bulletin des Sciences Mathematiques

SN - 0007-4497