Polynomial Vector Fields with Given Partial and First Integrals

2016 ◽  
pp. 41-85
Author(s):  
Jaume Llibre ◽  
Rafael Ramírez
Keyword(s):  
Vector Fields ◽  
First Integrals ◽  
Polynomial Vector ◽  
Polynomial Vector Fields

scholarly journals RATIONAL FIRST INTEGRALS FOR POLYNOMIAL VECTOR FIELDS ON ALGEBRAIC HYPERSURFACES OF ℝn+1

2012 ◽  
Vol 22 (11) ◽  
pp. 1250270 ◽  
Author(s):  
JAUME LLIBRE ◽  
YUDY BOLAÑOS
Keyword(s):  
Algebraic Geometry ◽  
Vector Field ◽  
Linear Algebra ◽  
Vector Fields ◽  
First Integral ◽  
First Integrals ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Algebraic Hypersurfaces ◽  
Rational First Integral

Using sophisticated techniques of Algebraic Geometry, Jouanolou in 1979 showed that if the number of invariant algebraic hypersurfaces of a polynomial vector field in ℝn of degree m is at least [Formula: see text], 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 ℝn+1, this extended result completes the standard results of the Darboux theory of integrability for polynomial vector fields on regular algebraic hypersurfaces of ℝn+1.

scholarly journals Efficient algorithms for computing rational first integrals and Darboux polynomials of planar polynomial vector fields

2015 ◽  
Vol 85 (299) ◽  
pp. 1393-1425 ◽  
Author(s):  
Alin Bostan ◽  
Guillaume Chèze ◽  
Thomas Cluzeau ◽  
Jacques-Arthur Weil
Keyword(s):  
Vector Fields ◽  
First Integrals ◽  
Efficient Algorithms ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Darboux Polynomials

scholarly journals On planar polynomial vector fields with elementary first integrals

2019 ◽  
Vol 267 (8) ◽  
pp. 4572-4588 ◽  
Author(s):  
Colin Christopher ◽  
Jaume Llibre ◽  
Chara Pantazi ◽  
Sebastian Walcher
Keyword(s):  
Vector Fields ◽  
First Integrals ◽  
Polynomial Vector ◽  
Polynomial Vector Fields
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