scholarly journals Invariant Algebraic Curves and Rational First Integrals for Planar Polynomial Vector Fields

2001 ◽  
Vol 169 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Javier Chavarriga ◽  
Jaume Llibre
Keyword(s):  
Vector Fields ◽  
Algebraic Curves ◽  
First Integrals ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Invariant Algebraic Curves

scholarly journals Inverse problems for invariant algebraic curves: explicit computations

2009 ◽  
Vol 139 (2) ◽  
pp. 287-302 ◽  
Author(s):  
Colin Christopher ◽  
Jaume Llibre ◽  
Chara Pantazi ◽  
Sebastian Walcher
Keyword(s):  
Inverse Problems ◽  
Algebraic Curve ◽  
Affine Plane ◽  
Vector Fields ◽  
Algebraic Curves ◽  
General Setting ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Algorithmic Approach ◽  
Invariant Algebraic Curves

Given an algebraic curve in the complex affine plane, we describe how to determine all planar polynomial vector fields which leave this curve invariant. If all (finite) singular points of the curve are non-degenerate, we give an explicit expression for these vector fields. In the general setting we provide an algorithmic approach, and as an alternative we discuss sigma processes.

scholarly journals Cofactors and equilibria for polynomial vector fields

2014 ◽  
Vol 144 (4) ◽  
pp. 753-760
Author(s):  
Antoni Ferragut ◽  
Jaume Llibre
Keyword(s):  
Vector Fields ◽  
Algebraic Curves ◽  
Equilibrium Points ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Differential Systems ◽  
Existence Of Equilibrium ◽  
Invariant Algebraic Curves ◽  
Exponential Factors

We present a relationship between the existence of equilibrium points of differential systems and the cofactors of the invariant algebraic curves and the exponential factors of the system.

scholarly journals Multiplicity of invariant algebraic curves in polynomial vector fields

2007 ◽  
Vol 229 (1) ◽  
pp. 63-117 ◽  
Author(s):  
Colin Christopher ◽  
Jaume Llibre ◽  
Jorge Vitório Pereira
Keyword(s):  
Vector Fields ◽  
Algebraic Curves ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Invariant Algebraic Curves

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
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