Displaying similar documents to “Generalized nonlinear superposition principles for polynomial planar vector fields.”

On algebraic solutions of algebraic Pfaff equations

Henryk Żołądek (1995)

Studia Mathematica

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We give a new proof of Jouanolou’s theorem about non-existence of algebraic solutions to the system ẋ = z s , ẏ = x s , ż = y s . We also present some generalizations of the results of Darboux and Jouanolou about algebraic Pfaff forms with algebraic solutions.

Some (non-)elimination results for curves in geometric structures

Serge Randriambololona, Sergei Starchenko (2011)

Fundamenta Mathematicae

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We show that the first order structure whose underlying universe is ℂ and whose basic relations are all algebraic subsets of ℂ² does not have quantifier elimination. Since an algebraic subset of ℂ² is either of dimension ≤ 1 or has a complement of dimension ≤ 1, one can restate the former result as a failure of quantifier elimination for planar complex algebraic curves. We then prove that removing the planarity hypothesis suffices to recover quantifier elimination: the structure with...

Basic algebro-geometric conceps in the study of planar polynomial vector fields.

Dana Schlomiuk (1997)

Publicacions Matemàtiques

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In this work we show that basic algebro-geometric concepts such as the concept of intersection multiplicity of projective curves at a point in the complex projective plane, are needed in the study of planar polynomial vector fields and in particular in summing up the information supplied by bifurcation diagrams of global families of polynomial systems. Algebro-geometric concepts are helpful in organizing and unifying in more intrinsic ways this information.

Nondegenerate linearizable centre of complex planar quadratic and symmetric cubic systems in C.

Colin Christopher, Christiane Rousseau (2001)

Publicacions Matemàtiques

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In this paper we consider complex differential systems in the plane, which are linearizable in the neighborhood of a nondegenerate centre. We find necessary and sufficient conditions for linearizability for the class of complex quadratic systems and for the class of complex cubic systems symmetric with respect to a centre. The sufficiency of these conditions is shown by exhibiting explicitly a linearizing change of coordinates, either of Darboux type or a generalization of it. ...