Algebraic solutions of plane vector fields

Coutinho, S. C.; Schechter, L. Menasché

doi:10.1016/j.jpaa.2008.06.003

Cited by 8 publications

(8 citation statements)

References 14 publications

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“…This property heavily constrains the possible closed invariant curves for (A 2 , v). For example, with refinements of this idea in [5], Coutinho and Menasché describe an algorithm to compute polynomial vectors fields on the plane without closed invariant curves.…”

Section: -Foliation Tangent To a Vector Fieldmentioning

confidence: 99%

Rational factors, invariant foliations and algebraic disintegration of compact mixing Anosov flows of dimension 3

Jaoui¹

2021

Confluentes Mathematici

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In this article, we develop a geometric framework to study the notion of semiminimality for the generic type of a smooth autonomous differential equation (X, v), based on the study of rational factors of (X, v) and of algebraic foliations on X, invariant under the Lie derivative of the vector field v.We then illustrate the effectiveness of these methods by showing that certain autonomous algebraic differential equation of order three defined over the field of real numbers -more precisely, those associated to mixing, compact, Anosov flows of dimension three -are generically disintegrated.

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Section: -Foliation Tangent To a Vector Fieldmentioning

confidence: 99%

Rational factors, invariant foliations and algebraic disintegration of compact mixing Anosov flows of dimension 3

Jaoui¹

2021

Confluentes Mathematici

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“…Existing algorithms for computing rational first integrals and Darboux polynomials of bounded degree. There is a vast literature regarding the computation of rational or elementary first integrals (see for example [FG10, MM97, Chè11, PS83, SGR90, Sin92, LZ10, DDdMS01, Poi91]) and Darboux polynomials (see for example [CMS09,CMS06,CGG05,Wei95,Dar78]). Note that, among these articles, very few restrict to the specific question of rational first integrals.…”

Section: 3mentioning

confidence: 99%

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

et al. 2015

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Abstract. We present fast algorithms for computing rational first integrals with bounded degree of a planar polynomial vector field. Our approach is inspired by an idea of Ferragut and Giacomini ([FG10]). We improve upon their work by proving that rational first integrals can be computed via systems of linear equations instead of systems of quadratic equations. The main ingredients of our algorithms are the calculation of a power series solution of a first order differential equation and the reconstruction of a bivariate polynomial annihilating a power series. This leads to a probabilistic algorithm with arithmetic complexityÕ(N 2ω ) and to a deterministic algorithm solving the problem iñ O(d 2 N 2ω+1 ) arithmetic operations, where N denotes the given bound for the degree of the rational first integral, and where d ≤ N is the degree of the vector field, and ω the exponent of linear algebra. We also provide a fast heuristic variant which computes a rational first integral, or fails, inÕ(N ω+2 ) arithmetic operations. By comparison, the best previous algorithm given in [Chè11] uses at least d ω+1 N 4ω+4 arithmetic operations. We then show how to apply a similar method to the computation of Darboux polynomials. The algorithms are implemented in a Maple package RationalFirstIntegrals which is available to interested readers with examples showing its efficiency.

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“…But it seems that the question of integrating factors was discussed for the first time in [27]. Recently an algorithmic approach for curves (in the affine setting) was proposed by Coutinho and Menasché Schechter [14].…”

Section: The Direct Problemsmentioning

confidence: 99%

Inverse Problems in Darboux’ Theory of Integrability

et al. 2012

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Abstract. The Darboux theory of integrability for planar polynomial differential equations is a classical field, with connections to Lie symmetries, differential algebra and other areas of mathematics. In the present paper we introduce the concepts, problems and inverse problems, and we outline some recent results on inverse problems. We also prove a new result, viz. a general finiteness theorem for the case of prescribed integrating factors. A number of relevant examples and applications is included.

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Algebraic solutions of plane vector fields

Cited by 8 publications

References 14 publications

Rational factors, invariant foliations and algebraic disintegration of compact mixing Anosov flows of dimension 3

Rational factors, invariant foliations and algebraic disintegration of compact mixing Anosov flows of dimension 3

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

Inverse Problems in Darboux’ Theory of Integrability

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