Algebraic Multiplicity and the Poincaré Problem

Jinzhi, Lei; Yang, Lei

doi:10.1007/3-7643-7429-2_9

Cited by 2 publications

(1 citation statement)

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“…In relationship with our approach let us also mention the work by Lei and Yang [13], where the bounds for irreducible invariant algebraic curves of certain dynamical systems were obtained in terms of algebraic multiplicities of the dynamical systems in question at the singular points, and the method for finding rational first integrals of two-dimensional polynomial vector fields introduced by Ferragut and Giacomini [14]. This method uses Tailor and Puiseux series near finite points.…”

Section: Introductionmentioning

confidence: 99%

From Puiseux series to invariant algebraic curves: the FitzHugh-Nagumo model

Demina

2018

Preprint

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A relationship between Puiseux series satisfying an ordinary differential equation corresponding to a polynomial dynamical system and degrees of irreducible invariant algebraic curves is studied. A bound on the degrees of irreducible invariant algebraic curves for a wide class of polynomial dynamical systems is obtained. It is demonstrated that the Puiseux series near infinity can be used to find irreducible algebraic curves explicitly. As an example, all irreducible invariant algebraic curves for the famous FitzHugh-Nagumo system are obtained.

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Section: Introductionmentioning

confidence: 99%

From Puiseux series to invariant algebraic curves: the FitzHugh-Nagumo model

Demina

2018

Preprint

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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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Algebraic Multiplicity and the Poincaré Problem

Cited by 2 publications

References 10 publications

From Puiseux series to invariant algebraic curves: the FitzHugh-Nagumo model

From Puiseux series to invariant algebraic curves: the FitzHugh-Nagumo model

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

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