Algebraic solutions of holomorphic foliations: An algorithmic approach

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

doi:10.1016/j.jsc.2005.11.002

Cited by 9 publications

(6 citation statements)

References 18 publications

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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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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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“…For a proof of this well-known result see [19] or [9]. Thus, to prove that a 1-form Ω, with rational coefficients, does not have invariant algebraic curves in P 2 \ L ∞ , it is enough to consider the special case in which the solution is defined by a polynomial of Q[x, y, z].…”

Section: Preliminariesmentioning

confidence: 99%

“…We thank the referee of [9], who suggested that the methods of that paper might also be applicable to vector fields of the affine plane, and J. V. Pereira who pointed out that, using extactic curves, our algorithm could be greatly improved. The first author was partially supported by grants from CNPq and PRONEX(Commutative algebra and algebraic geometry), and the second author by a scholarship from CNPq.…”

Section: Acknowledgementsmentioning

confidence: 99%

“…For more on the relation between the problem of the centre and invariant curves see [5]. This is precisely the problem we tackle in this paper: building on the ideas of [9] we present an algorithm that allows one to test a given vector field (with rational coefficients) for the existence of invariant algebraic curves. Although the algorithm may be unable to come to a conclusion (thus returning the algorithm failed) it is very efficient for vector fields defined by polynomials for which all coefficients are nonzero up to the top monomial of highest degree (often called dense polynomials).…”

Section: Introductionmentioning

confidence: 99%

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Algebraic solutions of plane vector fields

Coutinho

Schechter

2009

Journal of Pure and Applied Algebra

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“…Our interest in this problem is twofold. On the one hand, we want to use the Galois group techniques developed in [4] in order to give an efficient algorithm that can be used to compute first integrals of polynomial vector fields over the projective plane. On the other hand, as shown in [3], the existence of an algebraic solution implies that any nonholonomic D-module defined as a deformation of a vector field by a polynomial function always has a holonomic quotient module.…”

Section: Introductionmentioning

confidence: 99%