A recombination algorithm for the decomposition of multivariate rational functions

Chèze, Guillaume

doi:10.1090/s0025-5718-2012-02658-5

Cited by 2 publications

(2 citation statements)

References 37 publications

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“…In [MO04,Chè12b], the authors propose different algorithms for the decomposition of rational functions using properties of Darboux polynomials and rational first integrals of the Jacobian derivation. Lemma 9.…”

Section: Review On First Integrals Of Polynomial Vector Fieldsmentioning

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: Review On First Integrals Of Polynomial Vector Fieldsmentioning

confidence: 99%

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

et al. 2015

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“…Therefore, any fast factorization algorithm in terms of the dense size leads to a fast algorithm in terms of the convex size. Another important application of our Theorem 1.2, developed by Chèze in [Chè10], concerns the decomposition of multivariate rational functions.…”

Section: Introductionmentioning

confidence: 99%

Reduction of bivariate polynomials from convex-dense to dense, with application to factorizations

Berthomieu

Lecerf

2012

Math. Comp.

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International audienceIn this article we present a new algorithm for reducing the usual sparse bivariate factorization problems to the dense case. This reduction simply consists in computing an invertible monomial transformation that produces a polynomial with a dense size of the same order of magnitude as the size of the integral convex hull of the support of the input polynomial. This approach turns out to be very efficient in practice, as demonstrated with our implementation

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A recombination algorithm for the decomposition of multivariate rational functions

Cited by 2 publications

References 37 publications

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

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

Reduction of bivariate polynomials from convex-dense to dense, with application to factorizations

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