RationalFirstintegrals
swMATH ID:  14472 
Software Authors:  Bostan, Alin; Chèze, Guillaume; Cluzeau, Thomas; Weil, JacquesArthur 
Description:  Efficient algorithms for computing rational first integrals and Darboux polynomials of planar polynomial vector fields. We present fast algorithms for computing rational first integrals with bounded degree of a planar polynomial vector field. Our approach builds upon a method proposed by Ferragut and Giacomini, whose main ingredients are the calculation of a power series solution of a first order differential equation and the reconstruction of a bivariate polynomial annihilating this power series. We provide explicit bounds on the number of terms needed in the power series. This enables us to transform their method into a certified algorithm computing rational first integrals via systems of linear equations. We then significantly improve upon this first algorithm by building a probabilistic algorithm with arithmetic complexity 𝒪 ˜(N 2ω ) and a deterministic algorithm solving the problem in at most 𝒪 ˜(N 2ω+1 ) arithmetic operations, where N denotes the given bound for the degree of the rational first integral, and ω the exponent of linear algebra. We also provide a fast heuristic variant which computes a rational first integral, or fails, in 𝒪 ˜(N ω+2 ) arithmetic operations. By comparison, the best previously known complexity was 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. 
Homepage:  http://www.ams.org/journals/mcom/201685299/S002557182015030075/ 
Dependencies:  Maple 
Related Software:  Lsolver; Maple; PSsolver; gfun 
Referenced in:  9 Publications 
Standard Articles
1 Publication describing the Software, including 1 Publication in zbMATH  Year 

Efficient algorithms for computing rational first integrals and Darboux polynomials of planar polynomial vector fields. Zbl 1335.34002 Bostan, Alin; Chèze, Guillaume; Cluzeau, Thomas; Weil, JacquesArthur 
2016

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Referenced by 18 Authors
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Referenced in 7 Serials
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