RationalFirstintegrals swMATH ID: 14472 Software Authors: Bostan, Alin; Chèze, Guillaume; Cluzeau, Thomas; Weil, Jacques-Arthur 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/2016-85-299/S0025-5718-2015-03007-5/ Dependencies: Maple Related Software: Lsolver; Maple; PSsolver; gfun Cited 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 1335.34002Bostan, Alin; Chèze, Guillaume; Cluzeau, Thomas; Weil, Jacques-Arthur 2016 all top 5 Cited by 18 Authors 3 Chèze, Guillaume 2 Galindo Pastor, Carlos 2 Monserrat, Francisco 1 Bostan, Alin 1 Christopher, Colin J. 1 Cluzeau, Thomas 1 Combot, Thierry 1 Coutinho, S. C. 1 da Mota, Luis Antonio Campinho Pereira 1 Duarte, L. G. S. 1 Ferragut, Antoni 1 Llibre, Jaume 1 Pantazi, Chara 1 Pérez-Callejo, Elvira 1 Qu, Jingjia 1 Walcher, Sebastian 1 Weil, Jacques-Arthur 1 Yang, Shuangling all top 5 Cited in 7 Serials 3 Journal of Differential Equations 1 American Mathematical Monthly 1 Journal of Mathematical Analysis and Applications 1 Mathematics of Computation 1 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 1 Qualitative Theory of Dynamical Systems 1 Foundations of Computational Mathematics all top 5 Cited in 6 Fields 7 Ordinary differential equations (34-XX) 3 Computer science (68-XX) 1 Commutative algebra (13-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Partial differential equations (35-XX) 1 Optics, electromagnetic theory (78-XX) Citations by Year