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

Abstract

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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