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 ([FG10]) 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 Õ(N2ω) and a deterministic algorithm solving the problem in at most Õ(d2N2ω+1) arithmetic operations, where N denotes the given bound for the degree of the rational first integral, and where d 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, in Õ(Nω+2) arithmetic operations. By comparison, the best previously known complexity was dω+1 N4ω+4 arithmetic operations using the algorithm given in [Chè11]. 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.