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) over tilde (N-2 omega) and a deterministic algorithm solving the problem in at most (O) over tilde (N2 omega+1) arithmetic operations, where N denotes the given bound for the degree of the rational first integral, and omega the exponent of linear algebra. We also provide a fast heuristic variant which computes a rational first integral, or fails, in (O) over tilde (N omega+2) arithmetic operations. By comparison, the best previously known complexity was N4 omega+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 RATIONAL FIRST INTEGRALS which is available to interested readers with examples showing its efficiency.