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 ˜O(N2ω) and a deterministic algorithm solving the problem in at most ˜O(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 ...

• Type: Journal
• Year: 2016
• Where: MOC
• Authors: Alin Bostan, Guillaume Chèze, Thomas Cluzeau, Jacques-Arthur Weil
• Keywords: Information Technology, MOC 2016