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2016

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

4 years 3 months ago
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 ...
Alin Bostan, Guillaume Chèze, Thomas Cluzea
Added 08 Apr 2016
Updated 08 Apr 2016
Type Journal
Year 2016
Where MOC
Authors Alin Bostan, Guillaume Chèze, Thomas Cluzeau, Jacques-Arthur Weil
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