# RationalFirstintegrals

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 ˜(N 2ω ) and a deterministic algorithm solving the problem in at most 𝒪 ˜(N 2ω+1 ) arithmetic operations, where N denotes the given bound for the degree of the rational first integral, and ω the exponent of linear algebra. We also provide a fast heuristic variant which computes a rational first integral, or fails, in 𝒪 ˜(N ω+2 ) arithmetic operations. By comparison, the best previously known complexity was 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.

## References in zbMATH (referenced in 2 articles , 1 standard article )

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Sorted by year (- Bostan, Alin; Chèze, Guillaume; Cluzeau, Thomas; Weil, Jacques-Arthur: Efficient algorithms for computing rational first integrals and Darboux polynomials of planar polynomial vector fields (2016)
- Chèze, Guillaume: Bounding the number of remarkable values via Jouanolou’s theorem (2015)