Section: New Results
Efficient algorithms for rational first integrals
We presented in  fast algorithms for computing rational first integrals with degree bounded by of a planar polynomial vector field of degree . The main novelty is that such rational first integrals are obtained by computing via systems of linear equations instead of systems of quadratic equations. This leads to a probabilistic algorithm with arithmetic complexity and to a deterministic algorithm for solving the problem in arithmetic operations, where is the exponent of linear algebra. By comparison, the best previous algorithm uses at least arithmetic operations. Our new algorithms are moreover very efficient in practice.