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Section: New Results

Efficient algorithms for rational first integrals

We presented in [4] fast algorithms for computing rational first integrals with degree bounded by $N$ of a planar polynomial vector field of degree $d\le N$. 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 $Õ\left(N^{2\omega }\right)$ and to a deterministic algorithm for solving the problem in $Õ\left(d^2{N}^{2\omega +1}\right)$ arithmetic operations, where $\omega$ is the exponent of linear algebra. By comparison, the best previous algorithm uses at least $d^{\omega +1}{N}^{4\omega +4}$ arithmetic operations. Our new algorithms are moreover very efficient in practice.