## Section: New Results

### Efficient algorithms for rational first integrals

We presented in [14] fast algorithms for computing rational first integrals with bounded degree of a planar polynomial vector field. Our approach is inspired by an idea of Ferragut and Giacomini. We improve upon their work by proving that rational first integrals can be computed via systems of linear equations instead of systems of quadratic equations. This leads to a probabilistic algorithm with arithmetic complexity $\xd5\left({N}^{2\omega}\right)$ and to a deterministic algorithm solving the problem in $\xd5\left({d}^{2}{N}^{2\omega +1}\right)$ arithmetic operations, where $N$ denotes the given bound for the degree of the rational first integral, and where $d\le N$ is the degree of the vector field, and $\omega $ the exponent of linear algebra. By comparison, the best previous algorithm uses at least ${d}^{\omega +1}{N}^{4\omega +4}$ arithmetic operations. The new algorithms are very efficient in practice.