## Section: New Results

### Efficient algorithms for rational first integrals

We presented in [29] 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 $\xd5\left({N}^{2\omega}\right)$ and to a deterministic algorithm for solving the problem in $\xd5\left({d}^{2}{N}^{2\omega +1}\right)$ 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.