Section: New Results

A fast randomized geometric algorithm for computing Riemann-Roch spaces

Participants : Aude Le Gluher, Pierre-Jean Spaenlehauer [contact] .

In [16], we proposed a probabilistic Las Vegas variant of Brill-Noether's algorithm for computing a basis of the Riemann-Roch space $L\left(D\right)$ associated to a divisor $D$ on a projective plane curve $𝒞$ over a sufficiently large perfect field $k$. Our main result shows that this algorithm requires at most $O(max(deg{\left(𝒞\right)}^{2\omega },deg{\left(D_{+}\right)}^{\omega }))$ arithmetic operations in $k$, where $\omega$ is a feasible exponent for matrix multiplication and $D_{+}$ is the smallest effective divisor such that $D_{+}\ge D$. This improves the best known upper bounds on the complexity of computing Riemann-Roch spaces. Our algorithm may fail, but we showed that provided that a few mild assumptions are satisfied, the failure probability is bounded by $O(max(deg{\left(𝒞\right)}^4,deg{\left(D_{+}\right)}^2)/\left|E\right|)$, where $E$ is a finite subset of $k$ in which we pick elements uniformly at random. We provide a freely available C++/NTL implementation of the proposed algorithm, and experimental data. In particular, our implementation enjoys a speed-up larger than 9 on several examples compared to the reference implementation in the Magma computer algebra system. As a by-product, our algorithm also yields a method for computing the group law on the Jacobian of a smooth plane curve of genus $g$ within $O\left(g^{\omega }\right)$ operations in $k$, which slightly improves in this context the best known complexity $O\left(g^{\omega +\epsilon }\right)$ of Khuri-Makdisi's algorithm.