A fast randomized geometric algorithm for computing Riemann-Roch spaces

Abstract:

We propose a probabilistic variant of Brill-Noether’s algorithm for computing a basis of the Riemann-Roch space $L(D)$ associated to a divisor $D$ on a projective nodal plane curve $\mathbb {C}$ over a sufficiently large perfect field $k$. Our main result shows that this algorithm requires at most $O(\max (\deg (\mathbb {C})^{2\omega }, \deg (D_+)^\omega ))$ arithmetic operations in $k$, where $\omega$ is a feasible exponent for matrix multiplication and $D_+$ is the smallest effective divisor such that $D_+\geq D$. This improves the best known upper bounds on the complexity of computing Riemann-Roch spaces. Our algorithm may fail, but we show that provided that a few mild assumptions are satisfied, the failure probability is bounded by $O(\max (\deg (\mathbb {C})^4, \deg (D_+)^2)/\lvert \mathcal E\rvert )$, where $\mathcal E$ 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 we present experimental data. In particular, our implementation enjoys a speedup larger than 6 on many examples (and larger than 200 on some instances over large finite fields) 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 $O(g^\omega )$ operations in $k$, which equals the best known complexity for this problem.