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 associated to a divisor on a projective plane curve over a sufficiently large perfect field . Our main result shows that this algorithm requires at most arithmetic operations in , where is a feasible exponent for matrix multiplication and is the smallest effective divisor such that . 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 , where is a finite subset of 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 within operations in , which slightly improves in this context the best known complexity of Khuri-Makdisi's algorithm.