A FAST RANDOMIZED GEOMETRIC ALGORITHM FOR COMPUTING RIEMANN-ROCH SPACES

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 C over a sufficiently large perfect field k. Our main result shows that this algorithm requires at most O(max(deg(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+ >= 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(C)(4), deg(D+)(2))/vertical bar epsilon vertical bar), where epsilon 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.