A fast randomized geometric algorithm for computing Riemann-Roch spaces
Aude Le Gluher, Pierre-Jean Spaenlehauer · Mathematics of Computation · 2020
We propose a probabilistic variant of Brill-Noether’s algorithm for computing a basis of the Riemann-Roch space L ( D ) L(D) associated to a divisor D D on a projective nodal plane curve C \mathbb {C} over a sufficiently large perfect field k k . Our main result shows that this algorithm requires at most O ( max ( deg ( C ) 2 ω , deg ( D + ) ω ) ) O(\max (\deg (\mathbb {C})^{2\omega }, \deg (D_+)^\omega )) arithmetic operations in k k , where ω \omega is a feasible exponent for matrix multiplication and D + D_+ is the smallest effective divisor such that D + ≥ D 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 ( C ) 4 , deg