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 \(\mathcal C\) over a sufficiently large perfect field \(k\). Our main result shows that this algorithm requir...

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Bibliographic Details
Published in:arXiv.org
Main Authors: Aude Le Gluher, Pierre-Jean Spaenlehauer
Format: Paper
Language:English
Published: Ithaca Cornell University Library, arXiv.org 19.10.2020
Subjects:
Algorithms
Complexity
Computation
Computer algebra
Curves
Geometric algorithms
Magma
Set theory
Statistical analysis
Upper bounds
ISSN:2331-8422
Online Access:Get full text
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Summary: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 \(\mathcal C\) over a sufficiently large perfect field \(k\). Our main result shows that this algorithm requires at most \(O(\max(\mathrm{deg}(\mathcal C)^{2\omega}, \mathrm{deg}(D_+)^\omega))\) 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 \(O(\max(\mathrm{deg}(\mathcal C)^4, \mathrm{deg}(D_+)^2)/\lvert \mathcal E\rvert)\), 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.
Bibliography:SourceType-Working Papers-1
ObjectType-Working Paper/Pre-Print-1
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ISSN:2331-8422
DOI:10.48550/arxiv.1811.08237