Computing Riemann–Roch spaces via Puiseux expansions

Computing large Riemann–Roch spaces for plane projective curves still constitutes a major algorithmic and practical challenge. Seminal applications concern the construction of arbitrarily large algebraic geometry error correcting codes over alphabets with bounded cardinality. Nowadays such codes are...

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Bibliographic Details
Published inJournal of Complexity Vol. 73; p. 101666
Main Authors Abelard, Simon, Berardini, Elena, Couvreur, Alain, Lecerf, Grégoire
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.12.2022
Elsevier
Subjects
Algebraic curves
Algebraic Geometry
Algorithms
Complexity
Computer Science
Mathematics
Puiseux expansions
Riemann–Roch spaces
Symbolic Computation
Riemann–Roch spaces
Algebraic curves
Algorithms
Puiseux expansions
Complexity
Riemann-Roch spaces
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Summary:Computing large Riemann–Roch spaces for plane projective curves still constitutes a major algorithmic and practical challenge. Seminal applications concern the construction of arbitrarily large algebraic geometry error correcting codes over alphabets with bounded cardinality. Nowadays such codes are increasingly involved in new areas of computer science such as cryptographic protocols and “interactive oracle proofs”. In this paper, we design a new probabilistic algorithm of Las Vegas type for computing Riemann–Roch spaces of smooth divisors, in characteristic zero, and with expected complexity exponent 2.373 (a feasible exponent for linear algebra) in terms of the input size.
ISSN:0885-064X
1090-2708
DOI:10.1016/j.jco.2022.101666