Computing Riemann–Roch spaces via Puiseux expansions

Abstract

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.