Fast computation of Riemann-Roch spaces for singular curves

TL;DR

This paper introduces a new, fast deterministic algorithm for computing bases of Riemann-Roch spaces on singular algebraic curves, with broad applicability in algebraic geometry and related fields.

Contribution

The paper presents the fastest known algorithm for computing Riemann-Roch spaces on singular curves, without restrictions on the divisor's support, over any perfect field.

Findings

Algorithm is deterministic and efficient

Works for curves with arbitrary singularities

Applicable over any perfect field

Abstract

Let C be a projective curve defined over a field k and let D be a divisor of C. The Riemann-Roch space L(D) is the set of rational functions on C for which certain zeros are imposed and certain poles are allowed, with some multiplicities determined by D. Riemann-Roch spaces play a fundamental role in algebraic geometry due to the central place of the Riemann-Roch theorem. They have also important applications, such as coding theory or arithmetic of Jacobians of curves. In this article, we present what we believe is the fastest algorithm to date that computes a basis of a Riemann-Roch space for a curve with arbitrary singularities. Our algorithm is deterministic, works over any perfect field k, and works with no assumptions on the support of D.