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Fast computation of Riemann-Roch spaces for singular curves

Dounia Darkaoui, Martin Weimann

TL;DR

This article presents what they believe is the fastest algorithm to date that computes a basis of a Riemann-Roch space for a curve with arbitrary singularities and works over any perfect field k.

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.

Fast computation of Riemann-Roch spaces for singular curves

TL;DR

This article presents what they believe is the fastest algorithm to date that computes a basis of a Riemann-Roch space for a curve with arbitrary singularities and works over any perfect field k.

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.
Paper Structure (41 sections, 61 theorems, 103 equations, 2 algorithms)

This paper contains 41 sections, 61 theorems, 103 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Main result.
  3. Main lines of the proof.
  4. State of the art.
  5. Organization.
  6. Acknowledgments.
  7. Divisors and Riemann--Roch spaces
  8. Representation of places of a function field
  9. Places of a rational function field.
  10. Places vs prime ideals of Dedekind rings.
  11. OM-representations of places
  12. OM-representation of divisors
  13. Other usual representations of places
  14. Representation of places by Puiseux expansions.
  15. Two-elements representation of a place.
  16. ...and 26 more sections

Key Result

Theorem 1.1

There exists a deterministic algorithm that, given a projective curve $\mathcal{C}\subset \mathbb{P}^2_k$ of degree $n$ defined by an irreducible homogeneous polynomial $F\in k[X_0,X_1,X_2]$ monic and separable in $X_2$, and given a divisor $D\in \mathop{\mathrm{Div}}\nolimits(\mathcal{C})$ of non n operations in $k$, where $\delta(\mathcal{C})=\frac{(n-1)(n-2)}{2}-g$, with $g$ the geometric genus

Theorems & Definitions (147)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 3.1
  • ...and 137 more