1. A proof of the Brill-Noether method from scratch
- Author
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Berardini, Elena, Couvreur, Alain, Lecerf, Grégoire, Berardini, Elena, Algèbre, preuves, protocoles, algorithmes, courbes, et surfaces pour les codes et leurs applications - - BARRACUDA2021 - ANR-21-CE39-0009 - AAPG2021 - VALID, Department of Mathematics and Computer Science [Eindhoven], Eindhoven University of Technology [Eindhoven] (TU/e), Geometry, arithmetic, algorithms, codes and encryption (GRACE), Laboratoire d'informatique de l'École polytechnique [Palaiseau] (LIX), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), This paper is part of a project that has received funding from the French 'Agence de l'innovation de défense'. Elena Berardini has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 899987. Alain Couvreur is funded by the ANR Grant ANR-21-CE39-0009-BARRACUDA., and ANR-21-CE39-0009,BARRACUDA,Algèbre, preuves, protocoles, algorithmes, courbes, et surfaces pour les codes et leurs applications(2021)
- Subjects
[INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC] ,Computer Science - Symbolic Computation ,FOS: Computer and information sciences ,Algebraic curves ,[INFO.INFO-SC] Computer Science [cs]/Symbolic Computation [cs.SC] ,[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG] ,Newton polygons ,Symbolic Computation (cs.SC) ,Brill–Noether method ,Mathematics - Algebraic Geometry ,Riemann–Roch spaces ,FOS: Mathematics ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,Hensel lemmas ,Algebraic Geometry (math.AG) - Abstract
In 1874 Brill and Noether designed a seminal geometric method for computing bases of Riemann-Roch spaces. From then, their method has led to several algorithms, some of them being implemented in computer algebra systems. The usual proofs often rely on abstract concepts of algebraic geometry and commutative algebra. In this paper we present a short self-contained and elementary proof that mostly needs Newton polygons, Hensel lifting, bivariate resultants, and Chinese remaindering.
- Published
- 2022
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