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Symbolic Hamburger-Noether expressions of plane curves and applications to AG codes


Authors: A. Campillo and J. I. Farrán
Journal: Math. Comp. 71 (2002), 1759-1780
MSC (2000): Primary 14Q05; Secondary 11T71
Published electronically: December 4, 2001
MathSciNet review: 1933054
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Abstract: In this paper, we consider some practical applications of the symbolic Hamburger-Noether expressions for plane curves, which are introduced as a symbolic version of the so-called Hamburger-Noether expansions. More precisely, we give and develop in symbolic terms algorithms to compute the resolution tree of a plane curve (and the adjunction divisor, in particular), rational parametrizations for the branches of such a curve, special adjoints with assigned conditions (connected with different problems, like the so-called Brill-Noether algorithm), and the Weierstrass semigroup at $P$ together with functions for each value in this semigroup, provided $P$ is a rational branch of a singular plane model for the curve. Some other computational problems related to algebraic curves over perfect fields can be treated symbolically by means of such expressions, but we deal just with those connected with the effective construction and decoding of algebraic geometry codes.


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Additional Information

A. Campillo
Affiliation: Departamento de Algebra, Geometría y Topología, Universidad de Valladolid, Spain
Email: campillo@agt.uva.es

J. I. Farrán
Affiliation: Departamento de Matemática Aplicada a la Ingeniería, Universidad de Valladolid, Spain
Email: ignfar@eis.uva.es

DOI: http://dx.doi.org/10.1090/S0025-5718-01-01390-4
Keywords: Algebraic curves, singular plane models, desingularization, symbolic Hamburger-Noether expressions, adjoints, virtual passing conditions, Weierstrass semigroups, AG codes
Received by editor(s): October 13, 1999
Received by editor(s) in revised form: December 26, 2000
Published electronically: December 4, 2001
Additional Notes: Both authors are partially supported by DIGICYT PB97-0471.
Article copyright: © Copyright 2001 American Mathematical Society