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

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...

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Bibliographic Details
Published in:Mathematics of computation 2002-10, Vol.71 (240), p.1759-1780
Main Authors: CAMPILLO, A, FARRAN, J. I
Format: Article
Language:English
Subjects:
Adjoints
Algebra
Algebraic geometry
Algorithms
Branches
Coordinate systems
Exact sciences and technology
Mathematical functions
Mathematics
Number theory
Plane curves
Polynomials
Sciences and techniques of general use
Semigroups
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Summary: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.
ISSN:0025-5718
1088-6842
DOI:10.1090/S0025-5718-01-01390-4