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Computing in the jacobian of a plane algebraic curve

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Algorithmic Number Theory (ANTS 1994)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 877))

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Abstract

We describe an algorithm which extends the classical method of adjoints due to Brill and Noether for carrying out the addition operation in the Jacobian variety (represented as the divisor class group) of a plane algebraic curve defined over an algebraic number field K with arbitrary singularities. By working with conjugate sets of Puiseux expansions, we prove this method is rational in the sense that the answers it produces are defined over K. Given a curve with only ordinary multiple points and allowing precomputation of singular places, the running time of addition using this algorithm is dominated by M 7 coefficient operations in a field extension of bounded degree, where M is the larger of the degree and the genus of the curve.

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Authors and Affiliations

  1. UCLA Dept. of Mathematics and Center for Science and Art, USA

    Emil J. Volcheck

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  1. Emil J. Volcheck
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Leonard M. Adleman Ming-Deh Huang

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© 1994 Springer-Verlag Berlin Heidelberg

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Volcheck, E.J. (1994). Computing in the jacobian of a plane algebraic curve. In: Adleman, L.M., Huang, MD. (eds) Algorithmic Number Theory. ANTS 1994. Lecture Notes in Computer Science, vol 877. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58691-1_60

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  • DOI: https://doi.org/10.1007/3-540-58691-1_60

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-58691-3

  • Online ISBN: 978-3-540-49044-9

  • eBook Packages: Springer Book Archive

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