Algorithmic Number Theory

Volume 877 of the series Lecture Notes in Computer Science pp 221-233


Computing in the jacobian of a plane algebraic curve

Extended abstract
  • Emil J. VolcheckAffiliated withUCLA Dept. of Mathematics and Center for Science and Art

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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.