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Complexity bounds for the rational Newton-Puiseux algorithm over finite fields

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Abstract

We carefully study the number of arithmetic operations required to compute rational Puiseux expansions of a bivariate polynomial F over a finite field. Our approach is based on the rational Newton-Puiseux algorithm introduced by D. Duval. In particular, we prove that coefficients of F may be significantly truncated and that certain complexity upper bounds may be expressed in terms of the output size. These preliminary results lead to a more efficient version of the algorithm with a complexity upper bound that improves previously published results. We also deduce consequences for the complexity of the computation of the genus of an algebraic curve defined over a finite field or an algebraic number field. Our results are practical since they are based on well established subalgorithms, such as fast multiplication of univariate polynomials with coefficients in a finite field.

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Authors and Affiliations

  1. SALSA Project, INRIA, Paris-Rocquencourt Center, LIP6, UMR 7606, Université Pierre et Marie Curie/CNRS, Paris, France

    Adrien Poteaux

  2. XLIM, UMR 6172, Université de Limoges/CNRS, Limoges, France

    Adrien Poteaux & Marc Rybowicz

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  1. Adrien Poteaux
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  2. Marc Rybowicz
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Correspondence to Adrien Poteaux.

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Poteaux, A., Rybowicz, M. Complexity bounds for the rational Newton-Puiseux algorithm over finite fields. AAECC 22, 187–217 (2011). https://doi.org/10.1007/s00200-011-0144-6

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