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

Original Paper

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.

Keywords

Puiseux series Complexity Algebraic functions Genus 

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.SALSA Project, INRIA, Paris-Rocquencourt Center, LIP6, UMR 7606, Université Pierre et Marie Curie/CNRSParisFrance
  2. 2.XLIM, UMR 6172, Université de Limoges/CNRSLimogesFrance

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