Fine costs for Euclid's algorithm on polynomials and Farey maps

This paper studies digit-cost functions for the Euclid algorithm on polynomials with coefficients in a finite field, in terms of the number of operations performed on the finite field Fq. The usual bit-complexity is defined with respect to the degree of the quotients; we focus here on a notion of ‘f...

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Vydáno v:	Advances in applied mathematics Ročník 54; s. 27 - 65
Hlavní autoři:	Berthé, Valérie, Nakada, Hitoshi, Natsui, Rie, Vallée, Brigitte
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Elsevier Inc 01.03.2014 Elsevier
Témata:	Algorithms Asymptotic properties Bit-complexity Bivariate generating functions Combinatorial analysis Computational Complexity Computer Science Continued fractions Cost function Costs Data Structures and Algorithms Ergodic processes Farey map Finite field Functions (mathematics) Laurent formal power series Mathematical analysis Polynomials Finite field 68W40 Bit-complexity Bivariate generating functions Farey map 11J70 Continued fractions 11K50 11A05 Laurent formal power series 62E20 Cost function 11T55 Combinatorial analysis
ISSN:	0196-8858, 1090-2074
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Abstract	This paper studies digit-cost functions for the Euclid algorithm on polynomials with coefficients in a finite field, in terms of the number of operations performed on the finite field Fq. The usual bit-complexity is defined with respect to the degree of the quotients; we focus here on a notion of ‘fine’ complexity (and on associated costs) which relies on the number of their non-zero coefficients. It also considers and compares the ergodic behavior of the corresponding costs for truncated trajectories under the action of the Gauss map acting on the set of formal power series with coefficients in a finite field. The present paper is thus mainly interested in the study of the probabilistic behavior of the corresponding random variables: average estimates (expectation and variance) are obtained in a purely combinatorial way thanks to classical methods in combinatorial analysis (more precisely, bivariate generating functions); some of our costs are even proved to satisfy an asymptotic Gaussian law. We also relate this study with a Farey algorithm which is a refinement of the continued fraction algorithm for the set of formal power series with coefficients in a finite field: this algorithm discovers ‘step by step’ each non-zero monomial of the quotient, so its number of steps is closely related to the number of non-zero coefficients. In particular, this map is shown to admit a finite invariant measure in contrast with the real case. This version of the Farey map also produces mediant convergents in the continued fraction expansion of formal power series with coefficients in a finite field.
AbstractList	This paper studies digit-cost functions for the Euclid algorithm on polynomials with coefficients in a finite field, in terms of the number of operations performed on the finite field FqFq. The usual bit-complexity is defined with respect to the degree of the quotients; we focus here on a notion of ‘fine’ complexity (and on associated costs) which relies on the number of their non-zero coefficients. It also considers and compares the ergodic behavior of the corresponding costs for truncated trajectories under the action of the Gauss map acting on the set of formal power series with coefficients in a finite field. The present paper is thus mainly interested in the study of the probabilistic behavior of the corresponding random variables: average estimates (expectation and variance) are obtained in a purely combinatorial way thanks to classical methods in combinatorial analysis (more precisely, bivariate generating functions); some of our costs are even proved to satisfy an asymptotic Gaussian law.We also relate this study with a Farey algorithm which is a refinement of the continued fraction algorithm for the set of formal power series with coefficients in a finite field: this algorithm discovers ‘step by step’ each non-zero monomial of the quotient, so its number of steps is closely related to the number of non-zero coefficients. In particular, this map is shown to admit a finite invariant measure in contrast with the real case. This version of the Farey map also produces mediant convergents in the continued fraction expansion of formal power series with coefficients in a finite field. This paper studies digit-cost functions for the Euclid algorithm on polynomials with coefficients in a finite field, in terms of the number of operations performed on the finite field FqFq. The usual bit-complexity is defined with respect to the degree of the quotients; we focus here on a notion of 'fine' complexity (and on associated costs) which relies on the number of their non-zero coefficients. It also considers and compares the ergodic behavior of the corresponding costs for truncated trajectories under the action of the Gauss map acting on the set of formal power series with coefficients in a finite field. The present paper is thus mainly interested in the study of the probabilistic behavior of the corresponding random variables: average estimates (expectation and variance) are obtained in a purely combinatorial way thanks to classical methods in combinatorial analysis (more precisely, bivariate generating functions); some of our costs are even proved to satisfy an asymptotic Gaussian law. This paper studies digit-cost functions for the Euclid algorithm on polynomials with coefficients in a finite field, in terms of the number of operations performed on the finite field Fq. The usual bit-complexity is defined with respect to the degree of the quotients; we focus here on a notion of ‘fine’ complexity (and on associated costs) which relies on the number of their non-zero coefficients. It also considers and compares the ergodic behavior of the corresponding costs for truncated trajectories under the action of the Gauss map acting on the set of formal power series with coefficients in a finite field. The present paper is thus mainly interested in the study of the probabilistic behavior of the corresponding random variables: average estimates (expectation and variance) are obtained in a purely combinatorial way thanks to classical methods in combinatorial analysis (more precisely, bivariate generating functions); some of our costs are even proved to satisfy an asymptotic Gaussian law. We also relate this study with a Farey algorithm which is a refinement of the continued fraction algorithm for the set of formal power series with coefficients in a finite field: this algorithm discovers ‘step by step’ each non-zero monomial of the quotient, so its number of steps is closely related to the number of non-zero coefficients. In particular, this map is shown to admit a finite invariant measure in contrast with the real case. This version of the Farey map also produces mediant convergents in the continued fraction expansion of formal power series with coefficients in a finite field.
Author	Natsui, Rie Berthé, Valérie Nakada, Hitoshi Vallée, Brigitte
Author_xml	– sequence: 1 givenname: Valérie surname: Berthé fullname: Berthé, Valérie email: berthe@liafa.univ-paris-diderot.fr organization: LIAFA, Univ. Paris Diderot Paris 7 & CNRS, Case 7014, 75205 Paris Cedex 13, France – sequence: 2 givenname: Hitoshi surname: Nakada fullname: Nakada, Hitoshi email: nakada@math.keio.ac.jp organization: Department of Mathematics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan – sequence: 3 givenname: Rie surname: Natsui fullname: Natsui, Rie email: natsui@fc.jwu.ac.jp organization: Department of Mathematics, Japan Women's University, 2-8-1 Mejirodai, Bunkyou-ku, Tokyo 112-8681, Japan – sequence: 4 givenname: Brigitte surname: Vallée fullname: Vallée, Brigitte email: Brigitte.Vallee@unicaen.fr organization: GREYC, Université de Caen, Bd. Maréchal Juin, 14032 Caen Cedex, France
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Keywords	Finite field 68W40 Bit-complexity Bivariate generating functions Farey map 11J70 Continued fractions 11K50 11A05 Laurent formal power series 62E20 Cost function 11T55 Combinatorial analysis
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SubjectTerms	Algorithms Asymptotic properties Bit-complexity Bivariate generating functions Combinatorial analysis Computational Complexity Computer Science Continued fractions Cost function Costs Data Structures and Algorithms Ergodic processes Farey map Finite field Functions (mathematics) Laurent formal power series Mathematical analysis Polynomials
Title	Fine costs for Euclid's algorithm on polynomials and Farey maps
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