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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Bibliographic Details
Published in:Advances in applied mathematics Vol. 54; pp. 27 - 65
Main Authors: Berthé, Valérie, Nakada, Hitoshi, Natsui, Rie, Vallée, Brigitte
Format: Journal Article
Language:English
Published: Elsevier Inc 01.03.2014
Elsevier
Subjects:
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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Summary: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.
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ISSN:0196-8858
1090-2074
DOI:10.1016/j.aam.2013.11.001