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

Valérie Berthé, Hitoshi Nakada, Rie Natsui, Brigitte Vallée

Research output: Contribution to journalArticle

1 Citation (Scopus)

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.

Original languageEnglish
Pages (from-to)27-65
Number of pages39
JournalAdvances in Applied Mathematics
Volume54
Issue number1
DOIs
Publication statusPublished - 2014 Mar

Fingerprint

Euclidean algorithm
Polynomials
Galois field
Polynomial
Formal Power Series
Costs
Coefficient
Mediant
Quotient
Random variables
Cost functions
Combinatorial Analysis
Gauss Map
Continued Fraction Expansion
Trajectories
Monomial
Continued fraction
Digit
Invariant Measure
Generating Function

Keywords

  • Bit-complexity
  • Bivariate generating functions
  • Combinatorial analysis
  • Continued fractions
  • Cost function
  • Farey map
  • Finite field
  • Laurent formal power series

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

Fine costs for Euclid's algorithm on polynomials and Farey maps. / Berthé, Valérie; Nakada, Hitoshi; Natsui, Rie; Vallée, Brigitte.

In: Advances in Applied Mathematics, Vol. 54, No. 1, 03.2014, p. 27-65.

Research output: Contribution to journalArticle

Berthé, Valérie ; Nakada, Hitoshi ; Natsui, Rie ; Vallée, Brigitte. / Fine costs for Euclid's algorithm on polynomials and Farey maps. In: Advances in Applied Mathematics. 2014 ; Vol. 54, No. 1. pp. 27-65.
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