### 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 language | English |
---|---|

Pages (from-to) | 27-65 |

Number of pages | 39 |

Journal | Advances in Applied Mathematics |

Volume | 54 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2014 Mar |

### Fingerprint

### 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

*Advances in Applied Mathematics*,

*54*(1), 27-65. https://doi.org/10.1016/j.aam.2013.11.001

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

Research output: Contribution to journal › Article

*Advances in Applied Mathematics*, vol. 54, no. 1, pp. 27-65. https://doi.org/10.1016/j.aam.2013.11.001

}

TY - JOUR

T1 - Fine costs for Euclid's algorithm on polynomials and Farey maps

AU - Berthé, Valérie

AU - Nakada, Hitoshi

AU - Natsui, Rie

AU - Vallée, Brigitte

PY - 2014/3

Y1 - 2014/3

N2 - 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.

AB - 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.

KW - Bit-complexity

KW - Bivariate generating functions

KW - Combinatorial analysis

KW - Continued fractions

KW - Cost function

KW - Farey map

KW - Finite field

KW - Laurent formal power series

UR - http://www.scopus.com/inward/record.url?scp=84894072918&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84894072918&partnerID=8YFLogxK

U2 - 10.1016/j.aam.2013.11.001

DO - 10.1016/j.aam.2013.11.001

M3 - Article

VL - 54

SP - 27

EP - 65

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

SN - 0196-8858

IS - 1

ER -