Algebraic and badly approximable power series over a finite field
type: Item
instance of: scholarly article
title: Algebraic and badly approximable power series over a finite field
zbMATH Open document ID: 1013.11035
author: Jean-Jacques Ruch
author: Alain Lasjaunias
published in: Finite Fields and their Applications
publication date: 2002-07-04T00:00:00Z
review text: Cet article présente une construction ingénieuse et générale de fractions continuées dans \(\mathbb F_q((T^{-1}))\). Les quotients partiels sont de degré 1 de sorte que les séries \(\Theta = \sum_{n\leq k}\theta_n T^n\) obtenues sont mal approchables au sens où elles satisfont (pour la valeur absolue ultramétrique habituelle) \(|\Theta - P/Q|\geq c^{te}/|Q|^2\) pour tout couple de polynômes \((P, Q)\) de \(\mathbb F_q[T]\) tel que \(Q\neq 0\), mais ces séries ne sont pas toutes algébriques de degré 2 sur \(\mathbb F_q(T)\). Les travaux précédents de \textit{L. E. Baum} et \textit{M. M. Sweet} [Ann. Math. (2) 103, 593--610 (1976; Zbl 0312.10024) et ibid. 105, 573--588 (1977; Zbl 0352.10017)], \textit{W. H. Mills} et \textit{D. P. Robbins} [J. Number Theory 23, 388--404 (1986; Zbl 0591.10021)], \textit{A. Lasjaunas} [Acta Arith. 95, 139--165 (2000; Zbl 0972.11061)] donnaient de tels exemples, mais seulement pour des \(q\) premiers, alors qu'ici la construction est valable pour tous les \(q=p^s\).
Mathematics Subject Classification ID: 11J70
Mathematics Subject Classification ID: 11J61
Mathematics Subject Classification ID: 11T55
zbMATH DE Number: 1763663
zbMATH Keywords: fields of power series
zbMATH Keywords: finite fields
zbMATH Keywords: continued fractions
reviewed by: François Gramain
MaRDI profile type: Publication
full work available at URL: ffta.2000.0329
OpenAlex ID: W2023203562
cites work: Continued fractions of algebraic power series in characteristic 2
cites work: Badly approximable power series in characteristic 2
cites work: A survey of Diophantine approximation in fields of power series
cites work: Quartic power series in $
cites work: Continued fractions for certain algebraic power series
cites work: On continued fractions and diophantine approximation in power series fields
DOI: 10.1006/FFTA.2000.0329
description: scientific article; zbMATH DE number 1763663