Orthogonal sequences of polynomials over arbitrary fields
type: Item
instance of: scholarly article
title: Orthogonal sequences of polynomials over arbitrary fields
zbMATH Open document ID: 0916.12001
author: Simon R. Blackburn
published in: Journal of Number Theory
publication date: 1999-07-07T00:00:00Z
review text: Let \(f\) and \(g\) be polynomials over some field \(F\) with deg \(g<\text{deg } f\). The quotient \(g/f\) is called badly approximable if all the partial quotients of the continued fraction of \(g/f\) have degree 1. For a given \(f\), the order of the set \(\{g\in F[x] \mid \text{deg } g<\text{deg } f\) and \(g/f\) is badly approximable\} is called the orthogonal multiplicity of \(f\). In this paper the author gives a characterization of those polynomials with orthogonal multiplicity one: when \(| F| \neq 2\) then \(f\) has orthogonal multiplicity one if and only if \(f=1\), and when \(| F| =2\) then \(f\) has orthogonal multiplicity one if and only if \(f=X^{e_{0}}(X+1)^{e_{1}}\) where \(\binom{e_{0}+e_{1}}{e_{1}}\equiv 1\bmod 2\). This result generalises two results of \textit{H. Niederreiter} [Monatsh. Math. 103, 269-288 (1987; Zbl 0624.12011)]. The author also proves that if \(| F| \geq d(d+1)/2\) where \(d=\text{deg } f\), then \(f\) has positive orthogonal multiplicity.
reviewed by: Dingyi Pei
Mathematics Subject Classification ID: 12E05
Mathematics Subject Classification ID: 11T06
Mathematics Subject Classification ID: 94A55
Mathematics Subject Classification ID: 11K45
zbMATH DE Number: 1146322
zbMATH Keywords: continued fraction
zbMATH Keywords: convergent
zbMATH Keywords: badly approximable
zbMATH Keywords: orthogonal sequence of polynomial
MaRDI profile type: Publication
full work available at URL: jnth.1997.2201
OpenAlex ID: W2079347660
cites work: Badly approximable power series in characteristic 2
cites work: Q3216750
cites work: Shift-register synthesis and BCH decoding
cites work: Continued fraction expansions of rational expressions with irreducible denominators in characteristic 2
cites work: Q4724719
cites work: Rational functions with partial quotients of small degree in their continued fraction expansion
cites work: Q3797089
cites work: Q4011778
cites work: Q3264312
cites work: Random Numbers Generated by Linear Recurrence Modulo Two
DOI: 10.1006/JNTH.1997.2201
description: scientific article; zbMATH DE number 1146322