Finiteness and periodicity of continued fractions over quadratic number fields

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21340%2F22%3A00355955" target="_blank" >RIV/68407700:21340/22:00355955 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.24033/bsmf.2845" target="_blank" >https://doi.org/10.24033/bsmf.2845</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.24033/bsmf.2845" target="_blank" >10.24033/bsmf.2845</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Finiteness and periodicity of continued fractions over quadratic number fields

Popis výsledku v původním jazyce

We consider continued fractions with partial quotients in the ring of integers of a quadratic number field K and we prove a generalization to such continued fractions of the classical theorem of Lagrange. A particular example of these continued fractions is the β-continued fraction introduced by Bernat. As a corollary of our theorem we show that for any quadratic Perron number β, the β-continued fraction expansion of elements in Q(β) is either finite of eventually periodic. The same holds for β being a square root of an integer. We also show that for certain 4 quadratic Perron numbers β, the β-continued fraction represents finitely all elements of the quadratic field Q(β), thus answering questions of Rosen and Bernat. Based on the validity of a conjecture of Mercat, these are all quadratic Perron numbers with this feature.

Název v anglickém jazyce

Finiteness and periodicity of continued fractions over quadratic number fields

Popis výsledku anglicky

We consider continued fractions with partial quotients in the ring of integers of a quadratic number field K and we prove a generalization to such continued fractions of the classical theorem of Lagrange. A particular example of these continued fractions is the β-continued fraction introduced by Bernat. As a corollary of our theorem we show that for any quadratic Perron number β, the β-continued fraction expansion of elements in Q(β) is either finite of eventually periodic. The same holds for β being a square root of an integer. We also show that for certain 4 quadratic Perron numbers β, the β-continued fraction represents finitely all elements of the quadratic field Q(β), thus answering questions of Rosen and Bernat. Based on the validity of a conjecture of Mercat, these are all quadratic Perron numbers with this feature.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

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OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/EF16_019%2F0000778" target="_blank" >EF16_019/0000778: Centrum pokročilých aplikovaných přírodních věd</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2022

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Bulletin de la Société Mathématique de France

ISSN

0037-9484

e-ISSN

2102-622X

Svazek periodika

150

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

FR - Francouzská republika

Počet stran výsledku

33

Strana od-do

77-109

Kód UT WoS článku

000853257900003

EID výsledku v databázi Scopus

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