On multidimensional Diophantine approximation of algebraic numbers

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F61988987%3A17310%2F17%3AA1701LK7" target="_blank" >RIV/61988987:17310/17:A1701LK7 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.1016/j.jnt.2016.07.002" target="_blank" >http://dx.doi.org/10.1016/j.jnt.2016.07.002</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.jnt.2016.07.002" target="_blank" >10.1016/j.jnt.2016.07.002</a>

Jazyk výsledku

angličtina

Název v původním jazyce

On multidimensional Diophantine approximation of algebraic numbers

Popis výsledku v původním jazyce

In this article we develop algorithms for solving the dual problems of approximating linear forms and of simultaneous approximation in number fields F. Using earlier ideas for computing independent units by Buchmann, Pethő and later Pohst we construct sequences of suitable modules in F and special elements ? contained in them. The most important ingredient in our methods is the application of the LLL-reduction procedure to the bases of those modules. For LLL-reduced bases we derive improved bounds on the sizes of the basis elements. From those bounds it is quite straightforward to show that the sequence of coefficient vectors (x1,...,xn) of the presentation of ? in the module basis becomes periodic. We can show that the approximations which we obtain are close to being optimal. Moreover, it is periodic on bases of real number fields. Thus our algorithm can be considered as a generalization, within the framework of number fields, of the continued fraction algorithm.

Název v anglickém jazyce

On multidimensional Diophantine approximation of algebraic numbers

Popis výsledku anglicky

In this article we develop algorithms for solving the dual problems of approximating linear forms and of simultaneous approximation in number fields F. Using earlier ideas for computing independent units by Buchmann, Pethő and later Pohst we construct sequences of suitable modules in F and special elements ? contained in them. The most important ingredient in our methods is the application of the LLL-reduction procedure to the bases of those modules. For LLL-reduced bases we derive improved bounds on the sizes of the basis elements. From those bounds it is quite straightforward to show that the sequence of coefficient vectors (x1,...,xn) of the presentation of ? in the module basis becomes periodic. We can show that the approximations which we obtain are close to being optimal. Moreover, it is periodic on bases of real number fields. Thus our algorithm can be considered as a generalization, within the framework of number fields, of the continued fraction algorithm.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

V - Vyzkumna aktivita podporovana z jinych verejnych zdroju

Rok uplatnění

2017

Kód důvěrnosti údajů

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

Název periodika

J NUMBER THEORY

ISSN

0022-314X

e-ISSN

—

Svazek periodika

171

Číslo periodika v rámci svazku

Únor

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

27

Strana od-do

422-448

Kód UT WoS článku

000386418700023

EID výsledku v databázi Scopus

2-s2.0-84990866189