On multidimensional Diophantine approximation of algebraic numbers

作     者：Petho, Attila Pohst, Michael E. Bertok, Csanad 

作者机构：Univ Debrecen Dept Comp Sci POB 12 H-4010 Debrecen Hungary Univ Ostrava Fac Sci Dvorakova 7 Ostrava 70103 Czech Republic Tech Univ Berlin Inst Math Str 17 Juni 136 D-10623 Berlin Germany Univ Debrecen Inst Math POB 400 H-4002 Debrecen Hungary 

出 版 物：《JOURNAL OF NUMBER THEORY》 (数论杂志)

年 卷 期：2017年第171卷

页      面：422-448页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：OTKA [NK100339  NK104208] 

主　　题：Diophantine approximation Continued fractions Jacobi-Perron algorithm LLL-algorithm 

摘      要：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, Petho and later Pohst we construct sequences of suitable modules in F and special elements beta 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 (x(1),..., x(n)) of the presentation of beta 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. (C) 2016 Elsevier Inc. All rights reserved.