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Multiplicative dependence of rational values modulo approximate finitely generated groups

Part of: Multiplicative number theory Algebraic number theory: global fields

Published online by Cambridge University Press:  19 September 2024

ATTILA BÉRCZES ,
YANN BUGEAUD ,
KÁLMÁN GYŐRY ,
JORGE MELLO ,
ALINA OSTAFE  and
MIN SHA
ATTILA BÉRCZES
Affiliation:
Institute of Mathematics, University of Debrecen, H-4010 Debrecen, P.O. BOX 12, Hungary e-mail: berczesa@science.unideb.hu
YANN BUGEAUD
Affiliation:
Institut de Recherche Mathématique Avancée, U.M.R. 7501, Université de Strasbourg et C.N.R.S., 7, rue René Descartes, 67084 Strasbourg, France; Institut Universitaire de France e-mail: yann.bugeaud@math.unistra.fr
KÁLMÁN GYŐRY
Affiliation:
Institute of Mathematics, University of Debrecen, H-4010 Debrecen, P.O. Box 12, Hungary e-mail: gyory@science.unideb.hu
JORGE MELLO
Affiliation:
Department of Mathematics and Statistics, Oakland University, 48307 Michigan, United States e-mail: jorgedemellojr@oakland.edu
ALINA OSTAFE
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia e-mail: alina.ostafe@unsw.edu.au
MIN SHA
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China e-mail: min.sha@m.scnu.edu.cn
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Abstract

In this paper, we establish some finiteness results about the multiplicative dependence of rational values modulo sets which are ‘close’ (with respect to the Weil height) to division groups of finitely generated multiplicative groups of a number field K. For example, we show that under some conditions on rational functions $f_1, \ldots, f_n\in K(X)$, there are only finitely many elements $\alpha \in K$ such that $f_1(\alpha),\ldots,f_n(\alpha)$ are multiplicatively dependent modulo such sets.

MSC classification

Primary: 11N25: Distribution of integers with specified multiplicative constraints
Secondary: 11R04: Algebraic numbers; rings of algebraic integers

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 177 , Issue 1 , July 2024 , pp. 149 - 165
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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