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On Multiplicative Independent Bases for Canonical Number Systems in Cyclotomic Number Fields

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Number Theory – Diophantine Problems, Uniform Distribution and Applications

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Abstract

In the present paper we are interested in number systems in the ring of integers of cyclotomic number fields in order to obtain a result equivalent to Cobham’s theorem. For this reason we first search for potential bases. This is done in a very general way in terms of canonical number systems. In a second step we analyse pairs of bases in view of their multiplicative independence. In the last part we state an appropriate variant of Cobham’s theorem.

Dedicated to Professor Robert F. Tichy on the occasion of his 60th birthday

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Acknowledgements

We want to explicitly give thanks to the anonymous referees. Due to their hints and suggestions we were able to improve a lot the quality and readability of this article. The second author’s research was supported by the Austrian Research Foundation (FWF), Project P23990.

Author information

Authors and Affiliations

  1. Institut Elie Cartan de Lorraine, Université de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France

    Manfred G. Madritsch

  2. Institut Elie Cartan de Lorraine, CNRS, UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France

    Manfred G. Madritsch

  3. Institut für Mathematik, Universtiät für Bodenkultur (BOKU), Gregor-Mendel-Straße 33, 1180, Wien, Austria

    Paul Surer

  4. Institute of Mathematics, University of Salzburg, Hellbrunnerstrasse 34/I, 5020, Salzburg, Austria

    Volker Ziegler

Authors
  1. Manfred G. Madritsch
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  2. Paul Surer
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  3. Volker Ziegler
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Correspondence to Manfred G. Madritsch .

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Editors and Affiliations

  1. Institute of Analysis and Number Theory, Graz University of Technology, Graz, Austria

    Christian Elsholtz

  2. Institute of Analysis and Number Theory, Graz University of Technology, Graz, Austria

    Peter Grabner

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Madritsch, M.G., Surer, P., Ziegler, V. (2017). On Multiplicative Independent Bases for Canonical Number Systems in Cyclotomic Number Fields. In: Elsholtz, C., Grabner, P. (eds) Number Theory – Diophantine Problems, Uniform Distribution and Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-55357-3_16

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