The distribution of powers of integers in algebraic number fields

Werner Georg Nowak[1]; Johannes Schoißengeier[2]

  • [1] Institute of Mathematics Department of Integrative Biology BOKU - University of Natural Resources and Applied Life Sciences Peter Jordan-Straße 82 A-1190 Wien, Austria
  • [2] Institut für Mathematik der Universität Wien Nordbergstraße 15 A-1090 Wien, Austria

Journal de Théorie des Nombres de Bordeaux (2004)

  • Volume: 16, Issue: 1, page 197-214
  • ISSN: 1246-7405

Abstract

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For an arbitrary (not totally real) number field K of degree ≥ 3 , we ask how many perfect powers γ p of algebraic integers γ in K exist, such that μ ( τ ( γ p ) ) ≤ X for each embedding τ of K into the complex field. ( X a large real parameter, p ≥ 2 a fixed integer, and μ ( z ) = max ( | Re ( z ) | , | Im ( z ) | ) for any complex z .) This quantity is evaluated asymptotically in the form c p , K X n / p + R p , K ( X ) , with sharp estimates for the remainder R p , K ( X ) . The argument uses techniques from lattice point theory along with W. Schmidt’s multivariate extension of K.F. Roth’s result on the approximation of algebraic numbers by rationals.

How to cite

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Nowak, Werner Georg, and Schoißengeier, Johannes. "The distribution of powers of integers in algebraic number fields." Journal de Théorie des Nombres de Bordeaux 16.1 (2004): 197-214. <http://eudml.org/doc/249273>.

@article{Nowak2004,
abstract = {For an arbitrary (not totally real) number field $K$ of degree $\ge 3$, we ask how many perfect powers $\gamma ^p$ of algebraic integers $\gamma $ in $K$ exist, such that $\mu (\tau (\gamma ^p))\le X$ for each embedding $\tau $ of $K$ into the complex field. ($X$ a large real parameter, $p\ge 2$ a fixed integer, and $\mu (z)=\max (|\{\rm Re\}(z)|,|\{\rm Im\}(z)|)$ for any complex $z$.) This quantity is evaluated asymptotically in the form $c_\{p,K\} X^\{n/p\} + R_\{p,K\}(X)$, with sharp estimates for the remainder $R_\{p,K\}(X)$. The argument uses techniques from lattice point theory along with W. Schmidt’s multivariate extension of K.F. Roth’s result on the approximation of algebraic numbers by rationals.},
affiliation = {Institute of Mathematics Department of Integrative Biology BOKU - University of Natural Resources and Applied Life Sciences Peter Jordan-Straße 82 A-1190 Wien, Austria; Institut für Mathematik der Universität Wien Nordbergstraße 15 A-1090 Wien, Austria},
author = {Nowak, Werner Georg, Schoißengeier, Johannes},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {asymptotic results on algebraic numbers; lattice points},
language = {eng},
number = {1},
pages = {197-214},
publisher = {Université Bordeaux 1},
title = {The distribution of powers of integers in algebraic number fields},
url = {http://eudml.org/doc/249273},
volume = {16},
year = {2004},
}

TY - JOUR
AU - Nowak, Werner Georg
AU - Schoißengeier, Johannes
TI - The distribution of powers of integers in algebraic number fields
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2004
PB - Université Bordeaux 1
VL - 16
IS - 1
SP - 197
EP - 214
AB - For an arbitrary (not totally real) number field $K$ of degree $\ge 3$, we ask how many perfect powers $\gamma ^p$ of algebraic integers $\gamma $ in $K$ exist, such that $\mu (\tau (\gamma ^p))\le X$ for each embedding $\tau $ of $K$ into the complex field. ($X$ a large real parameter, $p\ge 2$ a fixed integer, and $\mu (z)=\max (|{\rm Re}(z)|,|{\rm Im}(z)|)$ for any complex $z$.) This quantity is evaluated asymptotically in the form $c_{p,K} X^{n/p} + R_{p,K}(X)$, with sharp estimates for the remainder $R_{p,K}(X)$. The argument uses techniques from lattice point theory along with W. Schmidt’s multivariate extension of K.F. Roth’s result on the approximation of algebraic numbers by rationals.
LA - eng
KW - asymptotic results on algebraic numbers; lattice points
UR - http://eudml.org/doc/249273
ER -

References

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