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Distribution of complex algebraic numbers


Authors: Friedrich Götze, Dzianis Kaliada and Dmitry Zaporozhets
Journal: Proc. Amer. Math. Soc. 145 (2017), 61-71
MSC (2010): Primary 11N45; Secondary 11C08
DOI: https://doi.org/10.1090/proc/13208
Published electronically: July 7, 2016
MathSciNet review: 3565360
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Abstract: For a region $ \Omega \subset \mathbb{C}$ denote by $ \Psi (Q;\Omega )$ the number of complex algebraic numbers in $ \Omega $ of degree $ \leq n$ and naive height $ \leq Q$. We show that

$\displaystyle \Psi (Q;\Omega )=\frac {Q^{n+1}}{2\zeta (n+1)}\int _\Omega \psi (z)\,\nu (dz)+O\left (Q^n \right ),\quad Q\to \infty , $

where $ \nu $ is the Lebesgue measure on the complex plane and the function $ \psi $ will be given explicitly.

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Additional Information

Friedrich Götze
Affiliation: Department of Mathematics, Bielefeld University, P.O. Box 10 01 31, 33501 Bielefeld, Germany
Email: goetze@math.uni-bielefeld.de

Dzianis Kaliada
Affiliation: Institute of Mathematics, National Academy of Sciences of Belarus, Surganova Str 11, 220072 Minsk, Belarus
Email: koledad@rambler.ru

Dmitry Zaporozhets
Affiliation: St. Petersburg Department of Steklov Institute of Mathematics, Fontanka 27, 191011 St. Petersburg, Russia
Email: zap1979@gmail.com

DOI: https://doi.org/10.1090/proc/13208
Keywords: Algebraic numbers, distribution of algebraic numbers, integral polynomials
Received by editor(s): October 16, 2015
Received by editor(s) in revised form: March 17, 2016
Published electronically: July 7, 2016
Additional Notes: This research was supported by CRC 701, Bielefeld University (Germany).
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2016 American Mathematical Society