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ALGEBRAIC NUMBERS WITH BOUNDED DEGREE AND WEIL HEIGHT

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  18 July 2018

ARTŪRAS DUBICKAS Open the ORCID record for ARTŪRAS DUBICKAS [Opens in a new window]
ARTŪRAS DUBICKAS*
Affiliation:
Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania email [email protected]
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Abstract

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For a positive integer $d$ and a nonnegative number $\unicode[STIX]{x1D709}$, let $N(d,\unicode[STIX]{x1D709})$ be the number of $\unicode[STIX]{x1D6FC}\in \overline{\mathbb{Q}}$ of degree at most $d$ and Weil height at most $\unicode[STIX]{x1D709}$. We prove upper and lower bounds on $N(d,\unicode[STIX]{x1D709})$. For each fixed $\unicode[STIX]{x1D709}>0$, these imply the asymptotic formula $\log N(d,\unicode[STIX]{x1D709})\sim \unicode[STIX]{x1D709}d^{2}$ as $d\rightarrow \infty$, which was conjectured in a question at Mathoverflow [https://mathoverflow.net/questions/177206/].

Keywords

Mahler measureWeil heightcounting functionirreducible polynomial

MSC classification

Primary: 11R06: PV-numbers and generalizations; other special algebraic numbers; Mahler measure
Secondary: 11R09: Polynomials (irreducibility, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 2 , October 2018 , pp. 212 - 220
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

This research was funded by the European Social Fund according to the activity Improvement of researchers qualification by implementing world-class R&D projects of Measure no. 09.3.3-LMT-K-712-01-0037.

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