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Dal'nevostochnyi Matematicheskii Zhurnal, 2015, Volume 15, Number 2, Pages 133–155
(Mi dvmg305)
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This article is cited in 2 scientific papers (total in 2 papers)
On regular systems of algebraic $p$-adic numbers of arbitrary degree in small cylinders
N. V. Budarinaa, F. Götzeb a Institute for Applied Mathematics, Khabarovsk Division, Far-Eastern Branch of the Russian Academy of Sciences, 680000 Khabarovsk, Russia, Dzerzhinsky st., 54
b Faculty of Mathematics, University of Bielefeld, P. O. Box 10 01 31, 33501 Bielefeld, Germany
Abstract:
In this paper we prove that
for any sufficiently large $Q\in{\mathbb N}$ there exist cylinders $K\subset{\mathbb Q}_p$ with Haar measure $\mu(K)\le \frac{1}{2}Q^{-1}$ which do not contain
algebraic $p$-adic numbers $\alpha$ of degree $\deg\alpha=n$ and height $H(\alpha)\le Q$.
The main result establishes in any cylinder $K$, $\mu(K)>c_1Q^{-1}$, $c_1>c_0(n)$,
the existence of
at least $c_{3}Q^{n+1}\mu(K)$ algebraic $p$-adic numbers $\alpha\in K$ of degree $n$ and $H(\alpha)\le Q$.
Key words:
integer polynomials, algebraic $p$-adic numbers, regular system, Haar measure.
Received: 22.09.2015
Citation:
N. V. Budarina, F. Götze, “On regular systems of algebraic $p$-adic numbers of arbitrary degree in small cylinders”, Dal'nevost. Mat. Zh., 15:2 (2015), 133–155
Linking options:
https://www.mathnet.ru/eng/dvmg305 https://www.mathnet.ru/eng/dvmg/v15/i2/p133
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