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 Dal'nevost. Mat. Zh., 2015, Volume 15, Number 2, Pages 133–155 (Mi dvmg305)

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

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UDC: 511.42
MSC: Primary 11K60; Secondary 11J61, 11J83
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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

Citation in format AMSBIB
\Bibitem{BudGot15} \by N.~V.~Budarina, F.~G\"otze \paper On regular systems of algebraic $p$-adic numbers of arbitrary degree in small cylinders \jour Dal'nevost. Mat. Zh. \yr 2015 \vol 15 \issue 2 \pages 133--155 \mathnet{http://mi.mathnet.ru/dvmg305} \elib{http://elibrary.ru/item.asp?id=25058090}