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Chebyshevskii Sb., 2015, Volume 16, Issue 4, Pages 90–99 (Mi cheb437)  

This article is cited in 2 scientific papers (total in 2 papers)

Correlations between real conjugate algebraic numbers

F. Götzea, D. Kaliadab, D. N. Zaporozhetsc

a Bielefeld University, Department of Mathematics
b Institute of Mathematics of the National Academy of Sciences of Belarus
c St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: For $B\subset\mathbb{R}^k$ denote by $\Phi_k(Q,B)$ the number of ordered $k$-tuples in $B$ of real conjugate algebraic numbers of degree $\leq n$ and naive height $\leq Q$. We show that
$$ \Phi_k(Q;B) = \frac{(2Q)^{n+1}}{2\zeta(n+1)} \int\limits_{B} \chi_k(\mathbf{x}) \prod_{1\le i < j \le k} |x_i - x_j| d\mathbf{x} + O(Q^n),\quad Q\to \infty, $$
where the function $\chi_k$ is continuous in $\mathbb{R}^k$ and will be given explicitly. If $n=2$, then an additional factor $\log Q$ appears in the reminder term. This relation may be regarded as a "repulsion" of real algebraic conjugates from each other.
The function
$$ \rho_k(\mathbf{x}):= \chi_k(\mathbf{x}) \prod_{1\le i < j \le k} |x_i - x_j| $$
coincides with a $k$-point correlation function of real zeros of a random polynomial of degree $n$ with independent coefficients uniformly distributed on $[-1,1]$.
Bibliography: 18 titles.

Keywords: conjugate algebraic numbers, correlations between algebraic numbers, distribution of algebraic numbers, integral polynomial, random polynomial.

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-00256
Russian Academy of Sciences - Federal Agency for Scientific Organizations
Universität Bielefeld CRC 701
Supported by CRC 701, Bielefeld University (Germany). The work of the third author is supported by the grant RFBR 13-01-00256 and by the Program of Fundamental Researches of Russian Academy of Sciences Modern Problems of Fundamental Mathematics.


Full text: PDF file (270 kB)
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UDC: 511.35, 511.75, 511.48, 519.218.5
Received: 09.11.2015
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Citation: F. Götze, D. Kaliada, D. N. Zaporozhets, “Correlations between real conjugate algebraic numbers”, Chebyshevskii Sb., 16:4 (2015), 90–99

Citation in format AMSBIB
\Bibitem{GotKolZap15}
\by F.~G\"otze, D.~Kaliada, D.~N.~Zaporozhets
\paper Correlations between real conjugate algebraic numbers
\jour Chebyshevskii Sb.
\yr 2015
\vol 16
\issue 4
\pages 90--99
\mathnet{http://mi.mathnet.ru/cheb437}
\elib{http://elibrary.ru/item.asp?id=25006095}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. D. Koleda, “On a way of ordering real algebraic numbers uniformly”, Proc. Steklov Inst. Math., 296, suppl. 2 (2017), 61–69  mathnet  crossref  crossref  isi  elib
    2. D. V. Koleda, “O raspredelenii veschestvennykh algebraicheskikh chisel ravnoi vysoty”, Dalnevost. matem. zhurn., 18:1 (2018), 56–70  mathnet
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