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Zap. Nauchn. Sem. POMI, 2015, Volume 441, Pages 144–153 (Mi znsl6230)  

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

Discriminant and root separation of integral polynomials

F. Götzea, D. Zaporozhetsb

a Faculty of Mathematics, Bielefeld University, P.O.Box 10 01 31, 33501 Bielefeld, Germany
b St. Petersburg Department of Steklov Institute of Mathematics, Fontanka 27, 191011 St. Petersburg, Russia

Abstract: Consider a random polynomial
$$ G_Q(x)=\xi_{Q,n}x^n+\xi_{Q,n-1}x^{n-1}+…+\xi_{Q,0} $$
with independent coefficients uniformly distributed on $2Q+1$ integer points $\{-Q,…,Q\}$. Denote by $D(G_Q)$ the discriminant of $G_Q$. We show that there exists a constant $C_n$, depending on $n$ only such that for all $Q\ge2$ the distribution of $D(G_Q)$ can be approximated as follows
$$ \sup_{-\infty\leq a\leq b\leq\infty}|\mathbf P(a\leq\frac{D(G_Q)}{Q^{2n-2}}\leq b)-\int_a^b\varphi_n(x) dx|\leq\frac{C_n}{\log Q}, $$
where $\varphi_n$ denotes the probability density function of the discriminant of a random polynomial of degree $n$ with independent coefficients which are uniformly distributed on $[-1,1]$. Let $\Delta(G_Q)$ denote the minimal distance between the complex roots of $G_Q$. As an application we show that for any $\varepsilon>0$ there exists a constant $\delta_n>0$ such that $\Delta(G_Q)$ is stochastically bounded from below/above for all sufficiently large $Q$ in the following sense
$$ \mathbf P(\delta_n<\Delta(G_Q)<\frac1{\delta_n})>1-\varepsilon. $$


Key words and phrases: distribution of discriminants, integral polynomials, polynomial discriminant, polynomial root separation.

Funding Agency Grant Number
Universität Bielefeld SFB 701
Russian Foundation for Basic Research 13-01-00256
Russian Academy of Sciences - Federal Agency for Scientific Organizations
The work was done with the financial support of the Bielefeld University (Germany) in terms of project SFB 701. The second author is supported by the RFBR grant 13-01-00256 and by the program of RAS “Modern problems of theoretical mathematics”.


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English version:
Journal of Mathematical Sciences (New York), 2016, 219:5, 700–706

Bibliographic databases:

UDC: 519.2
Received: 10.10.2015
Language:

Citation: F. Götze, D. Zaporozhets, “Discriminant and root separation of integral polynomials”, Probability and statistics. Part 22, Zap. Nauchn. Sem. POMI, 441, POMI, St. Petersburg, 2015, 144–153; J. Math. Sci. (N. Y.), 219:5 (2016), 700–706

Citation in format AMSBIB
\Bibitem{GotZap15}
\by F.~G\"otze, D.~Zaporozhets
\paper Discriminant and root separation of integral polynomials
\inbook Probability and statistics. Part~22
\serial Zap. Nauchn. Sem. POMI
\yr 2015
\vol 441
\pages 144--153
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6230}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3504502}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2016
\vol 219
\issue 5
\pages 700--706
\crossref{https://doi.org/10.1007/s10958-016-3139-9}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Beresnevich V., Bernik V., Goetze F., “Integral polynomials with small discriminants and resultants”, Adv. Math., 298 (2016), 393–412  crossref  mathscinet  zmath  isi  elib  scopus
    2. Koleda D.V., “On the Distribution of Polynomial Discriminants: Totally Real Case”, Lith. Math. J., 59:1, SI (2019), 67–80  crossref  mathscinet  isi  scopus
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