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

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
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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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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
2. Koleda D.V., “On the Distribution of Polynomial Discriminants: Totally Real Case”, Lith. Math. J., 59:1, SI (2019), 67–80
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