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5 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
$$
G_Q(x)=\xi_{Q,n}x^n+\xi_{Q,n-1}x^{n-1}+\dots+\xi_{Q,0}
$$
with independent coefficients uniformly distributed on
$2Q+1$ integer points
$\{-Q,\dots,Q\}$. 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
$$
\sup_{-\infty\leq a\leq b\leq\infty}\left|\mathbf P\left(a\leq\frac{D(G_Q)}{Q^{2n-2}}\leq b\right)-\int_a^b\varphi_n(x)\,dx\right|\leq\frac{C_n}{\log Q},
$$
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
$$
\mathbf P\left(\delta_n<\Delta(G_Q)<\frac1{\delta_n}\right)>1-\varepsilon.
$$
Key words and phrases:
distribution of discriminants, integral polynomials, polynomial discriminant, polynomial root separation.
UDC:
519.2 Received: 10.10.2015
Language: English