 96314 Mezincescu G.A., Bessis D., Fournier J.D., Mantica G., Aaron F.D.
 Distribution of roots of random real polynomials
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Jun 24, 96

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Abstract. The average density of zeros for real generalized polynomials with Gaussian
coefficients is expressed in terms of correlation functions of the polynomial
and its derivative. Due to the real character of the polynomials the average
density of roots has a regular component and a singular one. The regular
component, corresponding to the complex roots, goes to zero in the vicinity of
the real axis like $\hbox{\rm Im}\,z$. The singular one, representing the
real roots, is located on the real axis. We present the low and high disorder
asymptotic behaviors. Then we investigate the large $n$ limit of the average
density of complex roots of monic algebraic polynomials of the form
$P_n(z) = z^n +\sum_{k=0}^{n1} c_kz^k $ with real independent, identically
distributed Gaussian coefficients having zero mean and dispersion
$\delta = \frac 1{\sqrt{n\lambda}}$. We show that the average density tends
to a simple, {\em universal} function of $\xi={2n}{\log z}$ and $\lambda$
in the domain $\xi\coth \frac{\xi}{2}\ll n\sin \arg (z)$ where nearly all
the roots are located for large $n$.
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