Mezincescu G.A., Bessis D., Fournier J.-D., Mantica G., Aaron F.D.
Distribution of roots of random real polynomials
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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}^{n-1}  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 $n$.