96-314 Mezincescu G.A., Bessis D., Fournier J.-D., Mantica G., Aaron F.D.

Distribution of roots of random real polynomials (1295K, uuencoded gz compressed tarfile (Revtex source + 8 postscript figures)) 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}^{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 $n$.

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