DISTRIBUTION OF ROOTS OF RANDOM REAL GENERALIZED POLYNOMIALS

The average density of zeros for monic generalized polynomials, P-n(z) =phi(z)+Sigma(k=1)(n)c(k)f(k)(z), With real holomorphic phi, f(k) and real Gaussian coefficients is expressed in terms of correlation functi ons of the values of the polynomial and its derivative. We obtain comp act expressions For both the regular component (generated by the compl ex roots) and the singular one (real roots) of the average density of roots. The density of the regular component goes to zero in the vicini ty of the real axis like \Im z\. We present the low- and high-disorder asymptotic behaviors. Then we particularize to the large-n limit of t he average density of complex roots of monic algebraic polynomials of the form P-n(z)=z(n)+Sigma(k=1)c(k)z(n-k) with real independent, ident ically distributed Gaussian coefficients having zero mean and dispersi on delta=1/root n lambda. The average density tends to a simple, unive rsal function of xi=2n log \z\ and lambda in the domain xi coth(xi/2) much less than n \sin arg(z)\, where nearly all the roots are located for large n.