Expected density of complex zeros of random hyperbolic polynomials

There are many known asymptotic estimates for the expected number of real zeros of polynomial H-n (z) = eta(1) cosh zetaz + eta(2) cosh 2zetaz + ... + eta(n) cosh nzetaz, where eta(j), j = 1, 2, 3,..., n is a sequence of independent random variables. This paper provides the asymptotic formula for the expected density of complex zeros of H-n (z), where eta(j) = a(j) + ib(j) and a(j) and b(j), j = 1, 2, 3,,.., n are sequences of independent normally distributed random variables. It is shown that this asymptotic formula for the density of complex zeros remains invariant for other types of polynomials, for instance random trigonometric polynomials, previously studied. (C) 2002 Elsevier Science Ltd. All rights reserved.