Expected density of complex zeros of random hyperbolic polynomials

AbstractThere are many known asymptotic estimates for the expected number of real zeros of polynomial Hn(z) = η1 cosh ζz + η2 cosh 2ζz + ⋯ + ηn cosh nζz, where ηj, 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 Hn(z), where ηj = aj + ibj and aj and bj, 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