Expected density of complex zeros of random hyperbolic polynomials

Abstract

There are many known asymptotic estimates for the expected number of real zeros of polynomial H_n(z) = η₁ cosh ζz + η₂ cosh 2ζz + ⋯ + η_n cosh nζz, where η_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 η_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.