Expected density of complex zeros of random hyperbolic polynomials. (English) Zbl 1010.60046
Summary: There are many known asymptotic estimates for the expected number of real zeros of polynomial \(H_n(z) = \eta_1 \cosh \zeta z + \eta_2 \cosh 2\zeta z + \cdots+ \eta_n \cosh n\zeta z\), where \(\eta_j\), \(j = 1, 2, 3, \dots,n,\) 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 \(b_j\), \(j = 1, 2, 3,\dots,n\), 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.
MSC:
| 60G99 | Stochastic processes |
Keywords:
number of complex zeros; real roots; complex roots; random hyperbolic polynomials; random trigonometric polynomials; random algebraic polynomials; Jacobian of transformation; Adler’s theorem; coordinate transform; density of zerosReferences:
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