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Theory of Probability & Its Applications
We show that the expected number of real zeros of the nth degree polynomial with real independent identically distributed coefficients with common characteristic function φ(z) = e-A(ln|1/z|)^-a for 0 < |z| < 1 and φ(0) = 1, φ(z) ≡ 0 for 1 ≦ |z| < ∞, with 1 < a and A ≧ a(a-1), is (a-1)/(a-1/2) log(n) asymptotically as n → ∞.
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random polynomials, number of real zeros, real roots, Kac-Rice formula, characteristic function
Shepp, L. A., & Farahmand, K. (2011). Expected Number of Real Zeros of a Random Polynomial With Independent Identically Distributed Symmetric Long-Tailed Coefficients. Theory of Probability & Its Applications, 55 (1), 173-181. http://dx.doi.org/10.1137/S0040585X97984735
Date Posted: 27 November 2017
This document has been peer reviewed.