Expected Number of Real Zeros of a Random Polynomial with Independent Identically Distributed Symmetric Long-Tailed Coefficients

2011/01/01 by L. A. Shepp, L. Shepp, K. Farahmand · 1 citation
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Stochastic processes and statistical mechanics

paper · doi:10.1137/s0040585x97984735

crossref issued 2011/01/01 · crossref published 2011/01/01 · crossref published-print 2011/01/01 · openalex publication_date 2011/01/01 · crossref created 2011/03/07 · crossref deposited 2017/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11 · crossref indexed 2026/08/01

Abstract

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)) logn asymptotically as n → ∞.

Citations

• Mathematical Analysis of Random Noise

Cited by

• Real zeroes of random polynomials, I: Flip-invariance, Turán's lemma, and the Newton-Hadamard polygon

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