Strong result for real zeros of random polynomials

M. N. Mishra, N. N. Nayak and Swadeenananda Pattanayak

Vol. 103 (1982), No. 2, 509–522

DOI: 10.2140/pjm.1982.103.509

Abstract

Let N_n be the number of real zeros of Σ_r=0ⁿa_rX_rx^r = 0 where X_r’s are independent random variables identically distributed belonging to the domain of attraction of normal law; a₀,a₁,a₂⋯a_n are nonzero real numbers such that (k_n∕t_n) = o(log n) where k_n = max_0≦r≦n|a_r| and t_n = min_0≦r≦n|a_r|. Further we suppose that the coefficients have zero means and P{X_r≠0} > 0. Then there exists a positive integer n₀ such that

for n > n₀ and 1 > 𝜀 > 0 where D_n = (log n∕log(k_n∕t_n)log log n)^{(1−𝜀)∕2}.

Mathematical Subject Classification 2000

Primary: 60G99

Milestones

Received: 19 August 1980

Revised: 16 September 1981

Published: 1 December 1982