Research Article On the Lower Bound for the Number of Real Roots of a Random Algebraic Equation

where the aν(ω), ν= 0,1, . . . ,n, are random variables defined on a fixed probability space (Ω, ,Pr) assuming real values only. During the past 40–50 years, the majority of published researches on random algebraic polynomials has concerned the estimation of Nn(R,ω). Works by Littlewood and Offord [1], Samal [2], Evans [3], and Samal and Mishra [4–6] in the main concerned cases in which the random coefficients aν(ω) are independent and identically distributed. For dependent coefficients, Sambandham [7] considered the upper bound forNn(R,ω) in the case when the aν(ω), ν = 0,1, . . . ,n, are normally distributed with mean zero and joint density function |M|1/2(2π)−(n+1)/2 exp− (1/2)a′Ma, (1.2)