On the variance of the number of real roots of random algebraic polynomials

Real roots of random polynomials with coefficients of polynomial growth: asymptotics of the variance

“…While the condition P({ξ j = 0}) = 0 has been removed by O. Nguyen and V. Vu [38], there has been no other result of this type for other generalized Kac polynomials (even for the Gaussian setting when ξ j 's are all Gaussian). We mention here another result due to Sambandham, Thangaraj, and Bharucha-Reid [42], who proved an estimate of Var[N n (R)] for random Kac polynomials with dependent Gaussian coefficients. For generalized Kac polynomials, O. Nguyen and V. Vu [38] have recently proved the following lower bound…”

On the lower bound of the number of real roots of a random algebraic equation

“…Also, Uno and Negishi [8] obtained the same result as Sambandham in the case of the moment matrix with σ i = 1, ρ i j = ρ |i− j| , 2 Journal of Applied Mathematics and Stochastic Analysis (i = j), i, j = 0,1,...,n, where ρ j is a nonnegative decreasing sequence satisfying ρ 1 < 1/2 and ∞ j=1 ρ j < ∞ in (1.2). The lower bound for N n (R,ω) in the case of dependent normally distributed coefficients was estimated by Renganathan and Sambandham [9] and Nayak and Mohanty [10] under the same condition of Sambandham [7]. Uno [11] pointed out the defect in the proofs of the above papers and obtained the result for the lower bound.…”

“…For dependent coefficients, Sambandham [7] considered the upper bound for N n (R,ω) in the case when the a ν (ω), ν = 0,1,...,n, are normally distributed with mean zero and joint density function…”

Expected Number of Slope Crossings of Certain Gaussian Random Polynomials

“…The works of Logan and Shepp [7,8], Ibragimov and Maslova [5], Wilkins [14], Farahmand [3], and Sambandham [12,13] are other fundamental contributions to the subject. For various aspects on random polynomials see Bharucha-Reid and Sambandham [1], and Farahmand [4].…”