On the Number of Real Roots of a Random Algebraic Equation

On the Lower Bound of the Number of Real Roots of a Random Algebraic Equation with Infinite Variance. II

“…However, incidentally we find that the corresponding result of Evans [1] happens to be a special case of our result when a=2, although our exceptional set is larger than his.…”

“…Taking the coefficients as normally distributed, Evans [1] has shown that there exists an integer n0 such that for n > «0, Nn > iß log «)/(log log n) except for a set of measure at most (5 log log «0)/(log «0).…”

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

“…Uno [11] pointed out the defect in the proofs of the above papers and obtained the result for the lower bound. Additionally, Uno [12] estimated the strong result for this particular problem in the sense of Evans [3]. The term strong indicates that the estimation for the exceptional set is independent of the degree n.…”

“…During the past 40-50 years, the majority of published researches on random algebraic polynomials has concerned the estimation of N n (R,ω). Works by Littlewood and Offord [1], Samal [2], Evans [3], and Samal and Mishra [4][5][6] in the main concerned cases in which the random coefficients a ν (ω) are independent and identically distributed.…”

Strong result for real zeros of random algebraic polynomials

“…They [3]). Taking the coefficients as normal random variables, Evans [2] proved that there exists an integer n o such that Cqog log n 0 Clog n except for a set of measure at most for each n > no, Nn(R,w) > log log n log n 0 for a positive integer m, where 0 < pj < 1, j-1,2,...,m and a' is the transpose of the column vector a, and the bv's are positive numbers. However, the result of Uno is not the "strong" result for the lower bound.…”