The Number of Real Zeros of a Class of Random Algebraic Polynomials

Complex Zeros of Trigonometric Polynomials with Standard Normal Random Coefficients

“…There are many known results for the expected number of reals of a random algebraic polynomial of the form P n x ≡ P x = n j=1 η j x j These results include the pioneering works of Dunnage [4] and Sambandham [8], and the more recent works of Wilkins [10], which are 1 Work supported by a scholarship from the Vice-Chancellor of the University of Ulster. reviewed in a comprehensive book by Bharucha-Reid and Sambandham [3]. Also there are some results concerning the density of the complex roots of P x .…”

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

“…Equations of the type (2) have been considered by Dunnage [1]. But his results do not cover our case in which the variance is infinite.…”

“…In Samal and Mishra ( [3] and [4]) we have considered the lower bound of the number of real roots of the algebraic equation (1) fix) -So + hx + • • ' + inxn = 0…”

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

“…In our paper [2] we have considered the upper bound of the number of real roots of the algebraic equation (1) fix) = J fv*v = 0 whose coefficients £v's are identically distributed independent random variables with a common characteristic function exp(-C\t\") where C is a positive constant and a^l. In fact we have proved that (i) Pr( Sup A/J(log nf > p\ < pfnf-2^, 0 < ß < 1,1 Sol S 2;…”

“…In the place of equation (1) we shall consider the equation (2) have been considered by Dunnage [1], but his variance is finite whereas our variance is infinite for 1 <a<2.…”