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Algebraic Polynomials with Non-identical Random Coefficients

Authors:	Email author" target="_blank">K?Farahmand Email author Jay?Jahangiri

Institution:	(1) Department of Mathematics, University of Ulster at Jordanstown, Co. Antrim, BT37 0QB, UK;(2) Department of Mathematical Sciences, Kent State University, Burton, OH 44021, USA

Abstract:	The asymptotic estimate for the expected number of real zeros of a random algebraic polynomial $Q_n(x,\omega)=a_o(\omega){n\choose 0}+a_1(\omega){n\choose 1}x+a_2(\omega){n\choose 2}x^2+\cdots + a_n(\omega){n\choose n}x^n$ is known. The identical random coefficients a_j(ω) are normally distributed defined on a probability space $(\Omega, \Pr, \mathcal{A})$ , ω ∈Ω. The estimate for the expected number of zeros of the derivative of the above polynomial with respect to x is also known, which gives the expected number of maxima and minima of Q_n(x, ω). In this paper we provide the asymptotic value for the expected number of zeros of the integration of Q_n(x,ω) with respect to x. We give the geometric interpretation of our results and discuss the difficulties which arise when we consider a similar problem for the case of $a_0(\omega)+a_1(\omega)x+a_2(\omega)x^2+\cdots + a_n(\omega)x^n$ .

Keywords:	Number of real zeros real roots random algebraic polynomials Kac-Rice formula non-identical random variables
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