Algebraic Polynomials with Non-identical Random Coefficients

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 $$a_0(\omega)+a_1(\omega)x+a_2(\omega)x^2+\cdots + a_n(\omega)x^n$$ .