REAL ZEROS OF ALGEBRAIC POLYNOMIALS WITH STABLE RANDOM COEFFICIENTS

We consider a random algebraic polynomial of the form P(n,0,alpha)(t) = theta(0)xi(0) + theta(1)xi(1)t + ... + theta(n)xi(n)t(n), where xi(k), k = 0, 1, 2...., n have identical symmetric stable distribution with index alpha, 0 < alpha <= 2. First, for a general form of theta(k,alpha) equivalent to theta(k) we derive the expected number of real zeros of P(n,theta,alpha)(t). We then show that our results can be used for special choices of theta(k). In particular, we obtain the above expected number of zeros when theta(k) = (n k)(1/2). The latter generate a polynomial with binomial elements which has recently been of significant interest and has previously been studied only for Gaussian distributed coefficients. We see the effect of alpha on increasing the expected number of zeros compared with the special case of Gaussian coefficients.