Real Zeros of Algebraic Polynomials With Stable Random Coefficients

Abstract

We consider a random algebraic polynomial of the form P n,θ,α (t) = θ 0 ξ 0 + θ 1 ξ 1 t + · · · + θ n ξ n t n , where ξ k , k = 0, 1, 2, . . . , n have identical symmetric stable distribution with index α, 0 < α ≤ 2. First, for a general form of θ k,α ≡ θ k we derive the expected number of real zeros of P n,θ,α (t). We then show that our results can be used for special choices of θ k . In particular, we obtain the above expected number of zeros when θ 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 α on increasing the expected number of zeros compared with the special case of Gaussian coefficients.