Journal of the Australian Mathematical Society

Research Article

REAL ZEROS OF ALGEBRAIC POLYNOMIALS WITH STABLE RANDOM COEFFICIENTS

K. FARAHMANDa1

a1 Department of Mathematics, University of Ulster at Jordanstown, Country Antrim, BT37 0QB, United Kingdom (email: k.farahmand@ulster.ac.uk)

Abstract

We consider a random algebraic polynomial of the form Pn,θ,α(t)=θ0ξ0+θ1ξ1t+xs22EF+θnξntn, where ξk, k=0,1,2,…,n have identical symmetric stable distribution with index α, 0<α≤2. First, for a general form of θk,αθk we derive the expected number of real zeros of Pn,θ,α(t). We then show that our results can be used for special choices of θk. In particular, we obtain the above expected number of zeros when $\theta _k={n\choose k}^{1/2}$. 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.

(Received November 17 2006)

(Accepted January 05 2007)

2000 Mathematics subject classification

  • primary 60G99; secondary 60H99

Keywords and phrases

  • random polynomials;
  • number of real zeros;
  • real roots;
  • Kac–Rice formula;
  • stable random variables