On the Expected Number of Level Crossings of a Random Polynomial

There are many known asymptotic estimates for the expected number of K-level crossings of an algebraic polynomial a0qa1xqa2x2q???qanxn with normally distributed coefficients. The present paper provides the estimate for the expected number of such level crossings when the coefficients are independent identically Cauchy distributed random variables. Using a numerical approach we show that in the interval y1, 1.by increasing K the number of K-level crossings decreases, while outside this interval this number is invariant, as long as Kso ʹn.. Since the Cauchy distribution does not belong to the wide class of distributions of domain of attraction of the normal law, the polynomials with Cauchy distributed coefficients are interesting as