On algebraic polynomials with random coefficients

The expected number of real zeros and maxima of the curve representing alge braic polynomial of the form a(o)((n-1)(0))(1/2) + a(1)((n-1)(1))(1/2) x a(2)((n-1)(2))(1/2) x(2) +...+a(n-1) ((n-1)(n-1))(1/2) x(n-1) where a(j), j = 0, 1, 2,...,n-1, are independent standard normal random variables, are k nown. In this paper we provide the asymptotic value for the expected number of maxima which occur below a given level. We also show that most of the z ero crossings of the curve representing the polynomial are perpendicular to the x axis. The results show a significant difference in mathematical beha viour between our polynomial and the random algebraic polynomial of the for m ao + a(1)x + a(2)x(2) +...+ a(n-1)x(n-1) which was previously the most st udied.