On algebraic polynomials with random coefficients

The expected number of real zeros and maxima of the curve representing algebraic polynomial of the form

$a_0\binom{n-1}{0}^{1/2}+a_1\binom{n-1}{1}^{1/2}x +a_2\binom{n-1}{2}^{1/2}x^2+\cdots+a_{n-1}\binom{n-1}{n-1}^{1/2}x^{n-1}$where $a_{j}, j= 0, 1, 2, \ldots, n-1$, are independent standard normal random variables, are known. 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 zero crossings of the curve representing the polynomial are perpendicular to the $x$ axis. The results show a significant difference in mathematical behaviour between our polynomial and the random algebraic polynomial of the form $a_{0}+a_{1}x +a_{2}x^{2}+\cdots +a_{n- 1}x^{n-1}$which was previously the most studied.