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 ( 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 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 , … , n − 1 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 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 + ⋯ + a n − 1 x n − 1 a_{0}+a_{1}x +a_{2}x^{2}+\cdots +a_{n- 1}x^{n-1} which was previously the most studied.