On random algebraic polynomials

Abstract

This paper provides asymptotic estimates for the expected number of real zeros and K K -level crossings of a random algebraic polynomial of the form a 0 ( n − 1 0 ) 1 / 2 + a 1 ( n − 1 1 ) 1 / 2 x a_{0}\binom {n-1 }{0}^{1/2}+ a_{1}\binom {n-1}{1}^{1/2}x + a 2 ( n − 1 2 ) 1 / 2 x 2 +a_{2}\binom {n-1 }{2}^{1/2}x^{2} + ⋯ + a n − 1 ( n − 1 n − 1 ) 1 / 2 x n − 1 + \cdots +a_{n-1}\binom {n-1 }{ n-1}^{1/2}x^{n-1} , where a j ( j = 0 , 1 , … , n − 1 ) a_{j} (j=0, 1, \ldots ,n-1) are independent standard normal random variables and K K is a constant independent of x x . It is shown that these asymptotic estimates are much greater than those for algebraic polynomials of the form a 0 a_{0} + a 1 x +a_{1}x + a 2 x 2 + ⋯ + a n − 1 x n − 1 +a_{2}x^{2}+ \cdots +a_{n-1}x^{n-1} .