On random algebraic polynomials

Abstract:

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