An asymptotic expansion for the expected number of real zeros of a random polynomial

Abstract:

Let ${\nu _n}$ be the expected number of real zeros of a polynomial of degree $n$ whose coefficients are independent random variables, normally distributed with mean 0 and variance 1. We find an asymptotic expansion for ${\nu _n}$ of the form \[ \nu _n = \frac {2}{\pi } \log (n + 1) + \sum \limits _{p = 0}^\infty {{A_p}{{(n + 1)}^{ - p}}} \] in which ${A_0} = 0.625735818,{A_1} = 0,{A_2} = - 0.24261274,{A_3} = 0,{A_4} = - 0.08794067,{A_5} = 0$. The numerical values of ${\nu _n}$ calculated from this expansion, using only the first four, or six, coefficients, agree with previously tabulated seven decimal place values $(1 \leq n \leq 100)$ with an error of at most ${10^{ - 7}}$ when $n \geq 30$, or $n \geq 8$.