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

Source abstract

Let ν n {\nu _n} be the expected number of real zeros of a polynomial of degree n n whose coefficients are independent random variables, normally distributed with mean 0 and variance 1. We find an asymptotic expansion for ν n {\nu _n} of the form $$ν n = 2 π log ⁡ ( n + 1 ) + ∑ p = 0 ∞ A p ( n + 1 ) − p \nu _n = \frac {2}{\pi } \log (n + 1) + \sum \limits _{p = 0}^\infty {{A_p}{{(n + 1)}^{ - p}}}$$ in which A 0 = 0.625735818 , A 1 = 0 , A 2 = − 0.24261274 , A 3 = 0 , A 4 = − 0.08794067 , A 5 = 0 {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 ν n {\nu _n} calculated from this expansion, using only the first four, or six, coefficients, agree with previously tabulated seven decimal place values ( 1 ≤ n ≤ 100 ) (1 \leq n \leq 100) with an error of at most 10 − 7 {10^{ - 7}} when n ≥ 30 n \geq 30 , or n ≥ 8 n \geq 8 .