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


Author: J. Ernest Wilkins
Journal: Proc. Amer. Math. Soc. 103 (1988), 1249-1258
MSC: Primary 60G99
DOI: https://doi.org/10.1090/S0002-9939-1988-0955018-1
MathSciNet review: 955018
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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

$\displaystyle \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$.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0955018-1
Article copyright: © Copyright 1988 American Mathematical Society

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