Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
MathSciNet review: 955018
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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$.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 60G99

Retrieve articles in all journals with MSC: 60G99


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0955018-1
PII: S 0002-9939(1988)0955018-1
Article copyright: © Copyright 1988 American Mathematical Society