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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On random algebraic polynomials


Author: K. Farahmand
Journal: Proc. Amer. Math. Soc. 127 (1999), 3339-3344
MSC (1991): Primary 60H99; Secondary 42Bxx
Published electronically: May 6, 1999
MathSciNet review: 1610956
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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 $a_{0}$ $+a_{1}x$ $+a_{2}x^{2}+ \cdots +a_{n-1}x^{n-1}$.


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Additional Information

K. Farahmand
Affiliation: Department of Mathematics, University of Ulster, Jordanstown, Co. Antrim BT37 0QB, United Kingdom
Email: k.farahmand@ulst.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04912-6
PII: S 0002-9939(99)04912-6
Keywords: Number of real roots, real zeros, random algebraic polynomials, random trigonometric polynomials, Kac-Rice formula
Received by editor(s): July 9, 1997
Received by editor(s) in revised form: December 17, 1997, and February 5, 1998
Published electronically: May 6, 1999
Communicated by: Stanley Sawyer
Article copyright: © Copyright 1999 American Mathematical Society