On algebraic polynomials with random coefficients

Author:
K. Farahmand

Journal:
Proc. Amer. Math. Soc. **129** (2001), 2763-2769

MSC (2000):
Primary 60H99; Secondary 60G15

DOI:
https://doi.org/10.1090/S0002-9939-01-05836-1

Published electronically:
March 15, 2001

MathSciNet review:
1838800

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Abstract | References | Similar Articles | Additional Information

The expected number of real zeros and maxima of the curve representing algebraic polynomial of the form

where , are independent standard normal random variables, are known. In this paper we provide the asymptotic value for the expected number of maxima which occur below a given level. We also show that most of the zero crossings of the curve representing the polynomial are perpendicular to the axis. The results show a significant difference in mathematical behaviour between our polynomial and the random algebraic polynomial of the form which was previously the most studied.

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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:
https://doi.org/10.1090/S0002-9939-01-05836-1

Keywords:
Number of real roots,
real zeros,
number of maxima,
random algebraic polynomials,
Kac-Rice formula

Received by editor(s):
September 1, 1999

Received by editor(s) in revised form:
January 26, 2000

Published electronically:
March 15, 2001

Communicated by:
Claudia M. Neuhauser

Article copyright:
© Copyright 2001
American Mathematical Society