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Proceedings of the American Mathematical Society

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On algebraic polynomials with random coefficients


Author: K. Farahmand
Journal: Proc. Amer. Math. Soc. 129 (2001), 2763-2769
MSC (2000): Primary 60H99; Secondary 60G15
Published electronically: March 15, 2001
MathSciNet review: 1838800
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Abstract:

The expected number of real zeros and maxima of the curve representing 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 $a_{j}, j= 0, 1, 2, \ldots, n-1$, 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 $x$ axis. The results show a significant difference in mathematical behaviour between our polynomial and the random algebraic polynomial of the form $a_{0}+a_{1}x +a_{2}x^{2}+\cdots +a_{n- 1}x^{n-1}$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: http://dx.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