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

 

Algebraic polynomials with non-identical random coefficients


Authors: K. Farahmand and A. Nezakati
Journal: Proc. Amer. Math. Soc. 133 (2005), 275-283
MSC (2000): Primary 60G99; Secondary 60H99
Published electronically: July 26, 2004
MathSciNet review: 2086220
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Abstract: There are many known asymptotic estimates for the expected number of real zeros of a random algebraic polynomial $a_0 +a_1 x+ a_2 x^2+ \cdots +a_{n-1}x^{n-1}.$ The coefficients $a_j$ $(j=0, 1, 2, \dotsc, n-1)$ are mostly assumed to be independent identical normal random variables with mean zero and variance unity. In this case, for all $n$ sufficiently large, the above expected value is shown to be $O(\log n)$. Also, it is known that if the $a_j$ have non-identical variance $\binom{n-1}{j}$, then the expected number of real zeros increases to $O(\sqrt{n})$. It is, therefore, natural to assume that for other classes of distributions of the coefficients in which the variance of the coefficients is picked at the middle term, we would also expect a greater number of zeros than $O(\log n)$. In this work for two different choices of variance for the coefficients we show that this is not the case. Although we show asymptotically that there is some increase in the number of real zeros, they still remain $O(\log n)$. In fact, so far the case of $\mbox{var}(a_j)={\binom{n-1}{j}}$ is the only case that can significantly increase the expected number of real zeros.


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Additional Information

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

A. Nezakati
Affiliation: Department of Statistics, School of Mathematical Sciences, Ferdowsi University of Mashhad, P.O. Box 1159-91775, Mashhan, Iran
Email: Nezakati@math.um.ac.ir

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07501-X
PII: S 0002-9939(04)07501-X
Keywords: Number of real zeros, real roots, random algebraic polynomials, Kac-Rice formula, non-identical random variables
Received by editor(s): February 18, 2003
Received by editor(s) in revised form: September 8, 2003
Published electronically: July 26, 2004
Additional Notes: This work was completed while the second author was visiting the Department of Mathematics at the University of Ulster. The hospitality of the Department of Mathematics at the University of Ulster is appreciated. The financial support for this visit was supported by the Ministry of Science, Research and Technology of I. R. Iran
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2004 American Mathematical Society