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International Journal of Mathematics and Mathematical Sciences
Volume 2015, Article ID 261370, 7 pages
http://dx.doi.org/10.1155/2015/261370
Research Article

Real Zeros of a Class of Hyperbolic Polynomials with Random Coefficients

1Department of Mathematics, College of Basic Science and Humanities, OUAT, Bhubaneswar, India
2DPS Kalinga, Bhubaneswar, India
3Gopabandhu Science College, Athagad, India

Received 7 January 2015; Revised 21 May 2015; Accepted 27 May 2015

Academic Editor: Niansheng Tang

Copyright © 2015 Mina Ketan Mahanti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We have proved here that the expected number of real zeros of a random hyperbolic polynomial of the form , where is a sequence of standard Gaussian random variables, is . It is shown that the asymptotic value of expected number of times the polynomial crosses the level is also as long as does not exceed , where . The number of oscillations of about will be less than asymptotically only if , where or . In the former case the number of oscillations continues to be a fraction of and decreases with the increase in value of . In the latter case, the number of oscillations reduces to and almost no trace of the curve is expected to be present above the level if log .