Table of Contents
Journal of Applied Mathematics and Stochastic Analysis
Volume 2008, Article ID 189675, 8 pages
http://dx.doi.org/10.1155/2008/189675
Research Article

On Different Classes of Algebraic Polynomials with Random Coefficients

Department of Mathematics, University of Ulster, Jordanstown, County Antrim BT37 0QB, Northern Ireland, UK

Received 27 February 2008; Accepted 20 April 2008

Academic Editor: Lev Abolnikov

Copyright © 2008 K. Farahmand 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

The expected number of real zeros of the polynomial of the form 𝑎 0 + 𝑎 1 𝑥 + 𝑎 2 𝑥 2 + ⋯ + 𝑎 𝑛 𝑥 𝑛 , where 𝑎 0 , 𝑎 1 , 𝑎 2 , … , 𝑎 𝑛 is a sequence of standard Gaussian random variables, is known. For 𝑛 large it is shown that this expected number in ( − ∞ , ∞ ) is asymptotic to ( 2 / 𝜋 ) l o g 𝑛 . In this paper, we show that this asymptotic value increases significantly to √ 𝑛 + 1 when we consider a polynomial in the form 𝑎 0  𝑛 0  1 / 2 √ 𝑥 / 1 + 𝑎 1  𝑛 1  1 / 2 𝑥 2 / √ 2 + 𝑎 2  𝑛 2  1 / 2 𝑥 3 / √ 3 + ⋯ + 𝑎 𝑛  𝑛 𝑛  1 / 2 𝑥 𝑛 + 1 / √ 𝑛 + 1 instead. We give the motivation for our choice of polynomial and also obtain some other characteristics for the polynomial, such as the expected number of level crossings or maxima. We note, and present, a small modification to the definition of our polynomial which improves our result from the above asymptotic relation to the equality.