Real Zeros of a Class of Hyperbolic Polynomials with Random Coefficients

• Main Authors: Mina Ketan Mahanti, Amandeep Singh, Lokanath Sahoo
• Format: Article
• Language: English
• Published: Hindawi Limited 2015-01-01
• Series: International Journal of Mathematics and Mathematical Sciences
• Online Access: http://dx.doi.org/10.1155/2015/261370

We have proved here that the expected number of real zeros of a random hyperbolic polynomial of the form y=Pnt=n1a1cosh⁡t+n2a2cosh⁡2t+⋯+nnancosh⁡nt, where a1,…,an is a sequence of standard Gaussian random variables, is n/2+op(1). It is shown that the asymptotic value of expected number of times the polynomial crosses the level y=K is also n/2 as long as K does not exceed 2neμ(n), where μ(n)=o(n). The number of oscillations of Pn(t) about y=K will be less than n/2 asymptotically only if K=2neμ(n), where μ(n)=O(n) or n-1μ(n)→∞. In the former case the number of oscillations continues to be a fraction of n and decreases with the increase in value of μ(n). In the latter case, the number of oscillations reduces to op(n) and almost no trace of the curve is expected to be present above the level y=K if μ(n)/(n log n)→∞.