Real zeros of algebraic polynomials with nonidentical dependent random coefficients

Abstract

The expected number of real zeros of an algebraic polynomial $a_0+a_{1}x+a_2{x}^2+a_3{x}^3+....+a_{n-1}{x}^{n-1}$ depends on the types of random coefficients, with large $n.$ In this article, we show that when the random coefficients $\{a_i{\}}_{i=1}^{n-1}$ are assumed to be negatively dependent with $var(a_i)={\sigma }^{2i}$ and correlation between any two coefficients for $i\ne j,$ assumed to be ${\rho }_{ij}=-{\rho }^{|i-j|},$ where $0<\rho <\frac{1}{3}$, then the expected number of real zeros is asymptotically equal to $\frac{2}{\pi \sigma }logn.$

arXiv comment9 pages