Real zeros of algebraic polynomials with nonidentical dependent random coefficients

TL;DR

This paper investigates the asymptotic behavior of the expected number of real zeros of algebraic polynomials with dependent random coefficients, revealing a logarithmic growth rate under specific negative dependence conditions.

Contribution

It establishes the asymptotic formula for the expected number of real zeros for polynomials with negatively dependent coefficients and specified variance and correlation structures.

Findings

Expected zeros grow as (2/πσ) log n.

Negative dependence influences zero distribution.

Asymptotic behavior differs from independent coefficient cases.

Abstract

The expected number of real zeros of an algebraic polynomial $a_0+a_1x+a_2x^2+a_3x^3+....+a_{n-1}x^{n-1}$ 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 $i\neq j,$ 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 $\frac{2}{\pi \sigma}logn.$