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Real zeros of algebraic polynomials with nonidentical dependent random coefficients (1909.09411v3)
Published 20 Sep 2019 in math.FA, math.CA, and math.PR
Abstract: The expected number of real zeros of an algebraic polynomial $a_0+a_1x+a_2x2+a_3x3+....+a_{n-1}x{n-1}$ 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 $\frac{2}{\pi \sigma}logn.$