Probability of all real zeros for random polynomial with the exponential ensemble

On Classifications of Random Polynomials

 Let $ a_0 (omega), a_1 (omega), a_2 (omega), dots, a_n (omega)$ be a sequence of independent random variables defined on a fixed probability space $(Omega, Pr, A)$. There are many known results for the expected number of real zeros of a polynomial $ a_0 (omega) psi_0(x)+ a_1 (omega)psi_1 (x)+, a_2 (omega)psi_2 (x)+...

Inequalities for the polar derivative of a polynomial with $S$-fold zeros at the origin

‎Let $p(z)$ be a polynomial of degree $n$ and for a complex number $alpha$‎, ‎let $D_{alpha}p(z)=np(z)+(alpha-z)p'(z)$ denote the polar derivative of the polynomial p(z) with respect to $alpha$‎. ‎Dewan et al proved‎ ‎that if $p(z)$ has all its zeros in $|z| leq k, (kleq‎ ‎1),$ with $s$-fold zeros at the origin then for every‎ ‎$alphainmathbb{C}$ with $|alpha|geq k$‎, ‎begin{align*}‎ ‎max_{|z|=...

Random polynomials having few or no real zeros

Consider a polynomial of large degree n whose coefficients are independent, identically distributed, non-degenerate random variables having zero mean and finite moments of all orders. We show that such a polynomial has exactly k real zeros with probability n−b+o(1) as n → ∞ through integers of the same parity as the fixed integer k ≥ 0. In particular, the probability that a random polynomial of...

On the average number of sharp crossings of certain Gaussian random polynomials

Expected Number of Local Maxima of Some Gaussian Random Polnomials