Zeros of Random Tropical Polynomials, Random Polytopes and Stick-breaking

Domain of attraction of normal law and zeros of random polynomials

Let$ P_{n}(x)= sum_{i=0}^{n} A_{i}x^{i}$ be a random algebraicpolynomial, where $A_{0},A_{1}, cdots $ is a sequence of independent random variables belong to the domain of attraction of the normal law. Thus $A_j$'s for $j=0,1cdots $ possesses the characteristic functions $exp {-frac{1}{2}t^{2}H_{j}(t)}$, where $H_j(t)$'s are complex slowlyvarying functions.Under the assumption that there exist ...

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

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)+...

Random Polynomials with Prescribed Newton Polytope

The Newton polytope Pf of a polynomial f is well known to have a strong impact on its behavior. The Kouchnirenko-Bernstein theorem asserts that even the number of simultaneous zeros in (C∗)m of a system of m polynomials depends on their Newton polytopes. In this article, we show that Newton polytopes further have a strong impact on the distribution of mass and zeros of polynomials, the basic th...

Expected Discrepancy for Zeros of Random Polynomials

We study asymptotic clustering of zeros of random polynomials, and show that the expected discrepancy of roots of a polynomial of degree n, with not necessarily independent coefficients, decays like √ logn/n. Our proofs rely on discrepancy results for deterministic polynomials, and order statistics of a random variable. We also consider the expected number of zeros lying in certain subsets of t...