Random polynomials with prescribed Newton polytope, I

We consider random polynomials of the form $H_n(z)=\sum_{j=0}^n\xi_jq_j(z)$ where the $\{\xi_j\}$ are i.i.d non-degenerate complex random variables, and the $\{q_j(z)\}$ are orthonormal polynomials with respect to a compactly supported measure $\tau$ satisfying the Bernstein-Markov property on a regular compact set $K \subset \mathbb{C}$. We show that if $\mathbb{P}(|\xi_0|>e^{|z|})=o(|z|^{-1})$, then the normalized counting measure of the zeros of $H_n$ converges weakly in p...

Consider a random system $\mathfrak{f}_1(x)=0,\ldots,\mathfrak{f}_n(x)=0$ of $n$ random real polynomials in $n$ variables, where each $\mathfrak{f}_k$ has a prescribed set of exponent vectors in a set $A_k\subseteq \mathbb{Z}^n$ of size $t_k$. Assuming that the coefficients of the $\mathfrak{f}_k$ are independent Gaussian of any variance, we prove that the expected number of zeros of the random system in the positive orthant is bounded from above by $4^{-n} \prod_{k=1}^n t_k(...

This article is divided in two parts. In the first part we review some recent results concerning the expected number of real roots of random system of polynomial equations. In the second part we deal with a different problem, namely, the distribution of the roots of certain complex random polynomials. We discuss a recent result in this direction, which shows that the associated points in the sphere (via the stereographic projection) are surprisingly well-suited with respect t...

In this work, we study asymptotic zero distribution of random multi-variable polynomials which are random linear combinations $\sum_{j}a_jP_j(z)$ with i.i.d coefficients relative to a basis of orthonormal polynomials $\{P_j\}_j$ induced by a multi-circular weight function $Q$ satisfying suitable smoothness and growth conditions. In complex dimension $m\geq3$, we prove that $\Bbb{E}[(\log(1+|a_j|))^m]<\infty$ is a necessary and sufficient condition for normalized zero currents...

In this note, we prove a central limit theorem for smooth linear statistics of zeros of random polynomials which are linear combinations of orthogonal polynomials with iid standard complex Gaussian coefficients. Along the way, we obtain Bergman kernel asymptotics for weighted $L^2$-space of polynomials endowed with varying measures of the form $e^{-2n\varphi_n(z)}dz$ under suitable assumptions on the weight functions $\varphi_n$.

We give asymptotic large deviations estimates for the volume inside a domain U of the zero set of a random polynomial of degree N, or more generally, of a holomorphic section of the N-th power of a positive line bundle on a compact Kaehler manifold. In particular, we show that for all $\delta>0$, the probability that this volume differs by more than $\delta N$ from its average value is less than $\exp(-C_{\delta,U}N^{m+1})$, for some constant $C_{\delta,U}>0$. As a consequenc...

We study statistical properties of zeros of random polynomials and random analytic functions associated with the pseudoeuclidean group of symmetries SU(1,1), by utilizing both analytical and numerical techniques. We first show that zeros of the SU(1,1) random polynomial of degree $N$ are concentrated in a narrow annulus of the order of $N^{-1}$ around the unit circle on the complex plane, and we find an explicit formula for the scaled density of the zeros distribution along t...

Let f:=(f^1,\...,f^n) be a sparse random polynomial system. This means that each f^i has fixed support (list of possibly non-zero coefficients) and each coefficient has a Gaussian probability distribution of arbitrary variance. We express the expected number of roots of f inside a region U as the integral over U of a certain mixed volume form. When U = (C^*)^n, the classical mixed volume is recovered. The main result in this paper is a bound on the probability that the co...

We show that for Gaussian random SU(2)polynomials of a large degree $N$ the probability that there are no zeros in the disk of radius $r$ is less than $e^{-c_{1,r} N^2}$, and is also greater than $e^{-c_{2,r} N^2}$. Enroute to this result, we also derive a more general result: probability estimates for the event that the number of complex zeros of a random polynomial of high degree deviates significantly from its mean.

We study global distribution of zeros for a wide range of ensembles of random polynomials. Two main directions are related to almost sure limits of the zero counting measures, and to quantitative results on the expected number of zeros in various sets. In the simplest case of Kac polynomials, given by the linear combinations of monomials with i.i.d. random coefficients, it is well known that their zeros are asymptotically uniformly distributed near the unit circumference unde...