Related papers: Zeros of random polynomials undergoing the heat fl…

Let $x_1, \dots, x_n$ be $n$ independent and identically distributed random variables with mean zero, unit variance, and finite moments of all remaining orders. We study the random polynomial $p_n$ having roots at $x_1, \dots, x_n$. We…

Consider random polynomial $\sum_{i=0}^na_ix^i$ of independent mean-zero normal coefficients $a_i$, whose variance is a regularly varying function (in $i$) of order $\alpha$. We derive general criteria for continuity of persistence…

We establish the central limit theorem for the number of real roots of the Weyl polynomial $P_n(x)=xi_0 + xi_1 x+ ... + xi_n (n!)^{(-1/2)} x^n$, where $xi_i$ are iid Gaussian random variables. The main ingredients in the proof are new…

We study the density of the roots of the derivative of the characteristic polynomial Z(U,z) of an N x N random unitary matrix with distribution given by Haar measure on the unitary group. Based on previous random matrix theory models of the…

We show that with high probability the number of real zeroes of a random polynomial is bounded by the number of vertices on its Newton-Hadamard polygon times the cube of the logarithm of the polynomial degree. A similar estimate holds for…

We survey results and conjectures concerning the zero distribution of chromatic and flow polynomials of graphs, and characteristic polynomials of matroids.

In this paper we study the asymptotic behavior of a family of polynomials which are orthogonal with respect to an exponential weight on certain contours of the complex plane. The zeros of these polynomials are the nodes for complex Gaussian…

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…

Let $f_n$ be a random polynomial of degree $n\ge 2$ whose coefficients are independent and identically distributed random variables. We study the separation distances between roots of $f_n$ and prove that the set of these distances,…

We consider the polynomials $\Xi_n$ introduced in~\cite{TallaWaffo2025arxiv2511.02843} and studied in further details in\cite{TallaWaffo2026arxiv2602.16761}, which are expressed in terms of Eulerian polynomials of type~B, and study the zero…

Let $G_n(z)=\xi_0+\xi_1z+...+\xi_n z^n$ be a random polynomial with i.i.d. coefficients (real or complex). We show that the arguments of the roots of $G_n(z)$ are uniformly distributed in $[0,2\pi]$ asymptotically as $n\to\infty$. We also…

Motivated by Wilmshurst's conjecture, we investigate the zeros of harmonic polynomials. We utilize a certified counting approach which is a combination of two methods from numerical algebraic geometry: numerical polynomial homotopy…

We investigate the hydrodynamic behavior of zeroes of trigonometric polynomials under repeated differentiation. We show that if the zeroes of a real-rooted, degree $d$ trigonometric polynomial are distributed according to some probability…

Motivated by Hilbert's 16th problem we discuss the probabilities of topological features of a system of random homogeneous polynomials. The distribution for the polynomials is the Kostlan distribution. The topological features we consider…

We study distribution of zeros of a complex polynomial whose coefficients has been modified. We give a new proof of the theorem of Rubinstein, and with similar method we prove a new theorem that is not generalization of the previous…

Let $\mu$ be a probability measure on $\mathbb C$, and let $P_n$ be the random polynomial whose zeros are sampled independently from $\mu$. We study the asymptotic distribution of zeros of high-order derivatives of $P_n$. We show that, for…

In this paper we study the asymptotic zero distribution of eigenpolynomials for degenerate exactly-solvable operators. We present an explicit conjecture and partial results on the growth of the largest modulus of the roots of the unique and…

In this paper we determine the limiting distribution of the image of the Eichler--Shimura map or equivalently the limiting joint distribution of the coefficients of the period polynomials associated to a fixed cusp form. The limiting…

We introduce two probabilistic models of random log-concave polynomials, the uniform model and the beta model, and study the asymptotic distribution of their zeros in the complex plane. In the uniform model, we show that the empirical root…

We utilize Cauchy's argument principle in combination with the Jacobian of a holomorphic function in several complex variables and the first moment of a ratio of two correlated complex normal random variables to prove explicit formulas for…