Related papers: Distances between zeroes and critical points for r…

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…

Consider a degree-$d$ polynomial $f(\xi_1,\dots,\xi_n)$ of independent Rademacher random variables $\xi_1,\dots,\xi_n$. To what extent can $f(\xi_1,\dots,\xi_n)$ concentrate on a single point? This is the so-called polynomial…

Locating the zeros of quaternionic polynomials is a fundamental problem with significant implications across scientific and engineering disciplines, yet the noncommutative nature of quaternion multiplication makes it fundamentally more…

The curse of dimensionality is a common phenomenon which affects analysis of datasets characterized by large numbers of variables associated with each point. Problematic scenarios of this type frequently arise in classification algorithms…

We consider random orthonormal polynomials $$ P_{n}(x)=\sum_{i=0}^{n}\xi_{i}p_{i}(x), $$ where $\xi_{0}$, . . . , $\xi_{n}$ are independent random variables with zero mean, unit variance and uniformly bounded $(2+\ep_0)$-moments, and…

We study several related problems on polynomials with integer coefficients. This includes the integer Chebyshev problem, and the Schur problems on means of algebraic numbers. We also discuss interesting applications to approximation by…

We study the expected number of real zeros for random linear combinations of orthogonal polynomials. It is well known that Kac polynomials, spanned by monomials with i.i.d. Gaussian coefficients, have only $(2/\pi + o(1))\log{n}$ expected…

Mark Kac gave one of the first results analyzing random polynomial zeros. He considered the case of independent standard normal coefficients and was able to show that the expected number of real zeros for a degree n polynomial is on the…

The relationship between a polynomial's zeros and factors is well known. If a is a zero of f(x) then (x-a) is a factor of f(x). In this paper, we generalize this idea to polynomials of two variables and with real coefficients. We consider…

In this paper, we investigate the zero distributions of $q$-shift difference-differential polynomials of meromorphic functions with zero-order that extends and generalizes the classical Hayman results of the zeros of differential…

Let $\{\varphi_k\}_{k=0}^\infty $ be a sequence of orthonormal polynomials on the unit circle (OPUC) with respect to a probability measure $ \mu $. We study the variance of the number of zeros of random linear combinations of the form $$…

We establish a uniform approximation result for the Taylor polynomials of the xi function of Riemann which is valid in the entire complex plane as the degree grows. In particular, we identify a domain growing with the degree of the…

We give a survey concerning both very classical and recent results on the electrostatic interpretation of the zeros of some well-known families of polynomials, and the interplay between these models and the asymptotic distribution of their…

We study the annealed complexity of a random Gaussian homogeneous polynomial on the $N$-dimensional unit sphere in the presence of deterministic polynomials that depend on fixed unit vectors and external parameters. In particular, we…

This note will study complex polynomial maps of degree $n\ge 2$ with only one critical point.

A partition polynomial is a refinement of the partition number p(n) whose coefficients count some special partition statistic. Just as partition numbers have useful asymptotics so do partition polynomials. In fact, their asymptotics…

Number theorists have studied extensively the connections between the distribution of zeros of the Riemann $\zeta$-function, and of some generalizations, with the statistics of the eigenvalues of large random matrices. It is interesting to…

Assuming the Riemann hypothesis, we investigate the distribution of gaps between the zeros of \xi'(s). We prove that a positive proportion of gaps are less than 0.796 times the average spacing and, in the other direction, a positive…

We consider the logarithm of the characteristic polynomial of random permutation matrices, evaluated on a finite set of different points. The permutations are chosen with respect to the Ewens distribution on the symmetric group. We show…

We give an exact formula for the value of the derivative at zero of the gap probability in finite n x n Gaussian ensembles. As n goes to infinity our computation provides an asymptotic (with an explicit constant) of the order n^(1/2). As a…