Related papers: On asymptotic Gauss-Lucas theorem

We study the weak asymptotic behavior of the zeros of a family of a certain class of (generalized) hypergeometric polynomials, using the associated hypergeometric differential equation, as the parameters go to infinity. We describe the…

We study the asymptotic laws for the spatial distribution and the number of connected components of zero sets of smooth Gaussian random functions of several real variables. The primary examples are various Gaussian ensembles of real-valued…

Consider a sequence of polynomials of bounded degree evaluated in independent Gaussian, Gamma or Beta random variables. We show that, if this sequence converges in law to a nonconstant distribution, then (i) the limit distribution is…

In this paper, we give results that partially prove a conjecture which was discussed in our previous work (arXiv:1307.4991). More precisely, we prove that as $n\to \infty,$ the zeros of the polynomial$${}_{2}\text{F}_{1}\left[…

We consider transcendental entire solutions of linear $q$-difference equations with polynomial coefficients and determine the asymptotic behavior of their Taylor coefficients. We use this to show that under a suitable hypothesis on the…

Asymptotic expansions are derived for Gegenbauer (ultraspherical) polynomials for large order $n$ that are uniformly valid for unbounded complex values of the argument $z$, including the real interval $0 \leq z \leq 1$ in which the zeros in…

All linear operators $L:\mathcal{C}[z]\to \mathcal{C}[z]$ which decrease the diameter of the zero set of any $P\in\mathcal{C}[z]$ are found.

Loeb showed that a natural extension of the usual binomial coefficient to negative (integer) entries continues to satisfy many of the fundamental properties. In particular, he gave a uniform binomial theorem as well as a combinatorial…

Let $X_1,X_2,\ldots$ be independent and identically distributed random variables in $\mathbb{C}$ chosen from a probability measure $\mu$ and define the random polynomial $$ P_n(z)=(z-X_1)\ldots(z-X_n)\,. $$ We show that for any sequence $k…

We obtain some results on the asymptotic behaviour of Geometric polynomials in both the complex plane minus $[-1,0]$ and the interval $(-1,0)$. We also find the distance of consecutive zeros of these polynomials in the bulk of the interval…

Suppose $C \subset \mathbb{C}$ is compact. Let $q_k$ be a sequence of polynomials of degree $n_k \to \infty$, such that the locus of roots of all the polynomials is bounded, and the number of roots of $q_k$ in any closed set $L$ not meeting…

Sequences of discrete random variables are studied whose probability generating functions are zero-free in a sector of the complex plane around the positive real axis. Sharp bounds on the cumulants of all orders are stated, leading to…

We compute the variance asymptotics for the number of real zeros of trigonometric polynomials with random dependent Gaussian coefficients and show that under mild conditions, the asymptotic behavior is the same as in the independent…

A precise tie between a univariate spline's knots and its zeros abundance and dissemination is formulated. As an application, a conjecture formulated by De Concini and Procesi is shown to be true in the special univariate, unimodular case.…

In this paper, we present a correct proof of an $L_p$-inequality concerning the polar derivative of a polynomial with restricted zeros. We also extend Zygmund's inequality to the polar derivative of a polynomial.

Assuming an integral quadratic polynomial with nonsingular quadratic part has a nontrivial zero on an integer lattice outside of a union of finite-index sublattices, we prove that there exists such a zero of bounded norm and provide an…

Zeros of many ensembles of polynomials with random coefficients are asymptotically equidistributed near the unit circumference. We give quantitative estimates for such equidistribution in terms of the expected discrepancy and expected…

We consider random trigonometric polynomials with general dependent coefficients. We show that under mild hypotheses on the structure of dependence, the asymptotics as the degree goes to infinity of the expected number of real zeros…

We show that the zeros of random sequences of Gaussian systems of polynomials of increasing degree almost surely converge to the expected limit distribution under very general hypotheses. In particular, the normalized distribution of zeros…

Let $p:\mathbb{C} \rightarrow \mathbb{C}$ be a polynomial. The Gauss-Lucas theorem states that its critical points, $p'(z) = 0$, are contained in the convex hull of its roots. We prove a stability version whose simplest form is as follows:…