相关论文: Multiplicity of zeros and discrete orthogonal poly…

We prove that there is a universal measure on the unit circle such that any probability measure on the unit disk is the limit distribution of some subsequence of the corresponding orthogonal polynomials. This follows from an extension of a…

We obtain new upper bounds on the number of distinct roots of lacunary polynomials over finite fields. Our focus will be on polynomials for which there is a large gap between consecutive exponents in the monomial expansion.

We consider multivariate polynomials and investigate how many zeros of multiplicity at least $r$ they can have over a Cartesian product of finite subsets of a field. Here r is any prescribed positive integer and the definition of…

We consider the zeros of the sum of independent random polynomials as their degrees tend to infinity. Namely, let $p$ and $q$ be two independent random polynomials of degree $n$, whose roots are chosen independently from the probability…

The absolute separation of a polynomial is the minimum nonzero difference between the absolute values of its roots. In the case of polynomials with integer coefficients, it can be bounded from below in terms of the degree and the height…

The complex or non-hermitian orthogonal polynomials with analytic weights are ubiquitous in several areas such as approximation theory, random matrix models, theoretical physics and in numerical analysis, to mention a few. Due to the…

Systems of orthogonal polynomials whose recurrence coefficients tend to infinity are considered. A summability condition is imposed on the coefficients and the consequences for the measure of orthogonality are discussed. Also discussed are…

The real type of a finite family of univariate polynomials characterizes the combined sign behavior of the polynomials over the real line. We derive an explicit formula for the number of real types subject to given degree bounds. For the…

In this paper we relate the location of the complex zeros of the reliability polynomial to parameters at which a certain family of rational functions derived from the reliability polynomial exhibits chaotic behaviour. We use this connection…

We derive upper bounds for the smallest zero and lower bounds for the largest zero of Laguerre, Jacobi and Gegenbauer polynomials. Our approach uses mixed three term recurrence relations satisfied by polynomials corresponding to different…

Let $p_n(x)$ be orthogonal polynomials associated to a measure $d\mu$ of compact support in $R$. If $E\not\in supp(d\mu)$, we show there is a $\delta>0$ so that for all $n$, either $p_n$ or $p_{n+1}$ has no zeros in $(E-\delta, E+\delta)$.…

We study the density of complex zeros of a system of real random SO($m+1$) polynomials in several variables. We show that the density of complex zeros of this random polynomial system with real coefficients rapidly approaches the density of…

In this paper we investigate the asymptotic distribution of the zeros of polynomials $P_{n}(x)$ satisfying a first order differential-difference equation. We give several examples of orthogonal and non-orthogonal families.

We give a necessary and sufficient condition for the maximum multiplicity of a root of the matching polynomial of a tree to be equal to the minimum number of vertex disjoint paths needed to cover it.

The classical Mason-Stothers theorem deals with nontrivial polynomial solutions to the equation $a+b=c$. It provides a lower bound on the number of distinct zeros of the polynomial $abc$ in terms of the degrees of $a$, $b$ and $c$. We…

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…

In this note we initiate the probabilistic study of the critical points of polynomials of large degree with a given distribution of roots. Namely, let f be a polynomial of degree n whose zeros are chosen IID from a probability measure mu on…

A wide variety of problems in combinatorics and discrete optimization depend on counting the set $S$ of integer points in a polytope, or in some more general object constructed via discrete geometry and first-order logic. We take a tour…

For orthogonal polynomials defined by compact Jacobi matrix with exponential decay of the coefficients, precise properties of orthogonality measure is determined. This allows showing uniform boundedness of partial sums of orthogonal…

Exponential sums with monomials are highly related to many interesting problems in number theory and well studied by many literatures. In this paper, we consider the exponential sums with polynomials and prove a new upper bound. As an…