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Dynamical systems with quadratic or polynomial drift exhibit complex dynamics, yet compared to nonlinear systems in general form, are often easier to analyze, simulate, control, and learn. Results going back over a century have shown that…

Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…

Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…

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…

We study the probability distribution of the number of common zeros of a system of $m$ random $n$-variate polynomials over a finite commutative ring $R$. We compute the expected number of common zeros of a system of polynomials over $R$.…

We give the asymptotic behavior of the zeros of orthogonal polynomials, after appropriate scaling, for which the orthogonality measure is supported on the $q$-lattice $\{q^k, k=0,1,2,3,\ldots\}$, where $0 < q < 1$. The asymptotic…

We consider a set of electrostatic problems relevant for determining the real-space structure and the ground-state energy of a two-dimensional electron liquid subject to smooth external potentials. Three fundamental geometries are…

We analyze the polynomial solutions of the linear differential equation $p_2(x)y''+p_1(x)y'+p_0(x)y=0$ where $p_j(x)$ is a $j^{\rm th}$-degree polynomial. We discuss all the possible polynomial solutions and their dependence on the…

We propose a way to find the asymptotic distribution of zeros of orthogonal polynomials p_n(x) satisfying a difference equation of the form B(x)p_n(x+\delta)-C(x,n)p_n(x)+D(x)p_n(x-\delta)=0. We calculate the asymptotic distribution of…

We investigate completed interlacing of zeros for pairs of polynomial sequences that fail to interlace by exactly two points. Using a general mixed recurrence relation, we identify a quadratic polynomial whose zeros serve as the two extra…

Electrostatic forces play many important roles in molecular biology, but are hard to model due to the complicated interactions between biomolecules and the surrounding solvent, a fluid composed of water and dissolved ions. Continuum model…

We consider a problem of bounding the maximal possible multiplicity of a zero at of some expansions $\sum a_i F_i(x)$, at a certain point $c,$ depending on the chosen family $\{F_i \}$. The most important example is a polynomial with $c=1.$…

Via a generalization of the pseudospectral method for numerical solution of differential equations, a family of nonlinear algebraic identities satisfied by the zeros of a wide class of orthogonal polynomials is derived. The generalization…

In this paper we focus on two new families of polynomials which are connected with exponential polynomials and geometric polynomials. We discuss their generalizations and show that these new families of polynomials and their generalizations…

The purpose of the present paper is to examine the zeros of $R$-Bonacci polynomials and their derivatives. We confirm a conjecture about the zeros of $R$-Bonacci polynomials for some special cases. We also find explicit formulas of the…

An exponential polynomial is a finite linear sum of terms $P(z)e^{Q(z)}$, where $P(z)$ and $Q(z)$ are polynomials. The early results on the value distribution of exponential polynomials can be traced back to Georg P\'olya's paper published…

We identify a class of remarkable algebraic relations satisfied by the zeros of the Krall orthogonal polynomials that are eigenfunctions of linear differential operators of order higher than two. Given an orthogonal polynomial family…

We give results on zeros of a polynomial of $\zeta(s),\zeta'(s),\ldots,\zeta^{(k)}(s)$. First, we give a zero free region and prove that there exist zeros corresponding to the trivial zeros of the Riemann zeta function. Next, we estimate…

This paper is devoted to the analysis of the divergence of the electron self-energy in classical electrodynamics. To do so, we appeal to the theory of distributions and a method for obtaining corresponding extensions. At first sight,…

Information about the behavior of zeros of classical families of multiple or Hermite-Pad\'e orthogonal polynomials as functions of the intrinsic parameters of the family is scarce. We establish the interlacing properties of the zeros of…