Related papers: Open problems on polynomials, their zero distribut…

This paper is a write-up of the ideas that were presented, developed and discussed at the third International Workshop on QCD Challenges from pp to A-A, which took place in August 2019 in Lund, Sweden. The goal of the workshop was to focus…

We give an overview of the 2024 Computational Geometry Challenge targeting the problem \textsc{Maximum Polygon Packing}: Given a convex region $P$ in the plane, and a collection of simple polygons $Q_1, \ldots, Q_n$, each $Q_i$ with a…

In this article we combine two developments in polynomial optimization. On the one hand, we consider nonnegativity certificates based on sums of nonnegative circuit polynomials, which were recently introduced by the second and the third…

We derive a useful result about the zeros of the $k$-polar polynomials on the unit circle; in particular we obtain a ring shaped region containing all the zeros of these polynomials. Some examples are presented.

The zero sets of harmonic polynomials play a crucial role in the study of the free boundary regularity problem for harmonic measure. In order to understand the fine structure of these free boundaries a detailed study of the singular points…

We discuss the zero sets of two-variable polynomials as they relate to an approximation problem in the Hardy space on the bidisk.

The 2007 Midwest Geometry Conference included a panel discussion devoted to open problems and the general direction of future research in fields related to the main themes of the conference. This paper summarizes the comments made during…

Mhaskar-Saff found a kind of universal behavior for the bulk structure of the zeros of orthogonal polynomials for large $n$. Motivated by two plots, we look at the finer structure for the case of random Verblunsky coefficients and for what…

The goal of this paper is to attract attention of the reader to a dimension-free geometric inequality that can be proved using the classical needle decomposition. This inequality allows us to derive sharp dimension-free estimates for the…

In this paper we survey some recent contributions by the authors to the theory of null holomorphic curves in the complex Euclidean space $\mathbb C^3$, as well as their applications to null holomorphic curves in the special linear group…

The problem of extending partial geometric graph representations such as plane graphs has received considerable attention in recent years. In particular, given a graph $G$, a connected subgraph $H$ of $G$ and a drawing $\mathcal{H}$ of $H$,…

We investigate two distinct formulations of Laurent multiple orthogonal polynomials on the unit circle, introduced in arXiv:2410.12094 and arXiv:2601.04783 respectively. For the first formulation, we prove that all zeros lie strictly within…

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 $R$.…

This note is based on the plenary talk given by the second author at MACIS 2015, the Sixth International Conference on Mathematical Aspects of Computer and Information Sciences. Motivated by some of the work done within the Priority…

Let $p(z)=a_0+a_1z+a_2z^2+a_3z^3+\cdots+a_nz^n$ be a polynomial of degree $n,$ where the coefficients $a_j,$ $j \in \{0,1,2,\cdots n\},$ may be complex. We impose some restriction on the coefficients of the real part of the given polynomial…

We study the evolution of zeros of high polynomial powers under the heat flow. For any fixed polynomial $P(z)$, we prove that the empirical zero distribution of its heat-evolved $n$-th power converges to a distribution on the complex plane…

We study the root distribution of some univariate polynomials satisfying a recurrence of order two with linear and quadratic polynomial coefficients. We show that the set of non-isolated limits of zeros of the polynomials is the closure of…

We design new polynomials for representing threshold functions in three different regimes: probabilistic polynomials of low degree, which need far less randomness than previous constructions, polynomial threshold functions (PTFs) with…

We study the vanishing sets of slice regular polynomials in several quaternionic variables. We obtain a geometric description of the vanishing sets in two variables, which leads to a new version of the Strong Hilbert Nullstellensatz in the…

This volume contains selected papers presented at the 24th International Workshop on Unification, UNIF2010, which was held in Edinburgh on the 14th July 2010, as part of FLoC 2010 (Federated Logic Conferences).