Related papers: Para-orthogonal polynomials on the unit circle gen…
Quantum K-theory is a K-theoretic version of quantum cohomology, which was recently defined by Y.-P. Lee. Based on a presentation for the quantum K-theory of the classical flag variety Fl_n, we define and study quantum Grothendieck…
A compact subset $K$ of the complex plane $\C$ is a set of polynomial (respectively rational) approximation if $P(K)=A(K)$ (respectively $R(K)=A(K)$), where $P(K)$ (respectively $R(K)$) is the family of functions on $K$ which are uniform…
We evaluate the number of monic polynomials (of arbitrary degree $N$) the zeros of which equal their coefficients when these are allowed to take arbitrary complex values. In the following, we call polynomials with this property {\em…
Let $\Lambda$ be a finite set of nonnegative integers, and let $\mathcal P(\Lambda)$ be the linear hull of the monomials $z^k$ with $k\in\Lambda$, viewed as a subspace of $L^1$ on the unit circle. We characterize the extreme and exposed…
Using the cyclotomic identity we compute sums over d-tuples of monic polynomials in F_q[x] weighted by the multiplicity of their irreducible factors. As consequences we determine explicit expressions for the number of d-tuples of…
Let $D$ be a domain obtained by removing, out of the unit disk $\{z:|z|<1\}$, finitely many mutually disjoint closed disks, and for each integer $n\geq 0$, let $P_n(z)=z^n+\cdots$ be the monic $n$th-degree polynomial satisfying the planar…
Let $K$ be a number field generated by a root $\th$ of a monic irreducible trinomial $F(x) = x^n+ax^{m}+b \in \Z[x]$. In this paper, we study the problem of $K$. More precisely, we provide some explicit conditions on $a$, $b$, $n$, and $m$…
It was shown recently that associated with a pair of real sequences $\{\{c_{n}\}_{n=1}^{\infty}, \{d_{n}\}_{n=1}^{\infty}\}$, with $\{d_{n}\}_{n=1}^{\infty}$ a positive chain sequence, there exists a unique nontrivial probability measure…
We show that for any positive integer $h$, either $h$ or $h+1$ is a height of some cyclotomic polynomial $\Phi_n$, where $n$ is a product of three distinct primes.
In this paper, we study the theory of orthogonal trigonometric polynomials (OTP). We obtain asymptotics of OTP with positive and analytic weight functions by Riemann-Hilbert approach and find they have relations with orthogonal polynomials…
The purpose of this note is to establish, in terms of the primary coefficients in the framework of the tridiagonal theory developed by Delsarte and Genin in the environment of nonnegative definite Toeplitz matrices, necessary and sufficient…
The second author proved that the set of post-critically finite polynomials of given degree is a set of bounded height, up to change of variables. Motivated by an observation about unicritical polynomials, we complement this by proving that…
Among all states on the algebra of non-commutative polynomials, we characterize the ones that have monic orthogonal polynomials. The characterizations involve recursion relations, Hankel-type determinants, and a representation as a joint…
In this paper we discuss a couple of observations related to polynomial convexity. More precisely, (i) We observe that the union of finitely many disjoint closed balls with centres in $\cup_{\theta\in[0,\pi/2]}e^{i\theta}V$ is polynomially…
Polynomials with values in an irreducible module of the symmetric group can be given the structure of a module for the rational Cherednik algebra, called a standard module. This algebra has one free parameter and is generated by…
We study the postcritically-finite (PCF) maps in the moduli space of complex polynomials $\mathrm{MP}_d$. For a certain class of rational curves $C$ in $\mathrm{MP}_d$, we characterize the condition that $C$ contains infinitely many PCF…
Given a polynomial map $\psi:S^m\to \mathbb{R}^k$ with components of degree $d$, we investigate the structure of the semialgebraic set $Z\subseteq S^m$ consisting of those points where $\psi$ and its derivatives satisfy a given list of…
Given an integral $d \times n$ matrix $A$, the well-studied affine semigroup $\mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\}$ can be stratified by the number of lattice points inside the parametric polyhedra $P_A(b)=\{x:…
We investigate the zeros of a family of hypergeometric polynomials $_2F_1(-n,-x;a;t)$, $n\in\nn$ that are known as the Meixner polynomials for certain values of the parameters $a$ and $t$. When $a=-N$, $N\in\nn$ and $t=\frac1{p}$, the…
The main object of study in this paper is the completion Z[q]^N=\varprojlim_n Z[q]/((1-q)(1-q^2)...(1-q^n)) of the polynomial ring Z[q], which arises from the study of a new invariant of integral homology 3-spheres with values in Z[q]^N…