Related papers: Minimal polynomial dynamics on the set of 3-adic i…

We discuss asymptotics of the zeros of orthogonal polynomials on the real line and on the unit circle when the recursion coefficients are periodic. The zeros on or near the absolutely continuous spectrum have a clock structure with spacings…

This paper proves that the characteristic polynomial is a complete unitary invariant for pairs of projection matrices. Some special cases involving three or more projections are also considered.

The Bernstein polynomials with integer coefficients do not generally preserve monotonicity and convexity. We establish sufficient conditions under which they do. We also observe that they are asymptotically shape preserving.

We consider the problem of computing matrix polynomials $p(X)$, where $X$ is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let $\Pi_{2^{m}}^*$ represent the set of polynomials computable with…

We characterize all the multiple orthogonal threefold symmetric polynomial sequences whose sequence of derivatives is also multiple orthogonal. Such a property is commonly called the Hahn property and it is an extension of the concept of…

In these notes we investigate the rings of real polynomials in four variables, which are invariant under the action of the reflectiongroups [3,4,3] and [3,3,5]. It is well known that they are rationally generated in degree 2,6,8,12 and…

We study Nivat's conjecture on algebraic subshifts and prove that in some of them every low complexity configuration is periodic. This is the case in the Ledrappier subshift (the 3-dot system) and, more generally, in all two-dimensional…

The Ehrhart polynomial of an integral convex polytope counts the number of lattice points in dilates of the polytope. In math.CO/0402148, the authors conjectured that for any cyclic polytope with integral parameters, the Ehrhart polynomial…

We find the smallest characteristic factor and a limit formula for the multiple ergodic averages associated to any family of three polynomials and polynomial families of the form $\{l_1p,l_2p,...,l_kp\}$. We then derive several multiple…

Let $f(x) \in \bbz[x]$ and consider the index divisibility set $D = \{n \in \bbn : n \mid f^n(0)\}$. We present a number of properties of $D$ in the case that $(f^n(0))_{n=1}^\infty$ is a rigid divisibility sequence, generalizing a number…

We numerically investigate the dynamics of orbits in 3D circumbinary phase-space as a function of binary eccentricity and mass fraction. We find that inclined circumbinary orbits in the elliptically-restricted three-body problem display a…

In this paper, we obtain several new classes of irreducible polynomials having integer coefficients whose zeros lie inside an open disk around the origin or outside a closed annular region in the complex plane. Such irreducible polynomials…

Let $P$ be a bounded convex subset of $\mathbb R^n$ of positive volume. Denote the smallest degree of a polynomial $p(X_1,\dots,X_n)$ vanishing on $P\cap\mathbb Z^n$ by $r_P$ and denote the smallest number $u\geq0$ such that every function…

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…

Hyperbolic polynomials are real polynomials whose real hypersurfaces are nested ovaloids, the inner most of which is convex. These polynomials appear in many areas of mathematics, including optimization, combinatorics and differential…

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…

We present a new property of the trigonometric system arranged in a natural order. It is shown that the sequence of subspaces of trigonometric polynomials is optimal in the sense of order of convergence on convolution classes K*Up in Lq for…

In this paper we investigate analytic properties of sextet polynomials of hexagonal systems. For the pyrene chains, we show that zeros of the sextet polynomials $P_n(x)$ are real, located in the open interval $(-3-2\sqrt{2},-3+2\sqrt{2})$…

For a class of polynomials $f \in \mathbb{Z}[X]$, which in particular includes all quadratic polynomials, and also trinomials of some special form, we show that, under some natural conditions (necessary for quadratic polynomials), the set…

We survey all known examples of finite posets whose order polynomials have product formulas, and we propose the heuristic that these are the same posets with good dynamical behavior. Here the dynamics in question are the actions of…