Abstract:
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 distribution of operators on a Fock space.

Abstract:
This note is motivated by an old result of Kronecker on monic polynomials with integer coefficients having all their roots in the unit disc. We call such polynomials Kronecker polynomials for short. Let $k(n)$ denote the number of Kronecker polynomials of degree $n$. We describe a canonical form for such polynomials and use it to determine the sequence $k(n)$, for small values of $n$. The first step is to show that the number of Kronecker polynomials of degree $n$ is finite. This fact is included in the following theorem due to Kronecker. The theorem actually gives more: the non-zero roots of such polynomials are on the boundary of the unit disc. We use this fact later on to show that these polynomials are essentially products of cyclotomic polynomials.

Abstract:
We introduce a monic polynomial p_N(z) of degree N whose coefficients are the zeros of the N-th degree Hermite polynomial. Note that there are N! such different polynomials p_N(z), depending on the ordering assignment of the N zeros of the Hermite polynomial of order N. We construct two NxN matrices M_1 and M_2 defined in terms of the N zeros of the polynomial p_N(z). We prove that the eigenvalues of M_1 and M_2 are the first N integers respectively the first N squared-integers, a remarkable isospectral and Diophantine property. The technique whereby these findings are demonstrated can be extended to other named polynomials.

Abstract:
The notion of generations of monic polynomials such that the coefficients of the polynomials of the next generation coincide with the zeros of the polynomials of the current generation is introduced, and its relevance to the identification of endless sequences of new solvable many-body problems of "goldfish type" is demonstrated.

Abstract:
We study the stratification of the space of monic polynomials with real coefficients according to the number and multiplicities of real zeros. In the first part, for each of these strata we provide a purely combinatorial chain complex calculating (co)homology of its one-point compactification and describe the homotopy type by order complexes of a class of posets of compositions. In the second part, we determine the homotopy type of the one-point compactification of the space of monic polynomials of fixed degree which have only real roots (i.e., hyperbolic polynomials) and at least one root is of multiplicity $k$. More generally, we describe the homotopy type of the one-point compactification of strata in the boundary of the set of hyperbolic polynomials, that are defined via certain restrictions on root multiplicities, by order complexes of posets of compositions. In general, the methods are combinatorial and the topological problems are mostly reduced to the study of partially ordered sets.

Abstract:
In this paper, we consider $D=\mathbb{Z}[\theta]$, where $$\theta= \sqrt{-k} \,\,\,\, \mbox{if}\;\;\;-k\not\equiv 1 \;(\mbox{mod}\;4)\,\,\,\,\mbox{or}\,\,\,\, \theta=\frac{\sqrt{-k}+1}{2} \,\,\,\, \mbox{if}\;\;\;-k\equiv 1 \;(\mbox{mod}\;4),$$ $k\geq 2$ is a squarefree integer, and we proved that the number $R(y)$ of representations of a monic polynomial $f(x)\in \mathbb{Z}[\theta][x]$, of degree $d\geq 1$, as a sum of two monic irreducible polynomials $g(x)$ and $h(x)$ in $\mathbb{Z}[\theta][x]$, with the coefficients of $g(x)$ and $h(x)$ bounded in complex modulus by $y$, is asymptotic to $(4y)^{2d-2}$.

Abstract:
We study rational homology groups of one-point compactifications of spaces of complex monic polynomials with multiple roots. These spaces are indexed by number partitions. A standard reformulation in terms of quotients of orbit arrangements reduces the problem to studying certain triangulated spaces $X_{\lambda,\mu}$. We present a combinatorial description of the cell structure of $X_{\lambda,\mu}$ using the language of marked forests. As applications we obtain a new proof of a theorem of Arnold and a counterexample to a conjecture of Sundaram and Welker, along with a few other smaller results.

Abstract:
We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let ${\M}_n({\Z})$ denote the monic polynomials of degree $n$ with integer coefficients. A {\it monic integer Chebyshev polynomial} $M_n \in {\M}_n({\Z})$ satisfies $$ \| M_n \|_{E} = \inf_{P_n \in{\M}_n ({\Z})} \| P_n \|_{E}. $$ and the {\it monic integer Chebyshev constant} is then defined by $$ t_M(E) := \lim_{n \rightarrow \infty} \| M_n \|_{E}^{1/n}. $$ This is the obvious analogue of the more usual {\it integer Chebyshev constant} that has been much studied. We compute $t_M(E)$ for various sets including all finite sets of rationals and make the following conjecture, which we prove in many cases. \medskip\noindent {\bf Conjecture.} {\it Suppose $[{a_2}/{b_2},{a_1}/{b_1}]$ is an interval whose endpoints are consecutive Farey fractions. This is characterized by $a_1b_2-a_2b_1=1.$ Then} $$t_M[{a_2}/{b_2},{a_1}/{b_1}] = \max(1/b_1,1/b_2).$$ This should be contrasted with the non-monic integer Chebyshev constant case where the only intervals where the constant is exactly computed are intervals of length 4 or greater.

Abstract:
We prove a result on the representation of squares by second degree polynomials in the field of $p$-adic meromorphic functions in order to solve positively B\"uchi's $n$ squares problem in this field (that is, the problem of the existence of a constant $M$ such that any sequence $(x_n^2)$ of $M$ - not all constant - squares whose second difference is the constant sequence $(2)$ satisfies $x_n^2=(x+n)^2$ for some $x$). We prove (based on works by Vojta) an analogous result for function fields of characteristic zero, and under a Conjecture by Bombieri, an analogous result for number fields. Using an argument by B\"uchi, we show how the obtained results improve some theorems about undecidability for the field of $p$-adic meromorphic functions and the ring of $p$-adic entire functions.

Abstract:
We study the problem of finding nonconstant monic integer polynomials, normalized by their degree, with small supremum on an interval I. The monic integer transfinite diameter t_M(I) is defined as the infimum of all such supremums. We show that if I has length 1 then t_M(I) = 1/2. We make three general conjectures relating to the value of t_M(I) for intervals I of length less that 4. We also conjecture a value for t_M([0, b]) where 0 < b < 1. We give some partial results, as well as computational evidence, to support these conjectures. We define two functions that measure properties of the lengths of intervals I with t_M(I) on either side of t. Upper and lower bounds are given for these functions. We also consider the problem of determining t_M(I) when I is a Farey interval. We prove that a conjecture of Borwein, Pinner and Pritsker concerning this value is true for an infinite family of Farey intervals.