Related papers: On algebraic integers which are 2-Salem elements i…

We study systems of $n \geq 1$ discrete differential equations of order $k\geq1$ in one catalytic variable and provide a constructive and elementary proof of algebraicity of their solutions. This yields effective bounds and a systematic…

E. Bayer-Fluckiger gave a necessary and sufficient condition for a polynomial to be realized as the characteristic polynomial of a semisimple isometry of an even unimodular lattice, by describing the local-global obstruction, and the author…

Given a matroid or flag of matroids we introduce several broad classes of polynomials satisfying Deletion-Contraction identities, and study their singularities. There are three main families of polynomials captured by our approach:…

Suppose $\Lambda$ is a discrete infinite set of nonnegative real numbers. We say that $ {\Lambda}$ is type $1$ if the series $s(x)=\sum_{\lambda\in\Lambda}f(x+\lambda)$ satisfies a zero-one law. This means that for any non-negative…

In this paper we study ideals generated by quite general sets of 2-minors of an $m \times n$-matrix of indeterminates. The sets of 2-minors are defined by collections of cells and include 2-sided ladders. For convex collections of cells it…

For any finite reflection group $W$ on $\mathbb{R}^{N}$ and any irreducible $W$-module $V$ there is a space of polynomials on $\mathbb{R}^{N}$ with values in $V$. There are Dunkl operators parametrized by a multiplicity function, that is,…

Answering a question of Frank Calegari, we extend some of our earlier results on dimension of fixed point spaces of elements in irreducible linear groups. We consider characteristic polynomials rather than just fixed spaces.

An explicit lattice point realization is provided for the primary components of an arbitrary binomial ideal in characteristic zero. This decomposition is derived from a characteristic-free combinatorial description of certain primary…

We study linear divisibility sequences of order 4, providing a characterization by means of their characteristic polynomials and finding their factorization as a product of linear divisibility sequences of order 2. Moreover, we show a new…

Let $\Lambda$ be a finite dimensional algebra of type ${\mathbb A}_n$ over an algebraically closed field $K$ with the quiver $Q$ and let $|\Det(\Lambda)|$ be the number of the minimal right determiners of all irreducible morphisms between…

We introduce a class of ideals generated by a set of 2-minors of $m\times n$-matrix of indeterminates indexed by a pair of graphs. This class of ideals is a natural common generalization of binomial edge ideals and ideals generated by…

We determine simplicity criteria in characteristics 0 and $p$ for a ubiquitous class of iterated skew polynomial rings in two indeterminates over a base ring. One obstruction to simplicity is the possible existence of a canonical normal…

For a given integer $d\ge 1$, we consider $\binom{n+d-1}{d}$-color compositions of a positive integer $\nu$ for which each part of size $n$ admits $\binom{n+d-1}{d}$ colors. We give explicit formulas for the enumeration of such…

In this note, we consider matrices similar to $X$-form matrices, which are the matrices for which only the diagonal and the anti-diagonal elements can be different from zero. First, we give a characterization of these matrices using the…

We discuss the problem of constructing a small subset of a finite field containing primitive elements of the field. Given a finite field, $\mathbb{F}_{q^n}$, small $q$ and large $n$, we show that the set of all low degree polynomials…

The set $S(n)$ of all elementary symmetric polynomials in $n$ variables is a minimal generating set for the algebra of symmetric polynomials in $n$ variables, but over a finite field ${\mathbb F}_q$ the set $S(n)$ is not a minimal…

Departing from a class of infinite series with central binomial coefficients in the numerator and depending on a positive integer parameter, we first extend known identities to all complex parameters. Then we use various methods, including…

We study an analogue of a classical arithmetic problem over the ring of polynomials. We prove that $m = 5$ is the minimal number such that the sums of any two distinct polynomials in a set of $m$ polynomials over $\F_2[x]$ cannot all be of…

Let $A, B\subseteq \mathbb{R}^2$ be finite, nonempty subsets, let $s\geq 2$ be an integer, and let $h_1(A,B)$ denote the minimal number $t$ such that there exist $2t$ (not necessarily distinct) parallel lines,…

In this paper we characterize the set of polynomials $f\in\mathbb F_q[X]$ satisfying the following property: there exists a positive integer $d$ such that for any positive integer $\ell$ less or equal than the degree of $f$, there exists…