Related papers: On Certain Divisibility Property of Polynomials

Let $D$ be a domain with fraction field $K$, and let $M_n(D)$ be the ring of $n \times n$ matrices with entries in $D$. The ring of integer-valued polynomials on the matrix ring $M_n(D)$, denoted ${\rm Int}_K(M_n(D))$, consists of those…

We introduce the notion of numerical functors to generalise Eilenberg & MacLane's polynomial functors to modules over a binomial base ring. After shewing how these functors are encoded by modules over a certain ring, we record a precise…

We study Ore localisation of differential graded algebras. Further we define dg-prime rings, dg-semiprime rings, and study the dg-nil radical of dg-rings. Then, we define dg-essential submodules, dg-uniform dimension, and apply all this to…

We give a criterion when a polynomial $x^n-g$ is irreducible over a pseudofinite field. As an application we give an explicit description of algebraic closure of some pseudofinite fields of zero characteristic.

Let $D$ be a principal ideal domain with infinite spectrum such that for every nonzero prime ideal $M$ of $D$, the residue field $D/M$ is finite. Let $K$ be the quotient field of $D$. We investigate sets of lengths in the ring of…

The purpose of this note is to introduce a multiplication on the set of homogeneous polynomials of fixed degree d, in a way to provide a duality theory between monomial ideals of K[x_1,\ldots,x_d] generated in degrees \leq n and block…

In characteristic zero, Zinovy Reichstein and the author generalized the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative…

We classify the polynomials with integral coefficients that, when evaluated on a group element of finite order $n$, define a unit in the integral group ring for infinitely many positive integers $n$. We show that this happens if and only if…

Let $\mathbb{F}_q[t]$ denote the ring of polynomials over $\mathbb{F}_q$, the finite field of $q$ elements. We prove an estimate for fractional parts of polynomials over $\mathbb{F}_q[t]$ satisfying a certain divisibility condition…

The first author introduced a sequence of polynomials (\cite{8}, sequence A174531) defined recursively. One of the main results of this study is proof of the integrality of its coefficients.

A polynomial ring with rational coefficients is an irreducible representation of Lie algebras of endomorphisms of exterior powers of a infinite countable dimensional $\mathbb{Q}$-vector space. We give an explicit description of it, using…

$\DeclareMathOperator{\Int}{Int}\DeclareMathOperator{\IntR}{Int{}^\text{R}}$For a domain $D$, the ring $\Int(D)$ of integer-valued polynomials over $D$ is atomic if $D$ satisfies the ascending chain condition on principal ideals. However,…

Following work of Mazur-Tate and Satoh, we extend the definition of division polynomials to arbitrary isogenies of elliptic curves, including those whose kernels do not sum to the identity. In analogy to the classical case of division…

The aim of this paper is twofold. The first part is concerned with the associated and the so-called co-polynomials, i.e. new sequences obtained when finite perturbations of the recurrence coefficients are considered. In the second part we…

We extend results of Denef, Zahidi, Demeyer and the second author to show the following. (1) Rational integers have a single-fold Diophantine definition over the ring of integral functions of any function field of characteristic 0. (2)…

We state a kind of Euclidian division theorem: given a polynomial P(x) and a divisor d of the degree of P, there exist polynomials h(x),Q(x),R(x) such that P(x) = h(Q(x)) +R(x), with deg h=d. Under some conditions h,Q,R are unique, and Q is…

We establish a novel connection between the central binomial coefficients $\binom{2n}{n}$ and Gould's sequence through the construction of a specialized multivariate polynomial quotient ring. Our ring structure is characterized by ideals…

In this paper, we give some results on closed polynomials and factorially closed polynomial in $n$ variables. In particular, we give a characterization of factorially closed polynomials in $n$ variables over an algebraically closed field…

The set of periodic distributions, with usual addition and convolution, forms a ring, which is isomorphic, via taking a Fourier series expansion, to the ring ${\mathcal{S}}'({\mathbb{Z}}^d)$ of sequences of at most polynomial growth with…

We introduce the notion of multiplication kernels of birational and $D$-module type and give various examples. We also introduce the notion of a semi-classical multiplication kernel associated with an integrable system and discuss its…