Related papers: Divisibility and Arithmetic Properties of a Class …

We study an infinite class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. This generalizes a sequence of sparse polynomials which arises in a natural way as graph theoretic…

In this article, we give an account of some recent irreducibility testing criteria for polynomials having integer coefficients over the field of rational numbers.

Numerous results on self-reciprocal polynomials over finite fields have been studied. In this paper we generalize some of these to a-self reciprocal polynomials defined in [4]. We consider some properties of the divisibility of a-reciprocal…

Divisibility sequences are defined by the property that their elements divide each other whenever their indices do. The divisibility sequences that also satisfy a linear recurrence, like the Fibonacci numbers, are generated by polynomials…

In this paper, we study properties of polynomials over division rings. Moreover, we present formulas for finding roots of some polynomials

We introduce a subclass of linear recurrence sequences which we call poly-rational sequences because they are denoted by rational expressions closed under sum and product. We show that this class is robust by giving several…

Polynomials commute under composition are referred to as commuting polynomials. In this paper, we study division properties for commuting polynomials with rational (and integer) coefficients. As a consequence, we show an algebraic…

Motivated by a question of van der Poorten about the existence of infinite chain of prime numbers (with respect to some base), in this paper we advance the study of sequences of consecutive polynomials whose coefficients are chosen…

We examine the convergence properties of sequences of nonnegative real numbers that satisfy a particular class of recursive inequalities, from the perspective of proof theory and computability theory. We first establish a number of results…

In this article, we prove some factorization results for several classes of polynomials having integer coefficients, which in particular yield several classes of irreducible polynomials. Such classes of polynomials are devised by imposing…

In this paper, we study some properties of associated sequences of special polynomials. From the properties of associated sequences of polynomials, we derive some interesting identities of special polynomials.

We address some questions concerning indecomposable polynomials and their spectrum. How does the spectrum behave via reduction or specialization, or via a more general ring morphism? Are the indecomposability properties equivalent over a…

Simply put, a sparse polynomial is one whose zero coefficients are not explicitly stored. Such objects are ubiquitous in exact computing, and so naturally we would like to have efficient algorithms to handle them. However, with this compact…

Using polynomial evaluation, we give some useful criteria to answer questions about divisibility of polynomials. This allows us to develop interesting results concerning the prime elements in the domain of coefficients. In particular, it is…

We study the representations of a class of non-commutative polynomial algebras truncated at degree 3, with one additional relation. We determine the irreducible components of their varieties of representations. We do this by showing that…

We consider properties of polynomials with coefficients in division rings. A theorem on the decomposition of a polynomial with coefficients in an arbitrary division ring is obtained. It is shown that if a non-central element is not a root…

Let $S$ be a rational fraction and let $f$ be a polynomial over a finite field. Consider the transform $T(f)=\operatorname{numerator}(f(S))$. In certain cases, the polynomials $f$, $T(f)$, $T(T(f))\dots$ are all irreducible. For instance,…

Let $S \subset R$ be an arbitrary subset of a unique factorization domain $R$ and $\K$ be the field of fractions of $R$. The ring of integer-valued polynomials over $S$ is the set $\mathrm{Int}(S,R)= \{ f \in \mathbb{K}[x]: f(a) \in R\…

We establish how the coefficients of a sparse polynomial system influence the sum (or the trace) of its zeros. As an application, we develop numerical tests for verifying whether a set of solutions to a sparse system is complete. These…

We consider each of the three classes of representations of cyclic groups that arise in the study of rational sphere maps. We study the possible number of terms for invariant polynomials with non-negative coefficients that are constant on…