Irreducibility of integer-valued polynomials I

TL;DR

This paper investigates the conditions under which integer-valued polynomials over subsets of a unique factorization domain are irreducible, introducing $d$-sequences as a key tool for analysis and extending results to Dedekind domains.

Contribution

It introduces a new method using $d$-sequences to determine irreducibility of integer-valued polynomials over arbitrary subsets of UFDs and proposes a generalization to Dedekind domains.

Findings

Developed criteria for polynomial irreducibility using $d$-sequences.

Constructed explicit $d$-sequences in special cases.

Suggested a framework for extending results to Dedekind domains.

Abstract

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 $\mathrm{Int}(S,R)= \{ f \in \mathbb{K}[x]: f(a) \in R\ \forall\ a \in S \}.$ This article is an effort to study the irreducibility of integer-valued polynomials over arbitrary subsets of a unique factorization domain. We give a method to construct special kinds of sequences, which we call $d$-sequences. We then use these sequences to obtain a criteria for the irreducibility of the polynomials in $\mathrm{Int}(S,R).$ In some special cases, we explicitly construct these sequences and use these sequences to check the irreducibility of some polynomials in $\mathrm{Int}(S,R).$ At the end, we suggest a generalization of our results to an arbitrary subset of a Dedekind domain.