Irreducibility of integer-valued polynomials in several variables

TL;DR

This paper investigates the irreducibility of integer-valued polynomials in multiple variables over various domains, extending known results to cases where the domain is a UFD or Dedekind domain, which was previously unexplored.

Contribution

It is the first study of irreducibility of multivariable integer-valued polynomials over UFDs and extends results to Dedekind domains and more general domains.

Findings

Established irreducibility criteria for multivariable integer-valued polynomials over UFDs.

Extended the validity of these criteria to Dedekind domains.

Demonstrated the applicability of results to a broader class of domains.

Abstract

Let $\S $ be an arbitrary subset of $R^n$ where $R$ is a domain with the field of fractions $\K$. Denote the ring of polynomials in $n$ variables over $\K$ by $\K[\x].$ The ring of integer-valued polynomials over $\S,$ denoted by Int$(\S,R)$, is defined as the set of the polynomials of $\K[\x],$ which maps $\S$ to $R$. In this article, we study the irreducibility of the polynomials of Int$(\S,R)$ for the first time in the case when $R$ is a Unique Factorization Domain. We also show that our results remain valid when $R$ is a Dedekind domain or sometimes any domain.