Irreducibility of integer-valued polynomials in several variables

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.