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.

中文翻译：

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整数值多项式的不可约性 I

令 $S \subset R$ 是唯一因式分解域 $R$ 的任意子集，而 $\K$ 是 $R$ 的分数域。$S$ 上的整数值多项式环是集合 $\mathrm{Int}(S,R)= \{ f \in \mathbb{K}[x]: f(a) \in R\ \forall \ a \in S \}.$ 本文旨在研究整数值多项式在唯一分解域的任意子集上的不可约性。我们给出了一种构造特殊类型序列的方法，我们称之为 $d$-sequences。然后我们使用这些序列来获得 $\mathrm{Int}(S,R) 中多项式不可约性的标准。​​$ 在某些特殊情况下，我们显式构造这些序列并使用这些序列来检查某些多项式的不可约性在 $\mathrm{Int}(S,R).$ 最后，我们建议将我们的结果推广到 Dedekind 域的任意子集。