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Interpolation domains. (English) Zbl 0990.13014
Authors’ abstract: Call a domain $$D$$ with quotient field $$K$$ an interpolation domain if, for each choice of distinct arguments $$a_1, \dots, a_n$$ and arbitrary values $$c_1,\dots, c_n$$ in $$D$$, there exists an integer-valued polynomial $$f$$ (that is, $$f\in K[X]$$ with $$f(D)\subseteq (D))$$, such that $$f(a_i)= c_i$$ for $$1\leq i\leq n$$. We characterize completely the interpolation domains if $$D$$ is Noetherian or a Prüfer domain. In the first case, we show that $$D$$ is an interpolation domain if and only if it is one-dimensional, locally unibranched with finite residue fields, in the second one, if and only if the ring $$\text{Int} (D)=\{f\in K[X] \mid f(D) \subseteq D\}$$ of integer-valued polynomials is itself a Prüfer domain. We also show that an interpolation domain must satisfy a double-boundedness condition, and thereby simplify a recent characterization of the domains $$D$$ such that $$\text{Int}(D)$$ is a Prüfer domain [see K. A. Loper, Proc. Am. Math. Soc. 126, No. 3, 657-660 (1998; Zbl 0887.13010)].

##### MSC:
 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations 13G05 Integral domains
Zbl 0887.13010
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##### References:
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