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Integer-valued polynomials on a subset


Author: Paul-Jean Cahen
Journal: Proc. Amer. Math. Soc. 117 (1993), 919-929
MSC: Primary 13B25; Secondary 13F20, 13G05
DOI: https://doi.org/10.1090/S0002-9939-1993-1116252-5
MathSciNet review: 1116252
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Abstract: We let $ D$ be a local (noetherian) one-dimensional unibranched domain, $ K$ its quotient field, $ \mathfrak{m}$ its maximal ideal, $ {D'}$ its integral closure, and $ {\mathfrak{m}'}$ the maximal ideal of $ {D'}$. If $ E$ is a subset of $ K$, we let $ \operatorname{Int} (E,D)$ be the set of integer-valued polynomials on $ E$, thus $ \operatorname{Int} (E,D) = \{ f \in K[X]\vert f(E) \subset D\} $. For a fractional subset $ E$ of $ D$ (i.e., there is a nonzero element $ d$ of $ D$ such that $ dE \subset D$), we show that the prime ideals of $ \operatorname{Int} (E,D)$ above $ \mathfrak{m}$ are in one-to-one correspondence with the elements of the topological closure of $ E$ in the completion of $ K$ for the $ {\mathfrak{m}'}$-adic topology.


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DOI: https://doi.org/10.1090/S0002-9939-1993-1116252-5
Article copyright: © Copyright 1993 American Mathematical Society

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