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Integral-valued rational functions on valued fields. (English) Zbl 0771.12003
The paper contains extracts from the second author’s thesis written under the supervision of the first author. Let $$K$$ be a field, $$v$$ a non- trivial valuation ring of $$K$$, $${\mathfrak m}$$ the maximal ideal of $$v$$, $$(\widehat{K,v})$$ the completion of the valued field $$(K,v)$$ as a uniform space, $$n$$ a natural number. Denote by $$v[X]_{\text{sub}}:=\{f\in K[X]\mid f(v^ n)\subset v\}$$ the ring of integral-valued polynomials and by $$v(X)_{\text{sub}}:=\{{g\over h}\mid g,h\in K[X]$$ and $${g\over h}(a)\in v$$ for all $$a\in v^ n$$ such that $$h(a)\neq 0\}$$ the ring of integral-valued rational functions. It is proved that for rational functions of one variable the identity $v(X)_{\text{sub}}=v[X]_{\text{sub}}(1+{\mathfrak m}v[X]_{\text{sub}})^{-1}$ holds iff the completion $$(\widehat {K,v})$$ of $$(K,v)$$ is locally compact or algebraically closed. Using a theorem from [P.-J. Cahen and J. L. Chabert, Bull. Sci. Math., II. Sér. 95, 295-304 (1971; Zbl 0221.13006)] the authors prove that the identity $$v(X)_{\text{sub}}=v[X](1+{\mathfrak m}v[X])^{-1}$$ holds in any number of variables $$X=(X_ 1,\dots,X_ n)$$ if the completion $$(\widehat {K,v})$$ is algebraically closed. Conversely, if this identity holds for $$n=1$$, then $$(\widehat {K,v})$$ is algebraically closed.
Reviewer: G.Pestov (Tomsk)

##### MSC:
 12J10 Valued fields 12J12 Formally $$p$$-adic fields
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##### References:
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