Integral-valued rational functions on valued fields
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 [\textit{P.-J. Cahen} and \textit{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.
scientific article; zbMATH DE number 60098