Approximation types describing extensions of valuations to rational function fields

We introduce the notion of approximation type for the partial, and in certain cases the total description of extensions of a given valuation from a field K to the rational function field K ( x ). To every extension, a unique approximation type of x over K is associated, while x may be the limit of m...

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Bibliographic Details
Published in:Mathematische Zeitschrift 2022-07, Vol.301 (3), p.2509-2546
Main Author: Kuhlmann, F.-V.
Format: Article
Language:English
Subjects:
Approximation
Mathematical analysis
Mathematics
Mathematics and Statistics
Rational functions
Topology
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Summary:We introduce the notion of approximation type for the partial, and in certain cases the total description of extensions of a given valuation from a field K to the rational function field K ( x ). To every extension, a unique approximation type of x over K is associated, while x may be the limit of many pseudo Cauchy sequences. Approximation types also provide information in cases where the extensions are not immediate, and we prove that they correspond bijectively to the extensions when K is algebraically closed or, more generally, lies dense in its algebraic closure with respect to the topology induced by the valuation.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-022-02980-2