Distance functions on varieties over non-archimedian local fields
type: Item
instance of: scholarly article
title: Distance functions on varieties over non-archimedian local fields
zbMATH Open document ID: 0920.14008
author: José Felipe Voloch
published in: Rocky Mountain Journal of Mathematics
publication date: 1998-05-03T00:00:00Z
full work available at URL: CONT27-2.html
review text: For a variety \(X\) over a complete non-archimedean field \(K\), the author defines the distance of an \({\mathfrak O}\)-valued point \(P\) of \(X\) (where \({\mathfrak O}\) is the valuation ring of \(K)\) to a given subvariety \(Y\) of \(X\) as the minimal \(\varepsilon\leq 1\) such that the reduction \(P_\varepsilon\) of \(P\) modulo the ideal \(\{x\in K: | x|< \varepsilon\}\) lies in the reduction \(Y_\varepsilon\) (for fixed models of \(X\) and \(Y\) over \({\mathfrak O})\). He proves several properties of this distance and applies it to show that in some cases a subvariety \(Y\) of \(X\) without \(K\)-rational point can be approximated by \(K\)-rational points of \(X\); here \(K=k(t)\), \(k\) an algebraically closed field, and the valuation comes from the place (0).
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14F45
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 11G25
zbMATH DE Number: 1130578
zbMATH Keywords: distance function
zbMATH Keywords: variety over a complete nonarchimedean field
zbMATH Keywords: approximation by rational points
reviewed by: Frank Herrlich
MaRDI profile type: Publication
OpenAlex ID: W2016636920
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DOI: 10.1216/RMJM/1181071930
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description: scientific article; zbMATH DE number 1130578