Distance functions on varieties over non-archimedian local fields
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).
variety over a complete nonarchimedean field
scientific article; zbMATH DE number 1130578
scientific article