Valuations defined by Ostrowski nets
A family of elements \(\{a_ \beta\}\) of a valued field \((F,v)\) where the indices \(\beta\) run over a well ordered set \(B\) without a largest element, is called pseudo-convergent (or Ostrowski net) if whenever \(\alpha<\beta<\lambda\), \(v(a_ \alpha-a_ \beta) <v (a_ \beta - a_ \lambda)\). Such a sequence is called a transcendental net if for all polynomials \(f(x) \in F [X]\), \(v(f(a_ \beta))\) is eventually stationary (in the sense that there exists a \(\beta\) in \(B\) such that for all \(\alpha\) in \(B\) with \(\alpha \geq \beta\), \(v(f(a_ \beta)) = v(f(a_ \alpha)))\). A sequence which is not transcendental is called algebraic. Given a transcendental net, one can extend the valuation \(v\) to \(f[X]\), by setting \(w_ 0(f) = v(f(a_{\beta_ i}))\) and then to \(F(x)\). With an algebraic net, \(v\) can be extended to a suitable algebraic extension. The main results consider valuations \(v'\) of \(K \supset F\), which extend \(v\) of \(F\) and \(v'\)-Ostrowski nets in \(K\) that are \(F\)-transcendental. An explicit formula is given for valuations on \(F(x)\) by such a net. Let \(F \subset K\) and \( v\) be a valuation. A \(v\)-Ostrowski net \(\{a_ \beta\}\) in \(K\) is \(F\)-algebraic if and only if it is \(K\)-algebraic. If \(F\) is complete with respect to the valuation \(v\), then each \(v\)-Ostrowski net which is gauge cofinal in \(v(F^*)\) is \(F\) algebraic. In case \(v\) is discrete and \((F,v)\) is complete then each \(v\)-Ostrowski net in \(F\) is \(F\)-algebraic.
scientific article; zbMATH DE number 500584