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On a class of valuations on \(K(X)\). (English) Zbl 0872.12007

Let \(K\) be a field and \(\nu\) be a valuation on \(K\) of rank one. Denote by \(k_\nu\) the residue field, by \(G_\nu\) the value group. The aim of the paper is to describe all valuations \(u\) on \(K(X)\) which extend \(\nu\). To achieve the goal the authors use the general theory of extension of a valuation on \(K\) to a valuation on \(K(X)\), developed by the authors and also V. Alexandru, L. Popescu, C. Vraciu. Let \(w\) be an extension of \(\nu\) to \(K(X)\). The valuation \(w\) is said to be a residual transcendental extension of \(\nu\) if \(k_w/k_\nu\) is a transcendental extension. An element \((a,\delta)\) of \(K\times G\), where \(G\) is an ordered abelian group containing \(G_\nu\), will be called “a pair”. Denote by \(w_{(a,b)}\) the valuation on \(\overline{K}(X)\) defined in the following manner. If \(f\in K[X]\), \(f=a_0+a_1(X-a)+\dots+ a_n(X-a)^n\), then \(w_{(a,b)}(f)= \inf(\nu(a_i)+ i\delta)\). If \(\delta\in G_\nu\), then \(w_{(a,b)}\) is a valuation on \(\overline{K}(X)\) which may be canonically extended to \(\overline{K}(X)\). A pair \((a,\delta)\) from \(\overline{K}\times G\) is called minimal with respect to \(K\) if for every \(b\in\overline{K}\), the inequality \([K(b):K]< [K(a):K]\) implies \(\nu(a-b)<\delta\). It is shown that if \(w\) is a residual transcendental extension of \(\nu\) to \(\overline{K}(X)\), then there exists a pair \((a,\delta)\), minimal with respect to \(K\), such that \(w\) coincides with the restriction of \(w_{(a,b)}\) to \(K(X)\).
The minimal pairs and the minimal polynomials of the first component of minimal pairs play an important role in the extension of a valuation \(\nu\) on \(K\) to a valuation \(u\) on \(K(X)\). The main result of the paper under review deals with the case \(\text{rank }\nu=1\) and \(\text{rank }u=2\). The valuations on \(k(X,Y)\) of rank two and trivial on \(k\) are also investigated.
Reviewer: G.Pestov (Tomsk)

MSC:

12J25 Non-Archimedean valued fields
12F20 Transcendental field extensions
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