The space of real places on \(\mathbb{R}(x; y)\)
type: Item
instance of: scholarly article
title: The space of real places on \(\mathbb{R}(x; y)\)
zbMATH Open document ID: 1407.12001
author: Ron Brown
author: Jonathan Lee Merzel
published in: Annales Mathematicae Silesianae
publication date: 2018-10-02T00:00:00Z
review text: For any field \(F\), let \(M(F)\) be the space of real places of \(F\), i.e. places from \(F\) to the field \(\mathbb R\) of real numbers. This space is important for understanding the structure of the reduced Witt ring of quadratic forms over \(F\) (see [\textit{R. Brown} and \textit{M. Marshall}, Rocky Mt. J. Math. 11, 161--175 (1981; Zbl 0452.10022)]). It also plays a role in real algebraic geometry; given a formally real field \(F\), the points of \(M(F)\) correspond to the closed points of the real spectrum of the real holomorphy ring of \(F\) (see [\textit{H.-W. Schülting}, Contemp. Math. 8, 289--295 (1982; Zbl 0489.14012)]. It is known (see, e.g., Section 2 of [\textit{T. Y. Lam}, Orderings, valuations and quadratic forms. Providence, RI: American Mathematical Society (AMS) (1983; Zbl 0516.12001)] that \(M(F)\) has a structure of a topological space, such that the Harrison sets \(H(a) = \{\tau \in M(F): 0 < \tau (a) < \infty \}\), \(a \in F\), form a subbasis for the topology (see [\textit{K. Kuhlmann} and \textit{F.-V. Kuhlmann}, Fundam. Math. 218, 121--149 (2012; Zbl 1262.12002)]. When \(F\) is an algebraic function field in one variable over \(\mathbb R\), the structure of \(M(F)\) is well understood and the obtained results have an algebro-geometric flavour (see [\textit{M. Knebusch}, Math. Z. 150, 49--70 (1976; Zbl 0327.14009)]. Presently, no analogous results seem to be known for algebraic function fields in \(2\) variables over \(\mathbb R\), and even for the rational function field \(\mathbb R(x, y)\). It is known that \(M(\mathbb R(x, y))\) is compact, Hausdorff and connected (page 5 of [\textit{D. W. Dubois}, Diss. Math. 69, 43 pp. (1970; Zbl 0266.12103)], and Theorem 2.12 of [\textit{J. Harman}, Contemp. Math. 8, 141--174 (1982; Zbl 0509.12021)] but not whether it contains a disk (and much less whether it contains a torus). The main theorem of paper under review states that \(M(\mathbb R(x, y))\) is path-connected. The paper also determines the pos…
reviewed by: Ivan D. Chipchakov
Mathematics Subject Classification ID: 12D15
Mathematics Subject Classification ID: 12E30
Mathematics Subject Classification ID: 12J20
Mathematics Subject Classification ID: 14P99
zbMATH DE Number: 6946309
zbMATH Keywords: Real place
zbMATH Keywords: space of real places
zbMATH Keywords: strict system of polynomial extensions
zbMATH Keywords: Harrison set
zbMATH Keywords: path-connected
zbMATH Keywords: dense subset
MaRDI profile type: Publication
full work available at URL: amsil-2017-0017
OpenAlex ID: W2789608699
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DOI: 10.1515/AMSIL-2017-0017
description: scientific article; zbMATH DE number 6946309