The space of real places on \(\mathbb{R}(x; y)\)
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 possible value groups \(v(\mathbb R(x, y))\) when \(v\) runs across \(M(\mathbb R(x, y))\), and for each \(v \in M(\mathbb R(x, y))\), it proves that the set of those \(w \in M(\mathbb R(x, y))\), for which \(w(\mathbb R(x, y)) = v(\mathbb R(x, y))\), is dense in \(M(\mathbb R(x, y))\). For the proof of the main theorem, the authors show that: \(\mathbb R(x, y))\) has a retraction on a simple closed curve; each fiber of this retraction is homeomorhic to \(M(\mathbb R((x))(y))\) (here \(\mathbb R((x))(y)\) is the rational function field in \(y\) over the Laurent formal power series field \(\mathbb R((x))\)). This enables one to deduce the main result from the assertion that \(M(\mathbb R((x))(y))\) is path-connected. A key tool used for proving the assertion is a homeomorphism between \(M(\mathbb R((x))(y))\) and a certain space of sequences related to the ``signatures'' of the first author's paper [Trans. Am. Math. Soc. 174, 451--488 (1972; Zbl 0258.12103)], which themselves are shown in the present paper to be related to ``the strict systems of polynomial extensions'' in another article by the same author (see [Commun. Algebra 37, No. 7, 2169--2183 (2009; Zbl 1205.12006)]).
scientific article; zbMATH DE number 6946309
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