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Notes on the space of real places of a formally real field. (English) Zbl 0869.12002
Broglia, Fabrizio (ed.) et al., Real analytic and algebraic geometry. Proceedings of the international conference, Trento, Italy, September 21-25, 1992. Berlin: Walter de Gruyter. 21-46 (1995).
The objects of interest of the paper are the space of real places $$M(K)$$, the holomorphy ring $$H(K)$$ and the real spectrum Sper$$(K)= X(K)$$ of a formally real field $$K.$$ In the first two sections the authors discuss basic results, as well as various relations between these objects. Among others, the reader finds discussions on Min Sper $$H(K)$$, Max Sper $$H(K))$$ and results on the standard map $$\Lambda: X(K)\rightarrow M(K)$$ and its fibers. The main result of the third section (entitled ‘The space of real places in real algebraic geometry’) is an extended version of an unpublished result of L. Bröcker on the fibers of the center map. The last section is concerned with the connected components of $$M(K).$$ It includes a result of H.-W. Schülting [Commun. Algebra 10, 1239-1284 (1982; Zbl 0509.14026)] that says that the space of connected components $$\pi_0(M(K))$$ is homeomorphic to the maximal real spectrum of the Witt ring of $$H(K).$$
For the entire collection see [Zbl 0812.00016].

##### MSC:
 12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) 12J15 Ordered fields 14P99 Real algebraic and real-analytic geometry