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Real algebraic geometry. Transl. from the French. Rev. and updated ed. (English) Zbl 0912.14023
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 36. Berlin: Springer. ix, 430 p. (1998).
This is a revision, updating and translation into English of the authors’ previous book “Géométrie algébrique réelle” (1987; Zbl 0633.14016). The present edition presents basically the same material appending new results and developments over the last ten years. In particular, the chapters on Nash functions and real places are extended, the chapter on Hilbert’s 17th problem contains the recent Bröcker-Scheiderer theorem that a $$d$$-dimensional open semi-algebraic set can be defined by at most $$d$$ inequalities. The bibliography covers most of the publications in real algebraic geometry till 1998.

##### MSC:
 14Pxx Real algebraic and real-analytic geometry 14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry 12J15 Ordered fields 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 11E81 Algebraic theory of quadratic forms; Witt groups and rings 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 19E99 $$K$$-theory in geometry 19G12 Witt groups of rings 32C05 Real-analytic manifolds, real-analytic spaces 55R50 Stable classes of vector space bundles in algebraic topology and relations to $$K$$-theory 57N80 Stratifications in topological manifolds 58A07 Real-analytic and Nash manifolds 58A35 Stratified sets
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What is the topology on the set of field orders