Real algebraic geometry. Transl. from the French. Rev. and updated ed. (English) Zbl 0912.14023

Bochnak, Jacek; Coste, Michel; Roy, Marie-Françoise

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.

Reviewer: E.I.Shustin (Tel Aviv)

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

Keywords:

real closed fields; orders on rings or fields; real algebraic set; Nash functions; Hilbert’s 17th problem; Witt rings; semi-algebraic sets