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Géométrie algébrique réelle. (Real algebraic geometry). (French) Zbl 0633.14016
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Bd. 12, Berlin etc.: Springer-Verlag. X, 373 p. DM 188.00 (1987).
The primary geometric objects in real algebraic geometry are algebraic varieties over real closed fields: \(V=\{x\in R^ n|\) \(P_ 1(x)=...=P_ k(x)=0\}\) (R a real closed field, \(P_ 1,...,P_ k\in R[X_ 1,...,X_ n])\). Such zero sets of polynomials can be looked at from many different points of view. Each such point of view corresponds to the choice of a structure sheaf on V. One may use polynomial functions or regular functions (i.e. polynomial functions divided by nowhere vanishing functions) or Nash functions or semi-algebraic functions, just to name a few.
Although it is certainly impossible to give an account of all the recent development in real algebraic geometry within one volume, the present book treats enough of the most important concepts to give an impression of the peculiar “real” nature of the field and to show that each one of the above mentioned different points of view contributes to a better understanding of the subject. - The geometry of real algebraic sets and (to a lesser degree) of semi-algebraic sets is the main concern of the present book. The investigation of these geometric objects requires some algebraic background (e.g., real closed fields, real algebra, real places, Nash functions) which is provided in appropriate places throughout the book. In contrast to usual commutative algebra, partial or total orders on rings or field play an essential role in real algebra. These algebraic concepts are developed strictly for the geometric purposes of the book.
On the geometric side the book begins with some elementary results about semi-algebraic sets and goes on to discuss real algebraic varieties and Nash varieties. The topology of these varieties as well as vector bundles are studied. The theorem of Nash and Tognoli is proved that (over the field of real numbers) compact \(C^{\infty}\)-manifolds are diffeomorhic to real algebraic sets. Also induced are chapters about the real spectrum (which has become an indispensable tool in real algebraic geometry) and about functions between spheres. Finally the book concludes with applications of real algebraic geometry in the theory of Witt rings (theorems of Mahé and Brumfiel).
Reviewer: N.Schwartz

14Pxx Real algebraic and real-analytic geometry
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
13J25 Ordered rings
13K05 Witt vectors and related rings (MSC2000)
06F25 Ordered rings, algebras, modules
12J25 Non-Archimedean valued fields