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Semi-algebraic geometry. (Semialgebraische Geometrie.) (German) Zbl 0868.14031
A subset of $$\mathbb{R}^n$$ is said to be semi-algebraic if it can be defined by finitely many polynomial inequalities. These sets form the geometric foundation of semi-algebraic geometry. According to the paper, ‘semi-algebraic geometry is the geometry of semi-algebraic sets’. This is a very narrow view. Frequently the investigation of semi-algebraic subsets of $$\mathbb{R}^n$$ requires the use of semi-algebraic sets defined over arbitrary real closed fields or even of real spectra belonging to arbitrary rings. These more general concepts are also an indispensable and legitimate part of semi-algebraic geometry. Focussing on semi-algebraic subsets of $$\mathbb{R}^n$$, the attention of the paper is concentrated on the most concrete part of the theory. It is a lively and inspiring introduction exhibiting connections with several other fields in mathematics, such as model theory, algebraic topology, differential geometry, differential topology and integral geometry. Being a survey, it contains many remarkable results and only few proofs (or indications thereof). The main topics are:
– the finitary description of semi-algebraic sets (including model theoretic methods), in particular descriptions by few inequalities,
– topological and metric finiteness results growing out of the finitary description, and
– topological characterizations and properties of real algebraic varieties and semi-algebraic sets.

##### MSC:
 14P10 Semialgebraic sets and related spaces 14P25 Topology of real algebraic varieties 14P05 Real algebraic sets