A First Look at Semialgebraic Geometry

This chapter can be viewed as an introduction to the book and as motivation for the problems considered. It contains almost no proofs. In Section 1 we introduce the Tarski-Seidenberg theorem in several forms which are practical and sufficiently general, without entering too far into the terminology of model theory. In the next section we discuss some typical problems of semialgebraic geometry, trying to show how the topic of this book -the description of semialgebraic and more general sets by few generators- fits into the theory that we develop. For this kind of complexity problem we introduce in Section 3 the unifying terminology of real spaces, the spaces which occur in various contexts like semialgebraic geometry. semianalytic geometry, real spectra of rings and spaces of orderings of fields. This book deals with the relations between these. However, in Section 4 we first look at typical examples and illustrations in the semialgebraic situation. So this section is mostly recommended for motivation.