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## Inhaltsverzeichnis

### Introduction

Abstract
Let A be the ring of polynomials in n indeterminates over ℝ. Then any subset S of ℝ n which is the solution set of a polynomial system ƒ1(x) > 0, …, ƒ k (x) > 0 is also the solution set of a system of n inequalities g1(x) > 0, … g n (x) > 0, no matter how big k is. This observation, made about twelve years ago for n ≤ 3 and proved in full generality five years later is the starting point of the present book.
Carlos Andradas, Ludwig Bröcker, Jesús M. Ruiz

### Chapter I. A First Look at Semialgebraic Geometry

Summary
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.
Carlos Andradas, Ludwig Bröcker, Jesús M. Ruiz

### Chapter II. Real Algebra

Summary
We consider general properties of the real spectrum of a commutative ring with unit. In Sections 1 and 2 we collect the basic facts, and for more information we refer to [B-C-R] and [Kn-Schd]. In Section 3 valuation theory enters the scene. It is of fundamental importance for the whole work. As an application we obtain in Section 4 the first results on going-up and going-down in the real spectrum. In Section 5 we present the notion and basic properties of abstract semialgebraic functions on constructible sets of the real spectrum; this is done in two equivalent ways which have both their advantages. These functions are used in Section 6 to construct cylindrical decompositions with respect to systems of polynomials. Finally, in Section 7 we introduce real strict localizations, which are the real analogues of the strict localizations used in etale cohomology.
Carlos Andradas, Ludwig Bröcker, Jesús M. Ruiz

### Chapter III. Spaces of Signs

Summary
Spaces of signs are defined by imposing four suplementary axioms on real spaces; then, spaces of orderings are a special class of these spaces of signs. This is done in Section 1, where we also define subspaces and draw the first consequences of our definitions. Section 2 contains the fundamental properties of forms, mainly in the case of spaces of orderings. The important notion of fan is introduced in Section 3, together with its elementary properties. In Section 4 we consider local spaces of orderings and localizations, which behave very much like in ring theory. Localizations are used in Section 5 to show that the real space associated to a commutative ring with unit is actually a space of signs, and also in Section 6, to prove that the subspaces of a space of signs are again spaces of signs.
Carlos Andradas, Ludwig Bröcker, Jesús M. Ruiz

### Chapter IV. Spaces of Orderings

Summary
This chapter contains a fully detailed presentation of the theory of spaces of orderings. In Section 1 we reformulate the notion of space of orderings to stress the connection with the duality of topological groups and the theory of reduced quadratic forms. Section 2 is devoted to sums and extensions of spaces of orderings and their basic properties. In Section 3 we introduce spaces of finite type and their trees, which support the use of induction in many proofs. The fundamental fact that the chain length of a space of orderings is bigger than or equal to that of any subspace is proved in Section 4. Also in this section we define solid fans, impervious fans, and places, which are essential to prove in Section 5 that finite chain length is equivalent to finite type. This is the key technical result of the theory. We prove in Section 6 the local-global principle that reduces problems on forms from the whole space to its finite subspaces. In Section 7 we draw the consequences: the representation theorem, the generation formula and the stability formula. Using these, we bound each invariant s, $\bar s$, t, $\bar t$, w, l in terms of any of the others. In particular, all of them are finite or infinite at a time. The final result of the section and the chapter is a local-global separation principle.
Carlos Andradas, Ludwig Bröcker, Jesús M. Ruiz

### Chapter V. The Main Results

Summary
In an arbitrary space of signs X, generation of basic sets, stability indices, representation of functions by signatures and separation of closed sets reduce to the corresponding problems in the spaces of orderings V* associated to the subvarieties V of X. This general principle is proved in Sections 1, 2 and 3. Also, in Sections 1 and 2, we obtain criteria for a set to be basic open or to be principal open, and extend to X the inequalities among the invariants s, $\bar s$, t, $\bar t$, w and l, which were already known for spaces of orderings. Section 4 is devoted to the notions of real divisor and regularity in X. Using them we can bound from below s and $\bar s$. Moreover, we compare basicness of an open set and its closure, resp. of a closed set and its interior. Finally, Artin-Lang spaces are introduced in Section 5, jointly with the tilde operator: this is the notion that makes the abstract theory fruitful of geometric applications.
Carlos Andradas, Ludwig Bröcker, Jesús M. Ruiz

### Chapter VI. Spaces of Signs of Rings

Summary
In view of the global stability formulae (Corollary V.1.6), and the canonical decomposition (III.1.9), the computation of stability indices in the space of signs of a ring reduces to estimations of the size of fans of residue fields of that ring. We obtain in Section 1 such estimations, via real valuations, after proving the so-called trivialization theorem for fans. Then, in Section 2, we deduce upper bounds for the stability index of a field extension in terms of the ground field. These bounds are sharp when the ground field is real closed or the rational numbers field, as follows from the lower bounds discussed in Section 3. In section 4 we generalize the previous upper bounds to algebras. The results are specially good for algebras over a field, which are the matter of Section 5; again, we obtain the best estimations over a real closed field and over the rationals. Section 6 is devoted to totally archimedean rings, which are the abstract counterparts of compact spaces. These rings have two special features: firstly, their complexity bounds are low, and, secondly, generation of basic sets and separation are characterized by multilocal conditions. We end in Section 7 with the translation to concrete semialgebraic geometry of most of the abstract results obtained so far. We as well discuss several examples and counterexamples to questions raised in earlier chapters.
Carlos Andradas, Ludwig Bröcker, Jesús M. Ruiz

### Chapter VII. Real Algebra of Excellent Rings

Summary
This chapter is devoted to excellent rings, and contains the results that allow to extend what we have already seen for semialgebraic sets to semianalytic sets. In Sections 1 and 2 we collect the commutative algebra needed later. Very few proofs are given, since almost everything can be found in our general references [Mt] and the more elementary [At-Mc], [Bs-Is-Vg]; an important exception is our proof that local-ind-etale limits of excellent rings are again excellent. In addition, we state without proof the fundamental Rotthaus’s theorem on M. Artin’s approximation property. In Section 3 we characterize the extension of prime cones under completion, a crucial result for all that follows. The curve selection lemma which is proved in Section 4 has many important applications: existence theorems for valuations and fans (Section 5), and constructibility of closures (Section 6) are some. It is also needed in Section 7 for the proof of another key theorem: the real going-down for regular homomorphisms. After this, we characterize local constructibility of connected components in Section 8.
Carlos Andradas, Ludwig Bröcker, Jesús M. Ruiz

### Chapter VIII. Real Analytic Geometry

Summary
In this chapter we apply all the previous results to the study of semianalytic sets in real analytic manifolds. In Section 1 we settle the terminology concerning global analytic functions and sets. Sections 2 and 3 are devoted to the local theory, that is, to germs at points. We review there several classical results in the framework of real spaces, with some technical suplements that will be needed later. In Section 4 we obtain the algebraic properties of the various rings of global analytic functions that will be used in the sequel. Sections 5 to 7 are devoted to the Artin-Lang property, the complexity and the constructibility of topological operations. This is the concrete reward for all preceding abstract work. In Section 8 we put it all together for the nicest case, that of germs at compact sets.
Carlos Andradas, Ludwig Bröcker, Jesús M. Ruiz

### Backmatter

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