Spaces of Signs of Rings

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.