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Constructible sets in real geometry. (English) Zbl 0873.14044
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 33. Berlin: Springer. ix, 270 p. (1996).
Semi-algebraic subsets of \(\mathbb{R}^n\) are definable by finitely many polynomial inequalities. Thus, the properties of a semi-algebraic subset \(s\subseteq\mathbb{R}^n\) depend on a finite set of data. The complexity of numerous geometric and topological properties of \(S\) can be bounded in terms of the size of this finite set of data [for a beautiful survey, see L. Bröcker, “Semialgebraische Geometrie”, Jahresber. Dtsch. Math.-Ver. 97, No. 4, 130-156 (1995; Zbl 0868.14031)]. Therefore, a fundamental question is: Given a semi-algebraic subset \(S \subseteq \mathbb{R}^n\), what is the smallest set of data needed to define \(S\)? This question is somewhat vague as long as one does not specify the meaning of the word ‘smallest’. In about 1984, L. Bröcker suggested the following precise questions:
1. If \(S \subseteq \mathbb{R}^n\) is basic open, i.e., \(S\) can be defined by a finite number of strict inequalities \(f_i(x) >0\), \(i=1, \dots,l\), what is the smallest possible number of inequalities needed?
2. If \(S \subseteq \mathbb{R}^n\) is a finite union \(S=S_1 \cup \cdots \cup S_k\) of basic open sets, is there a bound on the number of sets in terms of the dimension of the ambient space?
Similar questions can be asked about basic closed sets (i.e., sets definable by finitely many inequalities \(f_i(x)\geq 0\), \(i=1, \dots,l)\) and unions of basic closed sets. All these questions have a positive answer: There exist functions \(s,t: \mathbb{N}\to \mathbb{N}\) such that each basic open set in \(\mathbb{R}^n\) needs at most \(s(n)\) inequalities and each union of basic open sets needs at most \(t(n)\) basic open building blocks. The corresponding bounding functions for basic closed and closed sets are denoted by \(\overline s\) and \(\overline t\). It is natural not only to ask for the existence of bounding functions, but rather for the smallest possible bounds. For \(s\) and \(\overline s\) these questions were settled by L. Bröcker and C. Scheiderer by showing that \(s(n)=n\) and \(\overline s(n)= {1\over 2} n(n +1)\) are the best possible bounds. For \(t\) and \(\overline t\) bounds are known, but it is not clear whether they are sharp.
The same questions can be asked in many different situations. For example, one may consider semianalytic sets or constructible subsets of real spectra instead of semi-algebraic sets. The aim of the book is first to set up axiomatically a framework for studying and answering these questions in the most general form conceivable, and then to apply this abstract theory to concrete situations, in particular to semi-algebraic and semianalytic sets. The book starts and ends with concrete geometric observations, results and examples. In between it shows impressively how an appropriate choice of axioms can lead to a clarification of the methods needed to prove a result and to a unification of analogous results in different situations.
The main axiomatic structures are real spaces and spaces of signs (as a special class of real spaces). They are pairs consisting of a set \(X\) together with a set \(G\) of maps \(X\to \{-1,0,1\}\) satisfying certain conditions. The set \(X\) is to be thought of as a geometric object, the elements of \(G\) are a collection of sign conditions at the points of \(X\). The most important example is provided by the real spectrum: If \(A\) is any ring then the real spectrum \(\text{Sper} (A)\) together with the sign functions associated with the elements of \(A\) is a space of signs. Marshall’s abstract spaces of orderings are special instances of spaces of signs. In fact, spaces of orderings play an indispensible part in the analysis of spaces of signs: Every space of signs has a canonical stratification as a disjoint union of spaces of orderings.
If \(X=(X,G)\) is a space of signs then a subset \(S \subseteq X\) is said to be basic open (principal basic open, or strictly open) if it is of the form \(S= \{x\in X; f_1(x)=1, \dots, f_l(x) =1\}\) with \(f_1, \dots, f_l\in G\) (if \(S\) is basic open with \(l=1\), or if \(S\) is a finite union of basic open sets). Similar definitions can be made with ‘open’ replaced by ‘closed.’ Given any subset \(S \subseteq X\) one may now ask:
How does one recognize whether \(S\) is basic open or principal basic open?
What is the lowest possible bound \(s(X)\) for the number of conditions needed to describe any basic open set in \(X\), or what is the lowest possible bound for the number \(t(X)\) of basic open sets needed to represent any given strictly open set as a union?
The obstruction for a set to be basic open or principal basic open can be formulated in terms of fans. These are distinguished subsets of \(X\). A fan is always contained in some stratum belonging to the canonical stratification of \(X\) by subspaces of orderings. The numerical information about the bounds is obtained from a careful analysis of the fans. Therefore, the answer to the questions is achieved in two steps: First, spaces of orderings and their fans are studied, then techniques are developed to put together the information about the different spaces of orderings to obtain results for spaces of signs. Examples of results showing the importance of fans are:
In any space of signs, \(X=(X,G)\), the bound \(s(X)\) can be determined as \[ \sup \{s;\;s=1 \text{ or there is a finite fan }F \subseteq X \text{ with }|F|=2^s\}. \] If \(s(X)\) is finite and if \(C \subseteq X\) is a constructible open subset then \(C\) is basic open if and only if \(|C\cap F |\neq 3\) for every fan \(F\subseteq X\) with \(|F|=4\).
Results of this type lead to bounds also for the invariants \(t(X)\), \(\overline s(X)\) and \(\overline t(X)\), as well as to a criterion deciding whether two closed constructible sets can be separated by the signs of a single function. The application of the axiomatic theory of spaces of signs and the complexity of constructible sets has a rather immediate application to the investigation of real spectra and semi-algebraic sets. One starts with a ring or with a ring of polynomial functions on a semi-algebraic set and proceeds in two steps: First the local situation (i.e., spaces of orderings of the residue fields at the prime ideals of the ring) is studied, afterwards the local results are globalized. The application to analytic geometry is far more difficult and requires a considerable amount of work. In particular, one needs to study the behavior of real spectra when passing to Henselizations and completions of local rings. Eventually it turns out that for germs of analytic sets at a point or for compact analytic manifolds the results are much like those in the semi-algebraic case. It is not surprising that the attempt to extend the theory beyond compact manifolds is only partially successful.

14Pxx Real algebraic and real-analytic geometry
13J07 Analytical algebras and rings
12-02 Research exposition (monographs, survey articles) pertaining to field theory
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra
12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)