On the Real Spectrum of a Ring and Its Application to Semialgebraic Geometry

Introduction. This paper is meant as an introduction and a guide to some recent developments in real algebraic geometry — more precisely, in semialgebraic geometry. In real algebraic geometry one is concerned with the set of real points V(R) of a variety V defined over R. More generally, one may replace the field of real numbers R by any real closed field. Real algebraic geometry is clearly a part of general algebraic geometry and therefore there seems to be no need for special considerations, i.e. special notions, tools, etc. However, in dealing with the set of real points V(R) one encounters new phenomena which are not, or at least not easily, treatable by the general methods of algebraic geometry. To give examples, let V be an affine variety over R. Then V(R) can be regarded as an algebraic subset of some suitable R, i.e., a subset defined by a finite set of polynomial equations Fx = 0 , . . . , Fr = 0 where Ft e R[ Xv..., XN]9 i = 1 , . . . , r. Consequently, V(R) carries the subspace topology inherited from R^. Even if V is irreducible it may happen that V(R) is not a connected topological space. Note that the corresponding set of complex points V(C) is always connected if V is irreducible. A typical example is provided by the elliptic curve E (Figure 1). In this example, is (R) has two components Q , C2, namely