Real closed valuation rings

The real closed valuation rings, i.e., convex subrings of real closed fields, form a proper subclass of the class of real closed domains. It is shown how one can recognize whether a real closed domain is a valuation ring. This leads to a characterization of the totally ordered domains whose real closure is a valuation ring. Real closures of totally ordered factor rings of coordinate rings of real algebraic varieties are very frequently valuation rings. In particular, the real closure of the coordinate ring of a curve is an SV-ring (i.e., the factor rings modulo prime ideals are valuation rings). Real closed valuation rings play a role in the definition of real closed rings, as well as in the construction of real closures of rings and porings. They can also be used for the study of univariate differentiable semi-algebraic functions. This leads to the notion of differentiablility of semi-algebraic functions along half branches of curves. Real closed rings were introduced in [17] (cf. [19]) and have been studied subsequently in a large number of publications. The class of real closed rings is closed under the formation of reduced factor rings (cf. [21], section 12). Thus, factor rings of real closed rings modulo prime ideals are real closed domains. Real closed valuation rings, i.e., convex subrings of real closed fields, were studied in [4] and [5] in connection with rings of continuous functions. It is well-known that not every real closed domain is a real closed valuation ring (cf. section 1 for the construction of examples). The present note studies the question how one can recognize whether a real closed domain is a real closed valuation ring. Theorem 1.1 gives a characterization of those real closed domains that are valuation rings. This leads to a characterization of those totally ordered domains whose real closure is a valuation ring. The criterion of Theorem 1.1 is not always easy to check in a concrete setting, in particular in a geometric context. Section 2 is devoted to totally ordered domains that arise in real algebraic geometry. Partially ordered finitely generated algebras over totally ordered fields are among the basic algebraic objects in real geometry. Totally ordered domains arise as residue rings at points of the real spectrum. So the following situation will be studied: There are a totally ordered field k,k ( ) and a totally ordered integral k-algebra A,A ( ) (which means, in particular, that k+ ! A+ " A+ ); the transcendence degree trdeg k A is finite. The quotient field of A is a totally ordered field, K ,K + ( ) , and the convex hull of A in K ,K + ( ) is a valuation ring. This valuation ring yields numerical invariants that contain detailed information about the transcendence degree. The numerical invariants lead to a charactization of those domains whose real closure is a valuation ring (Theorem 2.4). The numerical invariants are not always easy to compute. But they lead to practicable sufficient conditions for the real closure to be a valuation ring (Corollary 2.5). It follows that the real closure is always a valuation ring if the transcendence degree is at most 1 (Corollary 2.6). In the case of transcendence degree 2 there are real closed domains that are not valuation rings, but they can be described completely (Example 2.8). Suppose now that the field k is real closed and that X is a real algebraic variety, the coordinate ring is denoted by k X [ ] . The real spectrum of k X [ ] is partitioned into two subsets: P val X ( ) (valuation points) is the set of prime cones ! such that the real closure of the totally ordered domain k X [ ] ! is a valuation ring; P¬val X ( ) are the other prime cones. If