Rex represent an element of Specr(A). Then the associated prime cone P is the set of all f E A such that aU) ;::: O.

0 It follows immediately that if a semialgebraic set S C R2 is generically principal (resp. basic), then #{cPij E F IS E cPij} is even (resp. -=I- 3). This is exactly the same as in the discrete example of the preceding section. Regarding example b) we choose F like I so here #{cPij E F I S we choose F like so here #{ cPij E E cPij} F IS E cPij} = fIXX 1, hence S is not generically principal. For c) = 3, and these S's are not generically basic. This method is the right one to disprove the generic principal or basic property, since the fans, which we shall introduce in full generality in Chapter III, are the only obstructions to these properties.

7 Let a E Specr(A). A valuation ring V of K is said to be centered at a if the following conditions hold: 3. Real Valuations 41 a) V dominates the local ring Asupp(a)I and b) the total ordering a extends from I\:(supp(a)) to kv via the embedding I\:(supp(a)) C kv induced by a). Let us see how this situation occurs. 8) Construction of Valuations Centered at a Prime Cone. Let a E Specr(A) and (3 E Specr(K) be such that the restriction of (3 to A is a generization of a. Let V be the convex hull of Asupp(a) in K with respect to (3.