Spaces of Orderings and Separation of Connected Components of Real Varieties by Polynomials Original Research Article

The theory of spaces of orderings developed in [13-18] is useful in studying real varieties at the local and semi-local level; e.g., see [7]. On the other hand, N. Schwartz shows in [27] that in studying global separation of connected components of real varieties, more general sorts of structure arise. In general, the structures one is interested in can be described as follows: For any proper preordering T in a ring A (commutative with 1), there is associated a natural pairing GT × CXT → {1, −1}. XT is the topological space consisting of all orderings of A lying over T. CXT is the space of connected components of XT. GT is the factor group GT = {a set membership, variant A: a ≠ p 0 for all P set membership, variant XT}/{a set membership, variant A:a > p 0 for all P set membership, variant XT}. One can ask about the structure of this pairing. For example, under what conditions do elements of GT separate points in CXT? Under what conditions is CXT, GT) a space of orderings? As explained in [27], this has application to real algebraic geometry, to the question of existence of polynomials having prescribed signs on the connected components of a real algebraic variety.