Central orderings in fields of real meromorphic function germs

Abstract

The paper deals with orderings in the field K(X0) of meromorphic function germs on an irreducible analytic germ X0⊂Rn0 of dimension d. It is inspired by the theory of central points of real algebraic varieties (the set of central points is the closure of the set of smooth points of maximal dimension). In the case of germs, the central points are replaced by curves or, more precisely, half branch germs (actually even C∞ branch germs); the dimension of a half branch germ is the dimension of the smallest analytic germ which contains this half germ. An order on K(X0) is said to be centered on a half branch c if the set of its positive elements contains the functions which are positive on c. If Ωe is the set of orders on K(X0) centered on a half branch of dimension e, it is proved that all Ωe (e=1,⋯,d) and Ω∖Ω∗ (with Ω∗=Ω1∪⋯∪Ωd) are dense in Ω. Various other results and applications are given.