Central orderings in fields of real meromorphic function germs (Q793109)

The author relates the geometry of the maximum dimension locus of a real irreducible analytic germ \(X_ 0\subset {\mathbb{R}}^ n_ 0\) and the space of orders of the field of germs of meromorphic functions over \(X_ 0\), \({\mathcal O}(X_ 0)\). Results of the same nature are already known in the real algebraic case. A formal half branch is defined to be a germ of non-constant \(C^{\infty}\)-mapping from \(]0,\epsilon[\) to \(X_ 0,c\), such that, if \(\hat c\) is the jet of c, an analytic function f is such that \(f(\hat c)=0\) iff for \(\epsilon\) sufficiently small for all \(t\in]0,\epsilon [\): \(f(c(t))=0\) \((t\to(t,e^{-1/t}))\) is not a formal half branch of \({\mathbb{R}}^ 2_ 0\), for example). The dimension of a formal half-branch is the dimension of the smallest analytic germ containing the image of c. - An order \(\alpha\) of \({\mathcal O}(X_ 0)\) is centered at a formal half- branch of \(X_ 0\) if every germ of analytic function over \(X_ 0\) positive on the image of c is positive for \(\alpha\). This is a geometric illustration of the theory of specialization in the real spectrum of the ring of germs of analytic functions on \(X_ 0\). Let \(\Omega\) be the set of orders on \({\mathcal O}(X_ 0)\), \(\Omega^*\) (resp. \(\Omega_ e\), \(e=1,...,d=\dim X_ 0)\) be the set of orders centered at a formal half- branch (resp. of dimension e). The author shows that \(\Omega_ 1,...,\Omega_ d\), \(\Omega-\Omega^*\) are disjoint and dense in \(\Omega\). Then he establishes a bijection between the clopens of \(\Omega\) and the regularly closed semi-analytic germs of the maximum dimension locus \(X^*_ 0\) of \(X_ 0\), which gives, as in the real algebraic case, a solution to Hilbert 17th problem.