A remark on fields with the dense orbits property
Let K be a formally real field, G its automorphism group and \(\Omega\) its order space. Then G acts on \(\Omega\) in the natural way, and K is called D.O.P. if the orbits under this action are all dense in \(\Omega\). This notion was introduced by \textit{D. W. Dubois} and \textit{T. Récio} [Contemp. Math. 8, 265-288 (1982; Zbl 0484.12003)] and studied by \textit{J. M. Gamboa} and \textit{T. Récio} in [J. Pure Appl. Al Algebra 30, 237-246 (1983; Zbl 0533.12018)] and by \textit{J. M. Gamboa} in [Rocky Mt. J. Math. 14, 499-502 (1984)] specially in geometric contexts. In this note the following is proven: The field K of germs of meromorphic functions of a real irreducible analytic germ \(X_ 0\) of dimension \(>1\) is never D.O.P. The proof uses a theorem of \textit{S. S. Abhyankar} and \textit{M. van der Put} on homomorphisms of analytic rings [J. Reine Angew. Math. 242, 26-60 (1970; Zbl 0193.005)] and some previous results by the author on orderings in K [Manuscr. Math. 46, 193-214 (1984; Zbl 0538.14018)] and separation of semianalytic subgerms of \(X_ 0\) [Arch. Math 43, 422-426 (1984)].
scientific article; zbMATH DE number 3871507