A characterization of rational and elliptic real algebraic curves in terms of their space of orderings (Q802617)

Let K be a formally real field with space of orderings X(K). Then Aut(K) operates on X(K). K is said to have the ''dense orbits property'' if for any \(x\in X(K)\) the orbit of x is dense in X(K). Fields with the dense orbits property were introduced by \textit{D. W. Dubois} and \textit{T. Recio} in Contemp. Math. 8, 265-288 (1982; Zbl 0484.12003) under the name of \(Q_ 1\)-fields. They were further studied by the author and \textit{T. Recio} [J. Pure Appl. Algebra 30, 237-246 (1983; Zbl 0533.12018)]. In the present paper the dense orbits property is studied for function fields of real algebraic varieties. So, let V be a real algebraic variety over the field \({\mathbb{R}}\) of real numbers, \({\mathbb{R}}(V)\) the function field of V. It is proved that Aut(\({\mathbb{R}}(V))\) is infinite if \({\mathbb{R}}(V)\) has the dense orbits property. If V is a curve then \({\mathbb{R}}(V)\) has the dense orbits property if and only if V is a rational or elliptic curve.