A Characterization Of Rational And Elliptic Real Algebraic-Curves In Terms Of Their Space Of Orderings

Abstract

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 2 X(K) the orbit of
x is dense in X(K). Fields with the dense orbits property were introduced by D. W.
Dubois and T. Recio in Contemp. Math. 8, 265-288 (1982; Zbl 0484.12003) under the name of Q1-fields. They were further studied by the author and 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 R of real numbers, R(V ) the function field of V. It is proved that Aut(R(V )) is infinite if R(V ) has the dense orbits property. If V is a curve then R(V ) has the dense orbits property if and only if V is a rational or elliptic curve.