Real places in function-fields

Abstract

Let F/R be a function field over a real closed field R. The author proves the existence of real places with prescribed rank and dimension (where these numbers satisfy obvious conditions). The main tool, a Zariski-dense curve selection lemma, is interesting in its own right. Recent papers of F.-V. Kuhlmann and A. Prestel [J. Reine Angew. Math. 353, 181-195 (1984; Zbl 0535.12015] and of L. Br¨ocker and the referent [”Valuations of function fields from the geometrical point of view”, to appear in J. Reine Angew. Math.] extend these results to a more general situation.