Embedding theorems for spaces of $\mathbb R$-places of rational function fields and their products

Katarzyna Kuhlmann ¹   ; Franz-Viktor Kuhlmann ²

¹ Institute of Mathematics Silesian University 40-007 Katowice, Poland
² Department of Mathematics and Statistics University of Saskatchewan Saskatoon, SK, S7N 5E6, Canada

Fundamenta Mathematicae, Tome 218 (2012) no. 2, pp. 121-149

Résumé

We study spaces $M(R(y))$ of $\mathbb R$-places of rational function fields $R(y)$ in one variable. For extensions $F|R$ of formally real fields, with $R$ real closed and satisfying a natural condition, we find embeddings of $M(R(y))$ in $M(F(y))$ and prove uniqueness results. Further, we study embeddings of products of spaces of the form $M(F(y))$ in spaces of $\mathbb R$-places of rational function fields in several variables. Our results uncover rather unexpected obstacles to a positive solution of the open question whether the torus can be realized as a space of $\mathbb R$-places.

Détail

DOI : 10.4064/fm218-2-2

Keywords: study spaces mathbb r places rational function fields variable extensions formally real fields real closed satisfying natural condition embeddings prove uniqueness results further study embeddings products spaces form spaces mathbb r places rational function fields several variables results uncover rather unexpected obstacles positive solution question whether torus realized space mathbb r places