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

TL;DR

This paper investigates the structure and embeddings of spaces of $ $-places of rational function fields, revealing unexpected obstacles to realizing the torus as such a space.

Contribution

It establishes embeddings and uniqueness results for spaces of $ $-places in rational function fields and explores the complexities of embedding product spaces.

Findings

Embeddings of $M(R(y))$ into $M(F(y))$ are possible under certain conditions.

Uniqueness of these embeddings is proven for specific field extensions.

Obstacles to representing the torus as a space of $ $-places are identified.

Abstract

We study spaces $M(R(y))$ of $\R$-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 $\R$-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 $\R$-places.