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

Katarzyna Kuhlmann;  Franz-Viktor Kuhlmann — 2012

Fundamenta Mathematicae

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...

On the dimension of the space of ℝ-places of certain rational function fields

Taras Banakh;  Yaroslav Kholyavka;  Oles Potyatynyk;  Michał Machura;  Katarzyna Kuhlmann — 2014

Open Mathematics

We prove that for every n ∈ ℕ the space M(K(x 1, …, x n) of ℝ-places of the field K(x 1, …, x n) of rational functions of n variables with coefficients in a totally Archimedean field K has the topological covering dimension dimM(K(x 1, …, x n)) ≤ n. For n = 2 the space M(K(x 1, x 2)) has covering and integral dimensions dimM(K(x 1, x 2)) = dimℤ M(K(x 1, x 2)) = 2 and the cohomological dimension dimG M(K(x 1, x 2)) = 1 for any Abelian 2-divisible coefficient group G.

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