In formally real fields K sums of powers with arbitrary even exponents can be
studied by invoking the compact space M = M (K) of all real places, i.e. certain
maps λ : K → P 1 := R ∪ ∞, and the natural representation K → C(M, P 1 ), a 7→ â
based on the evaluation maps â : λ 7→ λ(a). The real holomorphy H = H(K) is
defined as the subring of K consisting of all elements a admitting a finite evaluation
map â : M → R. One is further led to study the series of representations
S n (H) → C(M, S n ), n ∈ N.
In the case of a real (=formally real) function field F over R this general approach
will be substantiated by relating M and H to the family of all smooth models of
F with a compact real locus X which is a real algebraic set and a C ∞ -manifold.
The investigation of line bundles on those manifolds leads to information on the
invertible ideals of H, recent results on continuous rational maps help to study
the image of S n (H) in C(M, S n ). All these informations can be used to derive
qualitative and quantitative results on the representation of an element as a sum of
powers with a given even exponent. Purely algebraic proofs for results of this type
have not been found so far.