The space S of all non-trivial real places on a real function field K|k of trascendence degree one, endowed with a natural topology analogous to that of Dedekind and Weber's Riemann surface, is shown to be a one-dimensional k-analytic manifold, which is homeomorphic with every bounded non-singular real affine model of K|k. The ground field k is an arbitrary ordered, real-closed Cantor field (definition below). The function field K|k is thereby represented as a field of real mappings of S which might be called meromorphic - each f in K|k has a convergent power series expansion at each of its finite points and a convergent Laurent series (finite negative order) in the vicinity of each of its finite set of infinities (or poles). The treatment is purely real up to the point where we want to show that K|k contains every meromorphic function on S. In order to do that we have had to take k = R, the field of all real numbers, and appeal to complex function theory.
@article{urn:eudml:doc:39838,
title = {Real commutative algebra. III. Dedekind-Weber-Riemann manifolds.},
journal = {Revista Matem\'atica Hispanoamericana},
volume = {40},
year = {1980},
pages = {157-167},
zbl = {0509.14025},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:39838}
}
Dubois, D. W.; Bukowski, A. Real commutative algebra. III. Dedekind-Weber-Riemann manifolds.. Revista Matemática Hispanoamericana, Tome 40 (1980) pp. 157-167. http://gdmltest.u-ga.fr/item/urn:eudml:doc:39838/