Real places in function fields

The Structure of Spaces of R-places of Rational Function Fields over Real Closed Fields

For arbitrary real closed fields R, we study the structure of the space M(R(y)) of R-places of the rational function field in one variable over R and determine its dimension to be 1. We determine small subbases for their topology and discuss a suitable metric in the metrizable case. In the case of non-archimedean R, we exhibit the rich variety of homeomorphisms of subspaces that can be found in...

Fq-rational places in algebraic function fields using Weierstrass semigroups

We present a new bound on the number of Fq-rational places in an algebraic function field. It uses information about the generators of theWeierstrass semigroup related to a rational place. As we demonstrate, the bound has implications to the theory of towers of function fields. © 2008 Elsevier B.V. All rights reserved.

Places of algebraic function fields in arbitrary characteristic

We consider the Zariski space of all places of an algebraic function field F |K of arbitrary characteristic and investigate its structure by means of its patch topology. We show that certain sets of places with nice properties (e.g., prime divisors, places of maximal rank, zerodimensional discrete places) lie dense in this topology. Further, we give several equivalent characterizations of field...

Places on ∗- fields and the real holomorphy ring

Metrizability of Spaces of R-places of Function Fields of Transcendence Degree 1 over Real Closed Fields

In this paper we discuss the question ”When do different orderings of the rational function field R(X) (where R is a real closed field) induce the same R-place?”. We use this to show that if R contains a dense real closed subfield R′, then the spaces of R-places of R(X) and R′(X) are homeomorphic. For the function field K = R(X) we prove that its space M(K) of R-places is metrizible if and only...