Places, cuts and orderings of function fields

TL;DR

This paper explores the relationship between orderings, cuts, and $ ext{R}$-places of algebraic function fields of one variable, establishing a homeomorphism between the space of cuts on a real curve and the space of orderings, and characterizing when two orderings correspond to a single $ ext{R}$-place.

Contribution

It introduces the space of cuts on a real curve, proves its homeomorphism to the space of orderings, and characterizes when two orderings correspond to a single $ ext{R}$-place.

Findings

The space of cuts on a real curve is homeomorphic to the space of orderings.

Two orderings correspond to a single $ ext{R}$-place if induced by a single ultrametric ball.

The paper provides a criterion for when two orderings define the same $ ext{R}$-place.

Abstract

In this paper we investigate the space of $\mathbb{R}$-places of an algebraic function field of one variable. We deal with the problem of determining when two orderings of such a field correspond to a single $\mathbb{R}$-place. To this end we introduce and study the space of cuts on a real curve and prove that the space is homeomorphic to the space of orderings. Finally, we prove that two cuts (consequently, two orderings) correspond to a single $\mathbb{R}$-place if they are induced by a single ultrametric ball.