Globally valued function fields: existential closure

TL;DR

This paper introduces the concept of globally valued fields (GVFs), a class of valued fields connected by a product formula, and proves that the canonical GVF structure on the algebraic closure of rational functions is existentially closed.

Contribution

It defines globally valued fields and proves the existential closure of the canonical GVF structure on algebraic closures of rational function fields.

Findings

The canonical GVF structure on $k(t)^{alg}$ is existentially closed.

Globally valued fields are connected by a product formula.

Varieties with a distinguished curve class approximate formulas in GVFs.

Abstract

These notes form part of a joint research project on the logic of fields with many valuations, connected by a product formula. We define such structures and name them {\em globally valued fields} (GVFs). This text aims primarily at a proof that {\em the canonical GVF structure on $k(t)^{alg}$ is existentially closed}. This can be read as saying that a variety {\em with a distinguished curve class} is a good approximation for a formula in the language of GVFs, in the same way that a variety is close to a formula for the theory ACF of algebraically closed fields.