Universal-existential theories of fields

TL;DR

This paper explores the universal-existential fragments of first-order theories of fields, focusing on function fields and valued fields, and examines how these theories relate across different types of fields.

Contribution

It provides new insights into the relationships and reductions between universal-existential theories of various fields, including rational function fields and Laurent series fields.

Findings

Identifies many-one reductions between fragments of theories of different fields.

Analyzes how the theory of a base field influences the theories of related function fields.

Provides a framework for understanding the logical structure of field theories.

Abstract

We study various universal-existential fragments of first-order theories of fields, in particular of function fields and of equicharacteristic henselian valued fields. For example we discuss to what extent the theory of a field k determines the universal-existential theories of the rational function field over k and of the field of Laurent series over k, and we find various many-one reductions between such fragments.