Elementary geometric local-global principles for fields

TL;DR

This paper introduces a new family of local-global principles for fields involving orderings and p-valuations, unifying several classes of fields and establishing their elementary and diophantine definability properties.

Contribution

It defines a comprehensive family of local-global principles for fields, showing they form an elementary class and possess diophantine and approximation properties.

Findings

Fields satisfying these principles form an elementary class.

Holomorphy domains are diophantine definable.

Orderings satisfy the strong approximation property.

Abstract

We define and investigate a family of local-global principles for fields involving both orderings and p-valuations. This family contains the PAC, PRC and PpC fields and exhausts the class of pseudo classically closed fields. We show that the fields satisfying such a local-global principle form an elementary class, admit diophantine definitions of holomorphy domains, and their orderings satisfy the strong approximation property.