Elementary geometric local-global principles for fields

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.

final version published in Annals of Pure and Applied Logic, Volume 164, Issue 10, October 2013, Pages 989-1008