Forking and Dividing in Fields with Several Orderings and Valuations

Abstract

We consider existentially closed fields with several orderings, valuations, and $p$-valuations. We show that these structures are NTP$_2$ of finite burden, but usually have the independence property. Moreover, forking agrees with dividing, and forking can be characterized in terms of forking in ACVF, RCF, and $p$CF.