Pseudo real closed field, pseudo p-adically closed fields and NTP2

TL;DR

This paper proves that the theory of bounded pseudo real closed and pseudo p-adically closed fields is NTP_2 and characterizes their model-theoretic complexity, including burden and forking behavior.

Contribution

It confirms a conjecture relating boundedness of PRC fields to NTP_2 and extends results to PpC fields, analyzing their burden and forking.

Findings

Bounded PRC fields have NTP_2 theories.

Bounded PpC fields also have NTP_2 theories.

The burden of theories of bounded PRC and PpC fields equals the number of orders or p-adic valuations.

Abstract

The main result of this paper is a positive answer to the Conjecture 5.1 by A. Chernikov, I. Kaplan and P. Simon: If M is a PRC field, then Th(M) is NTP_2 if and only if M is bounded. In the case of PpC fields, we prove that if M is a bounded PpC field, then Th(M) is NTP_2. We also generalize this result to obtain that, if M is a bounded PRC or PpC field with exactly n orders or p-adic valuations respectively, then Th(M) is strong of burden n. This also allows us to explicitly compute the burden of types, and to describe forking.