Definable types in algebraically closed valued fields

TL;DR

This paper investigates the conditions under which types over algebraically closed valued fields are definable, revealing limitations of the property for higher types and exploring implications in C-minimality.

Contribution

It characterizes pairs of algebraically closed valued fields where all types over a base are definable, and constructs counterexamples showing the property does not extend to higher types.

Findings

All 1-types over certain algebraically closed valued fields are definable.

Counterexamples exist where higher types are not all definable.

Discussion on the implications in C-minimality.

Abstract

Marker and Steinhorn shown that given two models $M\prec N$ of an o-minimal theory, if all 1-types over $M$ realized in $N$ are definable, then all types over $M$ realized in $N$ are definable. In this article we characterize pairs of algebraically closed valued fields satisfying the same property. Although it is true that if $M$ is an algebraically closed valued field such that all 1-types over $M$ are definable then all types over $M$ definable, we build a counterexample for the relative statement, \textit{i.e.}, we show for any $n\geq 1$ that there is a pair $M\prec N$ of algebraically closed valued fields such that all $n$-types over $M$ realized in $N$ are definable but there is an $n+1$-type over $M$ realized in $N$ which is not definable. Finally, we discuss what happens in the more general context of $C$-minimality.