Locally constant functions in C-minimal structures

TL;DR

This paper characterizes definable locally constant functions in C-minimal structures with complex canonical trees, extending known results from algebraically closed valued fields to a broader setting.

Contribution

It provides a detailed description of definable locally constant functions in C-minimal structures with infinitely branching, densely ordered canonical trees.

Findings

Describes definable subsets of the canonical tree in such structures.

Extends known algebraic results to a more general C-minimal context.

Abstract

Let $M$ be a $C$-minimal structure and $T$ its canonical tree (which corresponds in an ultrametric space to the set of closed balls with radius different than $\infty$ ordered by inclusion). We present a description of definable locally constant functions $f:M\rightarrow T$ in $C$-minimal structures having a canonical tree with infinitely many branches at each node and densely ordered branches. This provides both a description of definable subsets of $T$ in one variable and analogues of known results in algebraically closed valued fields.