Defining integer valued functions in rings of continuous definable functions over a topological field

TL;DR

This paper investigates how the ring of integers can be interpreted within rings of continuous definable functions over certain topological fields, revealing conditions under which Z is definable or interpretable.

Contribution

It establishes that under mild geometric and structural assumptions, the ring of integers Z is interpretable or definable in rings of continuous definable functions over various classes of topological fields.

Findings

Z is interpretable in C(X) under mild assumptions.

In o-minimal structures, Z is definably embedded in C(X) for certain X.

In P-minimal structures, a discrete subring Z is definable within C(X).

Abstract

Let K be an expansion of either an ordered field or a valued field. Given a definable set X $\subseteq$ K^m let C(X) be the ring of continuous definable functions from X to K. Under very mild assumptions on the geometry of X and on the structure K, in particular when K is o-minimal or P-minimal, or an expansion of a local field, we prove that the ring of integers Z is interpretable in C(X). If K is o-minimal and X is definably connected of pure dimension 2, then C(X) defines the subring Z. If K is P-minimal and X has no isolated points, then there is a discrete ring Z contained in K and naturally isomorphic to Z, such that the ring of functions f $\in$ C(X) which take values in Z is definable in C(X).