Triangulation in o-minimal fields with standard part map

TL;DR

This paper proves a triangulation theorem for definable sets in o-minimal fields with a standard part map, ensuring the triangulation of sets induces a triangulation of their standard parts, advancing geometric understanding in this setting.

Contribution

It introduces a triangulation result for definable sets in o-minimal fields with a standard part map, under mild assumptions, which was previously unresolved.

Findings

Triangulation of definable sets in o-minimal fields with standard part map

Induces triangulation of standard parts of sets

Advances geometric analysis in o-minimal structures

Abstract

In answering questions from arXiv:0901.2337v1 we prove a triangulation result that is of independent interest. In more detail, let R be an o-minimal field with a proper convex subring V, and let st: V \to k be the corresponding standard part map. Under a mild assumption on (R,V) we show that definable subsets X of V^n admit a triangulation that induces a triangulation of its standard part st(X).