Characterizing finitary functions over non-archimedean RCFs via a topological definition of OVF-integrality

TL;DR

This paper characterizes finitary rational functions over non-archimedean real closed fields using a topological approach to OVF-integrality, connecting valuation theory with geometric and topological methods.

Contribution

It introduces a topological characterization of OVF-integrality, enabling the application of a Ganzstellensatz to describe finitary functions in non-archimedean settings.

Findings

OVF-integrality admits a natural topological definition

Characterization of finitary rational functions via positivity and finiteness conditions

Insights into Kochen geometry related to OVF-integrality

Abstract

When $R$ is a non-archimedean real closed field we say that a function $f\in R(\bar{X})$ is finitary at a point $\bar{b}\in R^n$ if on some neighborhood of $\bar{b}$ the defined values of $f$ are in the finite part of $R$. In this note we give a characterization of rational functions which are finitary on a set defined by positivity and finiteness conditions. The main novel ingredient is a proof that OVF-integrality has a natural topological definition, which allows us to apply a known Ganzstellensatz for the relevant valuation. We also give some information about the Kochen geometry associated with OVF-integrality.