A ganzstellensatz for semi-algebraic sets in real closed valued fields

TL;DR

This paper provides an algebraic characterization of rational functions that are OVF-integral on definable open semi-algebraic sets in real closed valued fields, extending model theoretic methods.

Contribution

It introduces a ganzstellensatz for semi-algebraic sets in real closed valued fields, connecting algebraic properties with model theoretic frameworks.

Findings

Characterization of OVF-integral rational functions on semi-algebraic sets

Application of model theoretic framework to establish ganzstellensatz

Control of semi-sections and their relation to orderings

Abstract

Let $(K,\nu)$ be a real closed valued field, and let $S\subseteq K^n$ be a definable open semi-algebraic set. We find an algebraic characterization of rational functions which are OVF-integral on $S$. We apply the existing model theoretic framework for proving ganzstellens\"atze, and need to control semi-sections and their relations to orderings. (joint work with Noa Lavi)