A Ganzstellensatz for semi-algebraic sets and a boundedness criterion for rational functions

TL;DR

This paper characterizes rational functions that take values in the valuation ring on semi-algebraic sets using model theory, and provides a boundedness criterion for such functions on semi-algebraic subsets of varieties over real closed fields.

Contribution

It introduces an algebraic characterization of rational functions with valuation ring values on semi-algebraic sets and establishes a boundedness criterion over real closed and ordered fields.

Findings

Algebraic characterization of valuation ring-valued rational functions

Boundedness criterion for rational functions on semi-algebraic sets

Application to varieties over real closed and ordered fields

Abstract

Let $\langle K,\nu \rangle$ be a real closed valued field, and let $S\subseteq K^n$ be an open semi-algebraic set. Using tools from model theory, we find an algebraic characterization of rational functions which admit, on $S$, only values in the valuation ring. We use this result to deduce a criterion for a rational function to be bounded on an open semi algebraic subset of some irreducible variety over a real closed field or over an ordered field which is dense in its real closure.