Positivstellens\"atze for semi-algebraic sets in real closed valued fields

TL;DR

This paper characterizes polynomials and rational functions that are non-negative on definable sets in real closed valued fields, extending Positivstellensatz results using model theory and valuation theory.

Contribution

It generalizes Positivstellensatz to include valuation terms in real closed valued fields using model-theoretic methods.

Findings

Provides a characterization for non-negative polynomials and rational functions on definable sets

Extends Positivstellensatz to valuation-involved sets in real closed valued fields

Utilizes model theory and canonical valuations in the proofs

Abstract

The purpose of this paper is to give a characterization for polynomials and rational functions which admit only non-negative values on definable sets in real closed valued fields. That is, generalizing the relative positivstellens\"atze for sets defined also by valuation terms. For this, we use model theoretic tools, together with existence of canonical valuations.