# Archimedean Representation Theorem for modules over a commutative ring

## TL;DR

This paper extends the Archimedean Representation Theorem to modules over rings, unifying various Positivstellensatz results for matrix polynomials under a common theoretical framework.

## Contribution

It generalizes the Archimedean Representation Theorem to modules over rings, encompassing matrix polynomial Positivstellensatz results as special cases.

## Key findings

- Unified framework for Positivstellensatz for matrix polynomials
- Extension of the Archimedean Representation Theorem to modules over rings
- Connections between abstract theorems and concrete matrix polynomial results

## Abstract

P\'olya's Positivstellensatz and Handelman's Positivstellensatz are known to be concrete instances of the abstract Archimedean Representation Theorem for (commutative unital) rings. We generalise the Archimedean Representation Theorem to modules over rings. For example, consider the module of all symmetric matrices with entries in a polynomial ring, also known as matrix polynomials. P\'olya's Positivstellensatz and Handelman's Positivstellensatz had been generalised by Scherer and Hol, and L\^{e} and Du' respectively to matrix polynomials, using the method of effective estimates from analysis. We show that these two Positivstellens\"atze for matrix polynomials are concrete instances of our Archimedean Representation Theorem in the case of the module of symmetric matrix polynomials over the polynomial ring.