# Positivstellensatz\"e for noncommutative rational expressions

## TL;DR

This paper extends Positivstellensatz"e to noncommutative rational expressions, providing algebraic certificates for positivity on polynomially convex sets, especially when the set is convex, leading to a 'perfect' Positivstellensatz.

## Contribution

It generalizes Positivstellensatz"e from noncommutative polynomials to rational expressions, establishing conditions for positivity certificates on convex sets.

## Key findings

- Positivity on polynomially convex sets implies existence of algebraic certificates.
- Results are stronger under convexity assumptions, yielding a 'perfect' Positivstellensatz.
- Provides a framework for certifying positivity of noncommutative rational expressions.

## Abstract

We derive some Positivstellensatz\"e for noncommutative rational expressions from the Positivstellensatz\"e for noncommutative polynomials. Specifically, we show that if a noncommutative rational expression is positive on a polynomially convex set, then there is an algebraic certificate witnessing that fact. As in the case of noncommutative polynomials, our results are nicer when we additionally assume positivity on a convex set-- that is, we obtain a so-called "perfect Positivstellensatz" on convex sets.