Non-commutative Real Algebraic Geometry - Some Basic Concepts and First Ideas

TL;DR

This paper explores how fundamental concepts and results from real algebraic geometry can be extended to noncommutative *-algebras, including a noncommutative version of Stengle's Positivstellensatz for matrix polynomials.

Contribution

It introduces noncommutative analogs of key notions and proves a version of Stengle's Positivstellensatz for matrix polynomial algebras.

Findings

Generalization of quadratic modules and positive elements to noncommutative setting

Proof of a noncommutative Stengle's Positivstellensatz for matrix polynomials

Foundational ideas for noncommutative real algebraic geometry

Abstract

We propose and discuss how basic notions (quadratic modules, positive elements, semialgebraic sets, Archimedean orderings) and results (Positivstellensaetze) from real algebraic geometry can be generalized to noncommutative $*$-algebras. A version of Stengle's Positivstellensatz for $n \times n$ matrices of real polynomials is proved.