A Vergleichsstellensatz of Strassen's Type for a Noncommutative Preordered Semialgebra through the Semialgebra of its Fractions

TL;DR

This paper extends Strassen's Vergleichsstellensatz to noncommutative preordered semialgebras by defining a new semialgebra of fractions, enabling broader applications in real algebraic geometry and related fields.

Contribution

It introduces a noncommutative version of the Vergleichsstellensatz and defines the semialgebra of fractions for noncommutative semialgebras, generalizing existing concepts.

Findings

Characterizes the relaxed preorder via monotone homomorphisms to _+

Defines the semialgebra of fractions for noncommutative semialgebras

Provides equivalent conditions for the preorder on the fraction semialgebra

Abstract

Preordered semialgebras and semirings are two kinds of algebraic structures occurring in real algebraic geometry frequently and usually play important roles therein. They have many interesting and promising applications in the fields of real algebraic geometry, probability theory, theoretical computer science, quantum information theory, \emph{etc.}. In these applications, Strassen's Vergleichsstellensatz and its generalized versions, which are analogs of those Positivstellens\"atze in real algebraic geometry, play important roles. While these Vergleichsstellens\"atze accept only a commutative setting (for the semirings in question), we prove in this paper a noncommutative version of one of the generalized Vergleichsstellens\"atze proposed by Fritz [\emph{Comm. Algebra}, 49 (2) (2021), pp. 482-499]. The most crucial step in our proof is to define the semialgebra of the fractions of a noncommutative semialgebra, which generalizes the definitions in the literature. Our new Vergleichsstellensatz characterizes the relaxed preorder on a noncommutative semialgebra induced by all monotone homomorphisms to $\mathbb{R}_+$ by three other equivalent conditions on the semialgebra of its fractions equipped with the derived preorder, which may result in more applications in the future.