Positivstellens\"atze for Semirings

TL;DR

This paper advances the theory of Positivstellens"atze for semirings in real algebraic geometry, providing new theorems, reductions, and applications, including denominator-free results and examples.

Contribution

It introduces new Positivstellensatz results for semirings, reduces complex cases to quadratic modules, and explores applications with denominators and examples.

Findings

Reduced Archimedean Positivstellensatz to quadratic modules

Proved a general Positivstellensatz with denominators for filtered algebras

Derived a denominator-free Positivstellensatz for cylindrical extensions

Abstract

In this paper we develop a number of results and notions concerning Positivstellens\"atze for semirings (preprimes) of commutative unital real algebras. First we reduce the Archimedean Positivstellensatz for semirings to the corresponding result for quadratic modules. Various applications of the Archimedean Positivstellensatz for semirings are investigated. A general Positivstellensatz with denominators is proved for filtered algebras with semirings. As an application we derive a denominator-free Positivstellensatz for the cylindrical extension of an algebra with Archimedean semiring. A large number of illustrating examples are given.