A generalization of Strassen's Positivstellensatz

TL;DR

This paper generalizes and strengthens Strassen's Positivstellensatz by replacing the boundedness condition with polynomial growth and providing new equivalent characterizations of the induced preorder.

Contribution

It extends Strassen's theorem to a broader setting with polynomial growth, offering two new equivalent characterizations of the preorder induced by homomorphisms.

Findings

Generalized Positivstellensatz with polynomial growth

Two new equivalent characterizations of the homomorphism-induced preorder

Broader applicability to preordered semirings

Abstract

Strassen's Positivstellensatz is a powerful but little known theorem on preordered commutative semirings satisfying a boundedness condition similar to Archimedeanicity. It characterizes the relaxed preorder induced by all monotone homomorphisms to $\mathbb{R}_+$ in terms of a condition involving large powers. Here, we generalize and strengthen Strassen's result. As a generalization, we replace the boundedness condition by a polynomial growth condition; as a strengthening, we prove two further equivalent characterizations of the homomorphism-induced preorder in our generalized setting.