A generalization of Strassen’s Positivstellensatz

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.

Keywords:

• Ordered rings
• ordered semirings
• Positivstellensatz
• real spectrum
• semifield

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 06F25
• 16Y60
• Secondary: 12J15
• 14P10

Acknowledgments

Many thanks to Péter Vrana for pointing out an error in a previous version of this paper. Author thanks Max Daniel, Richard Küng, Rostislav Matveev, Tim Netzer, Luciano Pomatto, Konrad Schmüdgen, Markus Schweighofer, Matteo Smerlak, Volker Strassen, Arleta Szkoła, Omer Tamuz, and Péter Vrana for useful discussions and feedback, as well as David Handelman and Terence Tao for discussion at mathoverflow.net/questions/314378/when-can-a-function-be-made-positive-by-averaging. The author also thanks the referee for useful comments which have improved the manuscript substantially. Most of this work was conducted while the author was with the Max Planck Institute for Mathematics in the Sciences, which the author thanks for its outstanding research environment.

Notes

1 Marshall also requires ${\mathbb{Q}}_{+}\subseteq T,$ which is convenient for his purposes and essentially without loss of generality. But since this extra condition is not relevant in our context, we have decided to omit this extra condition, as have previous authors (e.g. [4]).

2 This method of proof is very common in real algebra, and e.g. analogous to the method of Becker and Schwartz for proving the Positivstellensatz of Krivine–Kadison–Dubois [1].

3 One way to see this is to note that the map $\mathbb{N}[X]\to \mathbb{N}$ given by $p\mapsto 2p(1)-p(0)$ is monotone with respect to the given preorder.

4 Technically, it would be enough for the proof to show that every extreme ray is a multiple of a monotone homomorphism. But the proof of the converse is instructive as well and may be of use in other contexts, which is why we nevertheless include it and prove the equivalence.

5 Here we are assuming that $1+a\ne 0,$ so that $1+a$ is indeed invertible. For if $1+a=0,$ then $1\ge 0$ implies $0=1+a\ge a.$ Since $a\ge 0$ as well, this implies $0\ge a^2=1.$ But then also $0\ge x$ for all $x\in F,$ making all four conditions of Theorem 3.1 trivially true. Hence we can assume $1+a\ne 0$ without loss of generality.

6 The case $f,g\in {\mathbb{N}}_{+}[\underset{¯}{X}]$ is not significantly different.

7 See [8, Proposition 3] and [4], and also [1] for a modern formulation with purely algebraic proof, and [12, Theorem 5.4.4] for a textbook treatment of a very general version.