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Abstract Vergleichsstellensätze for preordered semifields and semirings II

Tobias Fritz

Abstract

The present paper continues our foundational work on real algebra with preordered commutative semifields and semirings. We prove two abstract Vergleichsstellensätze for preordered commutative semirings of polynomial growth. These generalize the results of Part I by no longer assuming $1 \ge 0$. Such a generalization comes with substantial technical complications: our Vergleichsstellensätze now also need to take into account infinitesimal information encoded in the form of monotone derivations in addition to the monotone homomorphisms to the nonnegative reals and tropical reals. The auxiliary technical results we develop along the way include surprising implications between inequalities in preordered semifields and a type classification for multiplicatively Archimedean fully preordered semifields. Among other applications, two companion papers use these results in order to derive limit new results in probability and information theory; one on asymptotics of random walks on topological abelian groups, and the other on the asymptotics of matrix majorization.

Abstract Vergleichsstellensätze for preordered semifields and semirings II

Abstract

The present paper continues our foundational work on real algebra with preordered commutative semifields and semirings. We prove two abstract Vergleichsstellensätze for preordered commutative semirings of polynomial growth. These generalize the results of Part I by no longer assuming . Such a generalization comes with substantial technical complications: our Vergleichsstellensätze now also need to take into account infinitesimal information encoded in the form of monotone derivations in addition to the monotone homomorphisms to the nonnegative reals and tropical reals. The auxiliary technical results we develop along the way include surprising implications between inequalities in preordered semifields and a type classification for multiplicatively Archimedean fully preordered semifields. Among other applications, two companion papers use these results in order to derive limit new results in probability and information theory; one on asymptotics of random walks on topological abelian groups, and the other on the asymptotics of matrix majorization.
Paper Structure (9 sections, 43 theorems, 143 equations)

This paper contains 9 sections, 43 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Background from Part I
  3. Further basic definitions and observations
  4. Inequalities in preordered semirings and semifields
  5. Type classification of multiplicatively Archimedean fully preordered semifields
  6. The ambient preorder
  7. A stronger catalytic Vergleichsstellensatz
  8. A stronger asymptotic Vergleichsstellensatz
  9. The test spectrum

Key Result

Theorem 1.1

Let $S$ be a zerosumfree preordered semidomain with a power universal pair $u_-, u_+ \in S$ and such that: Let nonzero $x, y \in S$ with $x \sim y$ satisfy the following: Then there is nonzero $a \in S$ such that $a x \le a y$. Moreover, if $S$ is also a semialgebra, then it is enough to consider $\mathbb{R}_+$-linear derivations $D$ in the assumptions.

Theorems & Definitions (103)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: I.6.6
  • Theorem 2.2: I.4.2
  • Corollary 2.3: I.4.3
  • Theorem 2.4: I.7.15
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 93 more