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Dimension theory for the asymptotic couple of the field of logarithmic transseries

Allen Gehret, Elliot Kaplan, Nigel Pynn-Coates

TL;DR

This work characterizes all dimension functions on models of the theory $T_{ m log}$ of the asymptotic couple $(oldsymbol extGamma_{ m log},oldsymbololdsymbol\psi)$ by proving the Dimension Theorem and the Small Sets Theorem. It introduces a family of dimension functions $ ext{dim}_{oldsymbololdsymbol }$ indexed by $oldsymbololdsymbol \

Abstract

In this paper we completely characterize all dimension functions on all models of the theory $T_{\log}$ of the asymptotic couple of the field of logarithmic transseries (Dimension Theorem). This is done by characterizing the "small" $1$-variable definable sets (Small Sets Theorem). As a byproduct, we show that $T_{\log}$ is d-minimal and does not eliminate imaginaries. Separately, we provide an abstract criterion for d-minimality, which we use to observe some new examples of d-minimal expansions of valued fields.

Dimension theory for the asymptotic couple of the field of logarithmic transseries

TL;DR

This work characterizes all dimension functions on models of the theory of the asymptotic couple by proving the Dimension Theorem and the Small Sets Theorem. It introduces a family of dimension functions indexed by $oldsymbololdsymbol \

Abstract

In this paper we completely characterize all dimension functions on all models of the theory of the asymptotic couple of the field of logarithmic transseries (Dimension Theorem). This is done by characterizing the "small" -variable definable sets (Small Sets Theorem). As a byproduct, we show that is d-minimal and does not eliminate imaginaries. Separately, we provide an abstract criterion for d-minimality, which we use to observe some new examples of d-minimal expansions of valued fields.

Paper Structure

This paper contains 35 sections, 66 theorems, 87 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Outline of paper
  3. Conventions
  4. Overview of Tₗₒ₉
  5. More functions on the standard model
  6. The Lₗₒ₉-theory Tₗₒ₉
  7. The Ψ-set of a model
  8. Quotienting by ∆ᵩ
  9. Dimension, the Small Sets Theorem, and related results
  10. Dimension functions
  11. Pregeometries
  12. The Small Sets Theorem
  13. Proof of the Dimension Theorem from (1)↔(4)↔(5) of the Small Sets Theorem
  14. Higher arity definable sets
  15. Failure of elimination of imaginaries
  16. ...and 20 more sections

Key Result

Lemma 2.4

Suppose $(\Gamma_1,\psi_1)\models T_{\log}$ and $(\Gamma_0,\psi_0)\subseteq(\Gamma_1,\psi_1)$ is an $\mathcal{L}_{\log}$-substructure. Then $(\Gamma_0,\psi_0)\preccurlyeq(\Gamma_1,\psi_1)$, and thus $\psi_0(\Gamma_0^{\neq})=\Gamma_0\cap\psi_1(\Gamma^{\neq}_1)$.

Figures (4)

  • Figure 1: The standard model of $T_{\log}$.
  • Figure 2: Quotienting by $\Delta_\phi$.
  • Figure 3: A set $X$ with $X,X',X"\neq \varnothing$ and $X^{(3)} = \varnothing$.
  • Figure 4: The image in the quotient $\Gamma/\Delta_\phi$ of the set in Figure \ref{['CB3']} for the value $\phi=s^50$.

Theorems & Definitions (149)

  • proof : Answer
  • proof : Answer
  • proof : Answer
  • proof : Answer
  • proof : Answer
  • proof : Answer
  • Remark 2.1: Disclaimer about $\infty$
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • ...and 139 more