## Files in this item

FilesDescriptionFormat

application/pdf

GEHRET-DISSERTATION-2017.pdf (3MB)
(no description provided)PDF

## Description

 Title: Towards a model theory of logarithmic transseries Author(s): Gehret, Allen R Director of Research: van den Dries, Lou Doctoral Committee Chair(s): Hieronymi, Philipp Doctoral Committee Member(s): Aschenbrenner, Matthias; Nevins, Thomas Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Logarithmic transseries Model theory Abstract: The ordered valued differential field $\mathbb{T}_{\log}$ of logarithmic transseries is conjectured to have good model theoretic properties. This thesis records our progress in this direction and describes a strategy moving forward. As a first step, we turn our attention to the value group of $\mathbb{T}_{\log}$. The derivation on $\mathbb{T}_{\log}$ induces on its value group $\Gamma_{\log}$ a certain map $\psi$; together forming the pair $(\Gamma_{\log},\psi)$, the \emph{asymptotic couple of $\mathbb{T}_{\log}$}. We study the asymptotic couple $(\Gamma_{\log},\psi)$ and show that it has a nice model theory. Among other things, we prove that $\Th(\Gamma_{\log},\psi)$ has elimination of quantifiers in a natural language, is model complete, and has the non-independence property (NIP). As a byproduct of our work, we also give a complete characterization of when an $H$-field has exactly one or exactly two Liouville closures. Finally, we present an outline for proving a model completeness result for $\mathbb{T}_{\log}$ in a reasonable language. In particular, we introduce and study the notion of \emph{$\LD$-fields} and also the property of a differentially-valued field being \emph{$\Psi$-closed}. Issue Date: 2017-07-09 Type: Text URI: http://hdl.handle.net/2142/98343 Rights Information: Copyright 2017 Allen Gehret Date Available in IDEALS: 2017-09-29 Date Deposited: 2017-08
﻿