Towards a model theory of logarithmic transseries

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}.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2017

Author dc:creator

• Gehret, Allen R

Contributors dc:contributor

• van den Dries, Lou
• Hieronymi, Philipp
• Aschenbrenner, Matthias
• Nevins, Thomas

• Logarithmic transseries
• Model theory

Statement dc:rights

• Copyright 2017 Allen Gehret

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/98343

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F98343