Derivations on o-minimal fields

"Let $K$ be an o-minimal expansion of a real closed ordered field and let $T$ be the theory of $K$. In this thesis, we study derivations $\der$ on $K$. We require that these derivations be compatible with the \mathcal{C}¹-functions definable in $K$. For example, if $K$ defines an exponential function, then we require that $\der\exp(a) = \exp(a)\der a$ for all $a \in K$. We capture this compatibility with the notion of a $T$-derivation. Let T^\der be the theory of structures $(K,\der)$, where $K\models T$ and $\der$ is a $T$-derivation on $K$. We show that T^\der has a model completion T^\der_\mathcal{G}, in which derivation behaves ""generically."" The theory T^\der_\mathcal{G} is model theoretically quite tame; it is distal, it has o-minimal open core, and it eliminates imaginaries. Following our investigation of T^\der_\mathcal{G}, we turn our attention to $T$-convex $T$-differential fields. These are models $K\models T$ equipped with a $T$-derivation which is continuous with respect to a $T$-convex valuation ring of $K$, as defined by van den Dries and Lewenberg. We show that if $K$ is a $T$-convex $T$-differential field, then under certain conditions (including the necessary condition of power boundedness), $K$ has an immediate $T$-convex $T$-differential field extension which is spherically complete. In the penultimate chapter, we consider $T$-convex $T$-differential fields which are also $H$-fields, as defined by Aschenbrenner and van den Dries. We call these structures H_T-fields, and we show that if $T$ is power bounded, then every H_T-field $K$ has either exactly one or exactly two minimal Liouville closed H_T-field extensions up to $K$-isomorphism. We end with two theorems when T= T_{\operatorname{re}}, the theory of the real field expanded by restricted elementary functions. First, we prove a model completeness result for the expansion of the ordered valued differential field $\mathbb{T}$ of logarithmic-exponential transseries by its natural restricted elementary functions. We then use this result to prove that the theory of H_{T\operatorname{re}}-fields has a model companion."

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

2021

Author dc:creator

• Kaplan, Elliot Alexander

Contributors dc:contributor

• van den Dries, Lou
• Hieronymi, Philipp
• Tserunyan, Anush
• Chen, Ruiyuan

• model theory
• o-minimality
• differential algebra
• valued fields

Statement dc:rights

• Copyright 2021 Elliot Alexander Kaplan

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/110462

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/110462