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 Title: Derivations on o-minimal fields Author(s): Kaplan, Elliot Alexander Director of Research: van den Dries, Lou Doctoral Committee Chair(s): Hieronymi, Philipp Doctoral Committee Member(s): Tserunyan, Anush; Chen, Ruiyuan Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): model theory o-minimality differential algebra valued fields Abstract: 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. Issue Date: 2021-04-13 Type: Thesis URI: http://hdl.handle.net/2142/110462 Rights Information: Copyright 2021 Elliot Alexander Kaplan Date Available in IDEALS: 2021-09-17 Date Deposited: 2021-05
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