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 Title: On asymptotic valued differential fields with small derivation Author(s): Pynn-Coates, Nigel Adam Lucas Director of Research: van den Dries, Lou Doctoral Committee Chair(s): Hieronymi, Philipp Doctoral Committee Member(s): Tserunyan, Anush; Freitag, James Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): algebra valued differential fields asymptotic fields pre-H-fields differential-henselianity logic quantifier elimination model companion Abstract: This thesis is a contribution to the algebra and model theory of certain valued differential fields and ordered valued differential fields. We focus on those with small derivation, which is a strong form of continuity of the derivation with respect to the valuation topology, and especially on those that are also asymptotic, which is a weak valuation-theoretic analogue of l'Hôpital's Rule. The first component of this thesis concerns three conjectures for valued differential fields $K$ with small derivation and linearly surjective differential residue field: the uniqueness of maximal immediate extensions of $K$, the equivalence of differential-algebraic maximality and differential-henselianity for asymptotic $K$, and the existence and uniqueness of differential-henselizations of asymptotic $K$. First, we show that any two maximal immediate extensions of $K$ are isomorphic over $K$ whenever the value group of $K$ has only finitely many convex subgroups. More significantly, we also establish this conjecture when $K$ is asymptotic. Next, we show that if $K$ is asymptotic and differential-henselian, then it is differential-algebraically maximal; this is optimal, as Aschenbrenner, van den Dries, and van der Hoeven have shown that the asymptoticity assumption is necessary. They have also shown that if $K$ is differential-algebraically maximal, then it is differential-henselian, so this establishes the equivalence of differential-algebraic maximality and differential-henselianity for asymptotic $K$. Finally, we use this equivalence to show that if $K$ is asymptotic, then it has a differential-henselization, and that differential-henselizations are unique. The second component of this thesis builds on the first to study the model theory of pre-$H$-fields with gap 0, which are certain asymptotic ordered valued differential fields with small derivation that are transexponential in some sense. We show that the theory $T^*$ of differential-henselian, real closed pre-$H$-fields that have exponential integration and closed ordered differential residue field (such pre-$H$-fields necessarily have gap 0) has quantifier elimination in the language $\{+, -, \cdot, 0, 1, \leqslant, \preccurlyeq, \der\}$. From quantifier elimination, we deduce that this theory is complete and is the model completion of the theory of pre-$H$-fields with gap 0 (equivalently, it axiomatizes the class of existentially closed pre-$H$-fields with gap 0). Moreover, we show that it is combinatorially tame in the sense that it is distal, and hence has NIP. Finally, we consider a two-sorted structure with one sort for a model of $T^*$ and one sort for its residue field in a language $\mathcal{L}_{\res}$ expanding the language $\{+, -, \cdot, 0, 1, \leqslant, \der\}$ of ordered differential rings, and show that the theory of this two-sorted structure is model complete when the theory of the residue field is model complete in $\mathcal{L}_{\res}$. Issue Date: 2020-05-04 Type: Thesis URI: http://hdl.handle.net/2142/107953 Rights Information: Copyright 2020 Nigel Adam Lucas Pynn-Coates Date Available in IDEALS: 2020-08-26 Date Deposited: 2020-05
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