Monotone TT-convex TT-differential fields

Let $T$ be a complete, model complete o-minimal theory extending the theory of real closed ordered fields and assume that $T$ is power bounded. Let $K$ be a model of $T$ equipped with a $T$-convex valuation ring $\mathcal{O}$ and a $T$-derivation $\partial$ such that $\partial$ is monotone, i.e., weakly contractive with respect to the valuation induced by $\mathcal{O}$. We show that the theory of monotone $T$-convex $T$-differential fields, i.e., the common theory of such $K$, has a model completion, which is complete and distal. Among the axioms of this model completion, we isolate an analogue of henselianity that we call $T^{\partial}$-henselianity. We establish an Ax--Kochen/Ershov theorem and further results for monotone $T$-convex $T$-differential fields that are $T^{\partial}$-henselian.Comment: 26 page

Panorama of p-adic model theory

ABSTRACT. We survey the literature in the model theory of p-adic numbers since\ud Denef’s work on the rationality of Poincaré series. / RÉSUMÉ. Nous donnons un aperçu des développements de la théorie des modèles\ud des nombres p-adiques depuis les travaux de Denef sur la rationalité de séries de Poincaré,\ud par une revue de la bibliographie

Algebraically closed fields with characters; differential-henselian monotone valued differential fields

This thesis consists of two unrelated research projects. In the first project we study the model theory of the 2-sorted structure (F, C; χ), where F is an algebraic closure of a finite field of characteristic p, C is the field of complex numbers and χ ∶ F → C is an injective, multiplication preserving map. In the second project we study the model theory of the differential-henselian monotone valued differential fields. We also consider definability in differential-henselian monotone fields with c-map and angular component map

Valuation Theory and Its Applications

In recent years, the applications of valuation theory in several areas of mathematics have expanded dramatically. In this workshop, we presented applications related to algebraic geometry, number theory and model theory, as well as advances in the core of valuation theory itself. Areas of particular interest were resolution of singularities and Galois theory

TT-convex TT-differential fields and their immediate extensions

Let $T$ be a polynomially bounded o-minimal theory extending the theory of real closed ordered fields. Let $K$ be a model of $T$ equipped with a $T$-convex valuation ring and a $T$-derivation. If this derivation is continuous with respect to the valuation topology, then we call $K$ a $T$-convex $T$-differential field. We show that every $T$-convex $T$-differential field has an immediate strict $T$-convex $T$-differential field extension which is spherically complete. In some important cases, the assumption of polynomial boundedness can be relaxed to power boundedness.Comment: 26 page

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