Related papers: $T$-convex $T$-differential fields and their immed…

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…

We prove a dichotomy for o-minimal fields $\mathcal{R}$, expanded by a $T$-convex valuation ring (where $T$ is the theory of $\mathcal{R}$) and a compatible monomial group. We show that if $T$ is power bounded, then this expansion of…

Let $T$ be an o-minimal theory extending the theory of real closed ordered fields. An $H_T$-field is a model $K$ of $T$ equipped with a $T$-derivation such that the underlying ordered differential field of $K$ is an $H$-field. We study…

We show that, for a certain large class of power-bounded $o$-minimal $\mathcal{L}_T$-theories $T$ whose field of exponents is infinite-dimensional as a vector space over the rationals, any definable set in a $T$-convex valued field…

It is well known that ordered exponential fields with a compatible non-trivial valuation cannot be spherically complete, but there are some that are ``complete enough''. This paper gives analogues of Kaplansky's theorem on maximally valued…

I analyze $\mathcal{O}$-weakly immediate and $\mathcal{O}$-residual types in an o-minimal expansion of an ordered field $\mathbb{E}$, where $\mathcal{O}$ is a convex valuation ring. The main result is a characterization of those exponential…

We show that every valued differential field has an immediate strict extension that is spherically complete. We also discuss the issue of uniqueness up to isomorphism of such an extension.

Let $\mathcal M=\langle K;O\rangle$ be a real closed valued field and let $k$ be its residue field. We prove that every interpretable field in $\mathcal M$ is definably isomorphic to either $K$, $K(\sqrt{-1})$, $k$, or $k(\sqrt{-1})$. The…

I prove the statement in the title using results from arXiv:2404.07646(2). This shows that Question~1.1 in [1] has negative answer for certain expansions of a valued field.

We give a precise formulation of T-duality for Ramond-Ramond fields. This gives a canonical isomorphism between the "geometrically invariant" subgroups of the twisted differential K-theory of certain principal torus bundles. Our result…

We show that functions definable in power bounded $T$-convex fields have the (multidimensional) Jacobian property. Building on work of I. Halupczok, this implies that a certain notion of non-archimedean stratifications is available in such…

We compare the $T$-polynomial convexity of Guedj with holomorphic convexity away from the support of $T$. In particular we show an Oka--Weil theorem for $T$-polynomial convexity, as well as present a situation when the notions of…

We continue the effort of grokking the structure of power-bounded $T$-convex valued fields, whose theory is in general referred to as TCVF. In the present paper our focus is on certain expansion of it that is equipped with a tempered…

Extension problems for polynomial valuations on different cones of convex functions are investigated. It is shown that for the classes of functions under consideration, the extension problem reduces to a simple geometric obstruction on the…

We study the combination of two o-minimal extensions of the theory of real closed fields: one by a T-convex subring and the other by a T-derivation. Let T be a complete, model complete o-minimal extension of RCF. We show that the combined…

Let $\mathbb{T}$ be the differential field of logarithmic-exponential transseries. We show that the expansion of $\mathbb{T}$ by its natural exponential function is model complete and locally o-minimal. We give an axiomatization of the…

We study a class of tame $\mathcal{L}$-theories $T$ of topological fields and their $\mathcal{L}_\delta$-extension $T_{\delta}^*$ by a generic derivation $\delta$. The topological fields under consideration include henselian valued fields…

For every natural number $m$, the existentially closed models of the theory of fields with $m$ commuting derivations can be given a first-order geometric characterization in several ways. In particular, the theory of these differential…

We define a notion of residue field domination for valued fields which generalizes stable domination in algebraically closed valued fields. We prove that a real closed valued field is dominated by the sorts internal to the residue field,…

Let $T$ be the theory of an o-minimal field and $T_0$ a common reduct of $T$ and $T_{an}$. I adapt Mourgues' and Ressayre's constructions to deduce structure results for $T_0$-reducts of $T$-$\lambda$-spherical completion of models of…