Miniworkshop: Recent developments in the model theory of fields

Let $T$ be a complete, model-complete geometric dp-minimal $ℒ$-theory of topological fields of characteristic $0$ and let $T(\partial )$ be the theory of expansions of models of $T$ by a derivation ∂. We assume that $T(\partial )$ has a model-companion $T_{\partial }$.

Let $\mathrm{\Gamma }$ be a finite-dimensional ${ℒ}_{\partial }$-definable group in a model of $T_{\partial }$. Then we show that $\mathrm{\Gamma }$ definably and densely embeds in an $ℒ$-definable group $G$.

Further, using a $C^1$-cell decomposition result, we show that $\mathrm{\Gamma }$ definably and densely embeds in a definable $D$-group, generalizing the classical construction of Buium of algebraic $D$-groups and extending for that class of fields, results obtained in [1], [2]. Finally we will examine, mainly in the o-minimal case, transfer of algebraic properties between $\mathrm{\Gamma }$ and $G$.

[1] Peterzil K., Pillay, A. Point F., On definable groups in real-closed fields with a generic derivation, and related structures: I, ArXiv:2208.08293, submitted.

[2] Peterzil K., Pillay, A. Point F., On definable groups and $D$-groups in certain fields with a generic derivation, Canad. J. Math. Vol. 77 (2), 2025 pp. 459--480.

In joint work with Nigel Pynn-Coates, we consider o-minimal fields equipped with a compatible valuation and derivation, under the additional assumption that the derivation is monotone (that is, the valuation of any element is no larger than the valuation of its derivative). We develop a version of differential henselianity for these fields, and we use this to prove an Ax-Kochen/Ershov result. In the special case that the underlying o-minimal field is an elementary extension of the real field with restricted analytic functions, we show that any model which satisfies our analog of differential henselianity admits an elementary embedding into a power series model.

First, I will present a simplified proof of descent for stably dominated types in $\mathrm{ACVF}$. I will also state a more general version of descent for stably dominated types in any theory, dropping the hypothesis of the existence of invariant extensions. This first part is joint work with Pierre Simon.

In the second part, motivated by the study of the space of definable types orthogonal to the value group in a henselian valued field and their cohomology; I will present a theorem that states that over an algebraically closed base of imaginary elements, a global invariant type is residually dominated (essentially controlled by the residue field) if and only if it is orthogonal to the value group , if and only if its reduct in $\mathrm{ACVF}$ is stably dominated.

This is joint work with Pablo Cubides and Silvain Rideau-Kikuchi. The result extend to some valued fields with operators

The Ax–Kochen/Ershov Theorem, proved 60 years ago, axiomatized equicharacteric zero henselian valued fields in terms of their residue fields and value groups. This principle has been extended in many contributions, over the years, to include separably tame and finitely ramified henselian valued fields, and recently even perfectoid fields, by work of Jahnke and Kartas. In the finitely ramified case, one needs to identify the structure induced on the residue field. I will explain some recent developments in this rather classical story, with a focus on "fragmented" AKE principles: for a set $F$ of formulas (e.g. the set of existential formulas) the $F$-theory of a valued field depends only on the $F$-theories of the residue field and value group. This talk includes various joint works with a variety of coauthors: Dittmann, Fehm, Jahnke.

The theory of closed H-fields is model complete and axiomatizes the theory of transseries and maximal Hardy fields, as Aschenbrenner, Van den Dries, and Van der Hoeven have shown in a long series of works. To better understand large closed H-fields, such as maximal Hardy fields, I recently extended this model completeness to the theory of tame pairs of closed H-fields. Building on this work, I will explain how to extend differential-algebraic dimension on a closed H-field to tame pairs of closed H-fields so that it is a fibred dimension function in the sense of [L. van den Dries, "Dimension of definable sets, algebraic boundedness and Henselian fields", Ann. Pure Appl. Logic 45.2 (1989), 189–209] and the nonempty dimension zero definable sets are exactly the nonempty discrete definable sets.

In real or complex analysis, the Taylor expansion of a function at a given point contains all the local information of that function around that point, and Taylor series can be used both to study and to define analytic functions. In o-minimal geometry, it is usual to embed algebras of real-valued regular functions into algebras of functions defined on ordered fields of generalised power series, such as transseries or generalisations thereof. The Taylor expandableness of o-minimal real-valued functions should translate into the existence of formal Taylor expansions of their formal avatars.

I will show how to define Taylor expansions for functions over generalised power series, and show that composition laws on fields of generalised transseries can be understood and defined using such expansions. This is based on joint work with Vincenzo Mantova and the master student Shaun Dean (Konstanz).

By the results of Schmitt ([3]) and Cluckers-Halupczok ([2]), many first-order properties of ordered abelian groups can be reduced to corresponding properties of their spines, which are chains of uniformly definable convex subgroups. We investigate the property of augmentability by infinite elements for ordered abelian groups. Specifically, we say that an ordered abelian group $G$ is augmentable by infinite elements if there exists an ordered abelian group $H$ such that $G⪯H\oplus G$. Via reduction to the spines, we show that every non-trivial ordered abelian group is augmentable by infinite elements. This result has implications for the study of definable henselian valuations. In particular, we show that a field $k$ of characteristic zero is not t-henselian (i.e., not elementarily equivalent to any field admitting a non-trivial henselian valuation) if and only if all henselian valuations with residue field $k$ are ($\mathrm{\varnothing }$-)definable in the language of rings.

This is joint work with B. Boissonneau, F. Jahnke, and P. Touchard ([1]).

[1] B. Boissonneau, A. De Mase, F. Jahnke, P. Touchard. Growing spines ad infinitum, arXiv:2501.10531 [math.LO], 2025.

[2] R. Cluckers and I. Halupczok. Quantifier elimination in ordered abelian groups. Confluentes Math., 3(4):587–615, 2011.

[3] P. H. Schmitt. Model theory of ordered abelian groups, 1982. Habilitationsschrift.

We show that every ${\mathrm{NTP}}_2$ expansion of the field of real numbers by closed sets defines only finite Boolean combinations of closed sets (i.e. constructible sets). The same result holds for the field of $p$-adic numbers. More generally, the open core of every ${\mathrm{NTP}}_2$ uniform topological structure has quantifier elimination, under a certain definable $\sigma$-compactness condition. We ask which combinatorial phenomena can arise, in expansions of the real field, from the existence of a definable non-constructible $F_{\sigma }$ set.

I'll discuss some joint work with Gal Binyamini, Harry Schmidt and Margaret Thomas in which we prove a uniform Manin-Mumford result for products of CM elliptic curves. I'll try to briefly mention some ingredients for the proof (including a definability result), and then discuss a model theoretic application due to Andrew Harrison-Migochi.

We introduce a class of tame extensions of real closed valued fields, called 'residually constructible extensions', and show that these extensions are reasonably robust, while also having much simpler behaviour than arbitrary extensions. We then apply these results to answer a question of Marcus Tressl. (Joint work with Pietro Freni.)

We give a classification of 1-variable types in extensions of o-minimal expansions of ordered abelian groups and real closed fields. This is achieved by a valuation theoretic analysis of types, leading to the trichotomy:

(i) immediate transcendental (ii) value transcendental (iii) residue transcendental.

As application, we give necessary and sufficient conditions for a power bounded o-minimal expansion of a real closed field (in a language of arbitrary cardinality) to be $\kappa$-saturated. The conditions are in terms of the value group, residue field, and $\kappa$- bounded pseudo-Cauchy sequences of the natural valuation on the real closed field. A further application is a characterization of recursively saturated models. This provides a construction method for saturated and recursively saturated models, using fields of generalized power series. This is based on joint work with P. D'Aquino and K. Lange.

This is a report on joint work in progress with Rahim Moosa, but with origins in a paper with Jaoui and Jimenez.

Analogies between the finite rank part of ${\mathrm{DCF}}_0$ and $\mathrm{CCM}$ have been quite fruitful, with for example the constants corresponding to the sort $P^1$.

Some questions we would like to answer include, 1) what is the $\mathrm{CCM}$ analogue of a linear differential equation? 2) Which algebraic groups appear as binding groups for types internal to $P^1$?