We study groups definable in existentially closed geometric fields with commuting derivations. Our main result is that such a group can be definably embedded in a group interpretable in the underlying geometric field. Compared to earlier work of the first two authors toguether with K. Peterzil, the novelty is that we also deal with infinite dimensional groups.

We study finite-dimensional groups definable in models of the theory of real closed fields with a generic derivation (also known as CODF). We prove that any such group definably embeds in a semialgebraic group. We extend the results to several more general contexts; strongly model complete theories of large geometric fields with a generic derivation, model complete o-minimal expansions of RCF with a generic derivation, open theories of topological fields with a generic derivation. We also give a general theorem on recovering a definable group from generic data in the context of geometric structures.

We axiomatize a class of existentially closed differential expansions of exponential topological fields where the derivation is an E-derivation. We apply our results to differential expansions of, on the one hand the field of real numbers endowed with exp(x), the classical exponential function defined by its power series expansion, and on the other hand the field of p-adic numbers endowed with the function exp(px) defined on the subring of p-adic integers where pis a prime number strictly bigger than 2 (or with exp(4x) when p = 2). (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let T be a complete, model-complete, geometric dp-minimal ℒ-theory of topological fields of characteristic 0 and let T(∂) be the theory of expansions of models of T by a derivation ∂. We assume that T(∂) has a model-companion T_∂. Let Γ be a finite-dimensional ℒ_∂-definable group in a model of T_∂. Then we show that Γ densely and definably embeds in an ℒ-definable group G. Further, using a C^1-cell decomposition result, we show that Γ densely and definably 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 arXiv:2208.08293, arXiv:2305.16747.

We continue our study from Peterzil et al. (2022, Preprint, arXiv:2208.08293) of finite-dimensional definable groups in models of the theory T-partial derivative, the model companion of an o-minimal L-theory T expanded by a generic derivation partial derivative as in Fornasiero and Kaplan (2021, Journal of Mathematical Logic 21, 2150007). We generalize Buium's notion of an algebraic D-group to L-definable D-groups, namely (G,s), where G is an L-definable group in a model of T, and s:G ->tau(G) is an L-definable group section. Our main theorem says that every definable group of finite dimension in a model of T partial derivative is definably isomorphic to a group of the form (G,s)(partial derivative)= {g is an element of G:s(g)=del g}, for some L-definable D-group (G,s) (where del (g)=(g, partial derivative g)). We obtain analogous results when T is either the theory of p-adically closed fields or the theory of pseudo-finite fields of characteristic 0.

Outside of the framework of geometric theories, we exhibit complete, respectively model-complete theories of rings whose corresponding theory of pairs is complete, respectively model-complete, using transfer results proven in the seventies for boolean products of structures. It includes certain boolean products of pairs of dp-minimal fields of characteristic $0$. We also show, as in the case of pairs of fields, how it fits in the framework of differential rings.

The workshop brought together researchers with expertise in areas of mathematics where model theory has had interesting applications. The areas of expertise spanned from expansions of o-minimal structures preserving tame geometric properties to expansions of specified fields by classical operators that preserve neo-stability properties. There were presentations and discussions on recent developments in definable groups and decompositions in relatively tame setups, the interplay of different notions of dimension and closure operators, and applications of the model theory of differential fields to diophantine geometry.

We continue our earlier study of finite dimensional definable groups in models of the the model companion of an o-minimal L-theory T expanded by a generic derivation as in [F-K]. We generalize Buium's notion of an algebraic D-group to L-definable D-groups, namely (G,s), where G is a L-definable group in a model of T, and s is an L-definable group section into the prolongation of G. Our main theorem says that every definable group of finite dimension in a model of the theory is definably isomorphic to the ``sharp'' points of an L-definable D-group. We obtain analogous results when T is either the theory of p-adically closed fields or the theory of pseudo-finite fields of characteristic zero.

Let $(\Gamma,+,F)$ be a finitely generated $\mathbb Z[F]$-module where $F$ is an injective endomorphism of the abelian group $\Gamma$. We restrict ourselves to a finite automa presentable subclass, introduced by J. Bell and R. Moosa in "F-sets and finite automata. J. Th\'eor. Nombres Bordeaux 31 (2019), no. 1, 101-130" and define an expansion containing the $\mathcal F$-sets defined by R. Moosa and T. Scanlon in "Am. J. Math. 126 (2004), no. 3, p. 473-522", where every automatic subset is definable.

We study a class of tame L-theories T of topological fields and their Lδ-extension Tδ⁎ by a generic derivation δ. The topological fields under consideration include henselian valued fields of characteristic 0 and real closed fields. We show that the associated expansion by a generic derivation has L-open core (i.e., every Lδ-definable open set is L-definable) and derive both a cell decomposition theorem and a transfer result of elimination of imaginaries. Other tame properties of T such as relative elimination of field sort quantifiers, NIP and distality also transfer to Tδ⁎. As an application, we derive consequences for the corresponding theories of dense pairs. In particular, we show that the theory of pairs of real closed fields (resp. of p-adically closed fields and real closed valued fields) admits a distal expansion. This gives a partial answer to a question of P. Simon.

We prove a version of a Nullstellensatz for partial exponential fields $(K,E)$, even though the ring of exponential polynomials $K[X_1,\ldots,X_n]^E$ is not a Hilbert ring. We show that under certain natural conditions one can embed an ideal of $K[X_1,\ldots,X_n]^E$ into an exponential ideal. In case the ideal consists of exponential polynomials with one iteration of the exponential function, we show that these conditions can be met. We apply our results to the case of ordered exponential fields.

We axiomatize a class of existentially closed exponential fields equipped with an $E$-derivation. We apply our results to the field of real numbers endowed with $exp(x)$ the classical exponential function defined by its power series expansion and to the field of p-adic numbers endowed with the function $exp(px)$ defined on the $p$-adic integers where $p$ is a prime number strictly bigger than $2$ (or with $exp(4x)$ when $p=2$).

Let B be a commutative Bézout domain and let MSpec(B) be the maximal spectrum of B. We obtain a Feferman-Vaught type theorem for the class Mod-B of all (right) B-modules. We analyze the definable sets in terms, on the one hand, of the definable sets in the classes Mod-BM, where BM ranges over the localizations of B at M, M∈MSpec(B), and on the other hand, of the constructible subsets of MSpec(B). This allows us to derive decidability results for the class Mod-B, in particular when B is the ring Z˜ of algebraic integers or one of the rings Z˜∩R,Z˜∩Qp.

Call a (strictly increasing) sequence (rn) of natural numbers regular if it satisfies the following condition: rn+1/rn→θ∈R>1∪{∞} and, if θ is algebraic, then (rn) satisfies a linear recurrence relation whose characteristic polynomial is the minimal polynomial of θ. Our main result states that (Z,+,0,R) is superstable whenever R is enumerated by a regular sequence. We give two proofs of this result. One relies on a result of E. Casanovas and M. Ziegler and the other on a quantifier elimination result. We also show that (Z,+,0,<,R) is NIP whenever R is enumerated by a regular sequence that is ultimately periodic modulo m for all m>1.

We review results obtained on the model theory of valued modules over skew polynomial rings and Bezout domains in the definablility and decidability point of view.

The following strong form of density of definable types is introduced for theories T admitting a fibered dimension function d: given a model M of T and a definable set X subset of M-n, there is a definable type p in X, definable over a code for X and of the same d-dimension as X. Both o-minimal theories and the theory of closed ordered differential fields (CODF) are shown to have this property. As an application, we derive a new proof of elimination of imaginaries for CODF.

We will describe the Ziegler spectrum over the ring of entire complex valued functions.

Given a model-complete theory of topological fields, we considered its generic differential expansions and under a certain hypothesis of largeness, we axiomatised the class of existentially closed ones. Here we show that a density result for definable types over definably closed subsets in such differential topological fields. Then we show two transfer results, one on the VC-density and the other one, on the combinatorial property NTP2.

Given a dense additive subgroup $G$ of $\mathbb R$ containing $\mathbb Z$, we consider its intersection $\mathbb G$ with the interval $[0,1[$ with the induced order and the group structure given by addition modulo $1$. We axiomatize the theory of $\mathbb G$ and show it is model-complete, using a Feferman-Vaught type argument. We show that any sufficiently saturated model decomposes into a product of a "standard" part and two ordered semigroups of infinitely small and infinitely large elements.