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Jan Dobrowolski

Publications and source records attributed to Jan Dobrowolski.

At least 19 recordsLinked to original sources
arXiv2025

Automorphisms of valued fields: amalgamation and existential closedness

We study valued fields equipped with an automorphism. We prove that all of them have an extension admitting an equivariant cross-section of the valuation. In residual characteristic zero, and in the presence of such a cross-section, we show that amalgamation problems are solvable precisely when the induced residual problem is, characterise the existentially closed objects of this category, and prove that its positive theory does not have the tree property of the second kind. We prove analogous results with cross-sections replaced by angular components. Along the way, we show that array modelling does not require thickness.

math.LO↗
arXiv2025

Elekes-Szabó for collinearity on cubic surfaces

We study the orchard problem on cubic surfaces. We classify possibly reducible cubic surfaces $X\subseteq \mathbb{P}^3(\C)$ with smooth components on which there exist families of finite sets (of unbounded size) with quadratically many 3-rich lines which do not concentrate (in a natural sense) on any projective plane. Namely, we prove that such a family exists precisely when $X$ is a union of three planes sharing a common line. Along the way, we obtain a general result about nilpotency of groups admitting an algebraic action satisfying an Elekes-Szabó condition, and we prove the following purely algebrogeometric statement: if the composition of four Geiser involutions through sufficiently generic points $a,b,c,d$ on a smooth irreducible cubic surface has infinitely many fixed points, then a single plane contains $a,b,c,d$ and all but finitely many of the fixed points.

math.LO↗
arXiv2025

Fields with Lie-commuting and iterative operators

We introduce a general framework for studying fields equipped with operators, given as co-ordinate functions of homomorphisms into a local algebra $\mathcal{D}$, satisfying various compatibility conditions that we denote by $Γ$ and call such structures $\mathcal{D}^Γ$-fields. These include Lie-commutativity of derivations and $\mathfrak g$-iterativity of (truncated) Hasse-Schmidt derivations. Our main result is about the existence of principal realisations of $\mathcal{D}^Γ$-kernels. As an application, we prove companionability of the theory of $\mathcal{D}^Γ$-fields and denote the companion by $\mathcal{D}^Γ$-CF. In characteristic zero, we prove that $\mathcal{D}^Γ$-CF is a stable theory that satisfies the CBP and Zilber's dichotomy for finite-dimensional types. We also prove that there is a uniform companion for model-complete theories of large $\mathcal{D}^Γ$-fields, which leads to the notion of $\mathcal{D}^Γ$-large fields and we further use this to show that PAC substructures of $\mathcal{D}^Γ$-DCF are elementary.

math.LO↗
arXiv2024

Corrigendum to "Kim-independence in positive logic"

The proof of the Independence Theorem for Kim-independence in positive thick NSOP$_1$ theories from (Dobrowolski and Kamsma, 2022) contains a gap. The theorem is still true, and in this corrigendum we give a different proof.

math.LO↗
arXiv2023

Elekes-Szabó for groups, and approximate subgroups in weak general position

We show that with a suitable weak notion of general position, the Elekes-Szabó condition on the group operation of a connected complex algebraic group characterises nilpotence of the group. Along the way, we prove a Mordell-Lang result for generic finitely generated subgroups of commutative complex algebraic groups.

math.CO↗
arXiv2022

The Amalgamation Property for automorphisms of ordered abelian groups

We prove that the category of ordered abelian groups equipped with an automorphism has the Amalgamation Property, deduce that their inductive theory is NIP in the sense of positive logic, and initiate a development of the latter framework. As byproducts of the proof, we obtain a generalised version of the Hahn Embedding Theorem which allows to lift each automorphism of an ordered abelian group to one of an ordered real vector space, and we show that, on existentially closed structures, linear combinations of iterates of the automorphism have the Intermediate Value Property.

math.LO↗
arXiv2022

Kim-independence in positive logic

An important dividing line in the class of unstable theories is being NSOP$_1$, which is more general than being simple. In NSOP$_1$ theories forking independence may not be as well-behaved as in stable or simple theories, so it is replaced by another independence notion, called Kim-independence. We generalise Kim-independence over models in NSOP$_1$ theories to positive logic -- a proper generalisation of first-order logic where negation is not built in, but can be added as desired. For example, an important application is that we can add hyperimaginary sorts to a positive theory to get another positive theory, preserving NSOP$_1$ and various other properties. We prove that, in a thick positive NSOP$_1$ theory, Kim-independence over existentially closed models has all the nice properties that it is known to have in a first-order NSOP$_1$ theory. We also provide a Kim-Pillay style theorem, characterising which thick positive theories are NSOP$_1$ by the existence of a certain independence relation. Furthermore, this independence relation must then be the same as Kim-independence. Thickness is the mild assumption that being an indiscernible sequence is type-definable. In first-order logic Kim-independence is defined in terms of Morley sequences in global invariant types. These may not exist in thick positive theories. We solve this by working with Morley sequences in global Lascar-invariant types, which do exist in thick positive theories. We also simplify certain tree constructions that were used in the study of Kim-independence in first-order theories. In particular, we only work with trees of finite height.

math.LO↗
arXiv2021

On rank not only in NSOP1 theories

We introduce a family of local ranks DQ depending on a finite set Q of pairs of the form (φ(x,y),q(y)) where φ(x,y) is a formula and q(y) is a global type. We prove that in any NSOP1 theory these ranks satisfy some desirable properties; in particular, DQ(x=x)<ωfor any finite variable x and any Q, if q\supseteq p is a Kim-forking extension of types, then DQ(q)<DQ(p) for some Q, and if q\supseteq p is a Kim-non-forking extension, then DQ(q)=DQ(p) for every Q that involves only invariant types whose Morley powers are \ind^K-stationary. We give natural examples of families of invariant types satisfying this property in some NSOP1 theories. We also answer a question of Granger about equivalence of dividing and dividing finitely in the theory T_\infty of vector spaces with a generic bilinear form. We conclude that forking equals dividing in T_\infty, strengthening an earlier observation that T_\infty satisfies the existence axiom for forking independence. Finally, we slightly modify our definitions and go beyond NSOP1 to find out that our local ranks are bounded by the well-known ranks: the inp-rank (burden), and hence, in particular, by the dp-rank. Therefore, our local ranks are finite provided that the dp-rank is finite, for example if T is dp-minimal. Hence, our notion of ranks identifies a non-trivial class of theories containing all NSOP1 and NTP2 theories.

math.LO↗
arXiv2020

Sets, groups, and fields definable in vector spaces with a bilinear form

We study definable sets, groups, and fields in the theory $T_\infty$ of infinite-dimensional vector spaces over an algebraically closed field equipped with a nondegenerate symmetric (or alternating) bilinear form. First, we define an ($\mathbb{N}\times \mathbb{Z},\leq_{lex}$)-valued dimension on definable sets in $T_\infty$ enjoying many properties of Morley rank in strongly minimal theories. Then, using this dimension notion as the main tool, we prove that all groups definable in $T_\infty$ are (algebraic-by-abelian)-by-algebraic, which, in particular, answers a question of Granger. We conclude that every infinite field definable in $T_\infty$ is definably isomorphic to the field of scalars of the vector space. We derive some other consequences of good behaviour of the dimension in $T_\infty$, e.g. every generic type in any definable set is a definable type; every set is an extension base; every definable group has a definable connected component. We also consider the theory $T^{RCF}_\infty$ of vector spaces over a real closed field equipped with a nondegenerate alternating bilinear form or a nondegenerate symmetric positive-definite bilinear form. Using the same construction as in the case of $T_\infty$, we define a dimension on sets definable in $T^{RCF}_\infty$, and using it we prove analogous results about definable groups and fields: every group definable in $T^{RCF}_{\infty}$ is (semialgebraic-by-abelian)-by-semialgebraic (in particular, it is (Lie-by-abelian)-by-Lie), and every field definable in $T^{RCF}_{\infty}$ is definable in the field of scalars, hence it is either real closed or algebraically closed.

math.LO↗
arXiv2019

Elementary Equivalence Theorem for PAC structures

We generalize a well-known theorem binding the elementary equivalence relation on the level of PAC fields and the isomorphism class of their absolute Galois groups. Our results concern two cases: saturated PAC structures and non-saturated PAC structures.

math.LO↗
arXiv2019

Independence over arbitrary sets in NSOP$_1$ theories

We study Kim-independence over arbitrary sets. Assuming that forking satisfies existence, we establish Kim's lemma for Kim-dividing over arbitrary sets in an NSOP$_{1}$ theory. We deduce symmetry of Kim-independence and the independence theorem for Lascar strong types.

math.LO↗
arXiv2017

Left-ordered inp-minimal groups

We prove that any left-ordered inp-minimal group is abelian, and we provide an example of a non-abelian left-ordered group of dp-rank 2.

math.LO↗
arXiv2017

The Lascar groups and the 1st homology groups in model theory

Let $p$ be a strong type of an algebraically closed tuple over $B=\acl^{\eq}(B)$ in any theory $T$. Depending on a ternary relation $\indo^*$ satisfying some basic axioms (there is at least one such, namely the trivial independence in $T$), the first homology group $H^*_1(p)$ can be introduced, similarly to \cite{GKK1}. We show that there is a canonical surjective homomorphism from the Lascar group over $B$ to $H^*_1(p)$. We also notice that the map factors naturally via a surjection from the `relativised' Lascar group of the type (which we define in analogy with the Lascar group of the theory) onto the homology group, and we give an explicit description of its kernel. Due to this characterization, it follows that the first homology group of $p$ is independent from the choice of $\indo^*$, and can be written simply as $H_1(p)$. As consequences, in any $T$, we show that $|H_1(p)|\geq 2^{\aleph_0}$ unless $H_1(p)$ is trivial, and we give a criterion for the equality of stp and Lstp of algebraically closed tuples using the notions of the first homology group and a relativised Lascar group. We also argue how any abelian connected compact group can appear as the first homology group of the type of a model.

math.LO↗
arXiv2014

Topologies induced by group actions

We introduce some canonical topologies induced by actions of topological groups on groups and rings. For $H$ being a group [or a ring] and $G$ a topological group acting on $H$ as automorphisms, we describe the finest group [ring] topology on $H$ under which the action of $G$ on $H$ is continuous. We also study the introduced topologies in the context of Polish structures. In particular, we prove that there may be no Hausdorff topology on a group $H$ under which a given action of a Polish group on $H$ is continuous.

math.GN↗