arXiv subjects

Moshe Kamensky

Publications and source records attributed to Moshe Kamensky.

14 recordsLinked to original sources

arXiv2025

Binding groups for algebraic dynamics

A binding group theorem is proved in the context of quantifier-free internality to the fixed field in difference-closed fields of characteristic zero. This is articulated as a statement about the birational geometry of isotrivial algebraic dynamical systems, and more generally isotrivial $σ$-varieties. It asserts that if $(V,ϕ)$ is an isotrivial $σ$-variety then a certain subgroup of the group of birational transformations of $V$, namely those that preserve all the relations between $(V,ϕ)$ and the trivial dynamics on the affine line, is in fact an algebraic group. Several application are given including new special cases of the Zariski Dense Orbit Conjecture and the Dixmier-Moeglin Equivalence Problem in algebraic dynamics, as well as finiteness results about the existence of nonconstant invariant rational functions on cartesian powers of $σ$-varieties. These applications give algebraic-dynamical analogues of recent results in differential-algebraic geometry.

math.LO↗

arXiv2025

A transformal transcendence result for algebraic difference equations

Given an algebraic difference equation of the form \[σ^n(y)=f\big(y, σ(y),\dots,σ^{n-1}(y)\big)\] where $f$ is a rational function over a field $k$ of characteristic zero on which $σ$ acts trivially, it is shown that if there is a nontrivial algebraic relation amongst any number of $σ$-disjoint solutions, along with their $σ$-transforms, then there is already such a relation between three solutions. Here ``$σ$-disjoint" means $a\neqσ^r(b)$ for any integer $r$. A weaker version of the theorem, where ``three" is replaced by $n+4$, is also obtained when $σ$ acts non-trivially on $k$. Along the way a number of other structural results about primitive rational dynamical systems are established. These theorems are deduced as applications of a detailed model-theoretic study of finite-rank quantifier-free types in the theory of existentially closed difference fields of characteristic zero. In particular, it is also shown that the degree of non-minimality of such types over fixed-field parameters is bounded by $2$.

math.LO↗

arXiv2023

Higher internal covers

We define and study a higher-dimensional version of model theoretic internality, and relate it to higher-dimensional definable groupoids in the base theory.

math.LO↗

arXiv2021

Peterzil-Steinhorn subgroups and $μ$-stabilizers in ACF

We consider $G$, a linear group defined over $k$, an algebraically closed field. By considering $k$ as an embedded residue field of an algebraically closed valued field $K$, we can associate to it a compact $G$-space $S^μ_G(k)$, consisting of $μ$-types on $G$. We showed that for each $p_μ\in S^μ_G(k)$, $\text{Stab}^μ(p)=\text{Stab}(p_μ)$ is a solvable infinite algebraic group when $p_μ$ is centered at infinity and residually algebraic. Moreover we give a description of the dimension $\text{Stab}(p_μ)$ in terms of dimension of $p$.

math.LO↗

arXiv2019

Model theory of fields with free operators in positive characteristic

We give algebraic conditions about a finite algebra $B$ over a perfect field of positive characteristic, which are equivalent to the companionability of the theory of fields with "$B$-operators" (i.e. the operators coming from homomorphisms into tensor products with $B$). We show that, in the most interesting case of a local $B$, these model companions admit quantifier elimination in the "smallest possible" language and they are strictly stable. We also describe the forking relation there.

math.LO↗

arXiv2018

Picard--Vessiot structures

We demonstrate existence and uniqueness of Picard--Vessiot extensions satisfying prescribed properties, for systems of linear differential equations over a field satisfying the same properties, under some closure assumptions on the field of constants. An example includes the case of a equations over a real field, with a real-closed field of constants. The result is obtained through a model theoretic approach.

math.CA↗

arXiv2017

Imaginaries in separably closed valued fields

We show that separably closed valued fields of finite imperfection degree (either with lambda-functions or commuting Hasse derivations) eliminate imaginaries in the geometric language. We then use this classification of interpretable sets to study stably dominated types in those structures. We show that separably closed valued fields of finite imperfection degree are metastable and that the space of stably dominated types is strict pro-definable.

math.LO↗

arXiv2016

Interpretations and differential Galois extensions

We give model theoretic accounts and proofs of the existence and uniqueness of differential Galois extensions with no new constants, for logarithmic differential equations over a differential field K, when the field C of constants of K is not necessarily algebraically closed, under a variety of assumptions on C and K.

math.AG↗

arXiv2012

Tannakian formalism over fields with operators

We develop a theory of tensor categories over a field endowed with abstract operators. Our notion of a "field with operators", coming from work of Moosa and Scanlon, includes the familiar cases of differential and difference fields, Hasse-Schmidt derivations, and their combinations. We develop a corresponding Tannakian formalism, describing the category of representations of linear groups defined over such fields. The paper extends the previously know (classical) algebraic and differential algebraic Tannakian formalisms.

math.RT↗

arXiv2010

Model theory and the Tannakian formalism

We draw the connection between the model theoretic notions of internality and the binding group on one hand, and the Tannakian formalism on the other. More precisely, we deduce the fundamental results of the Tannakian formalism by associating to a Tannakian category a first order theory, and applying the results on internality there. We also formulate the notion of a differential tensor category, and a version of the Tannakian formalism for differential linear groups, and show how the same techniques can be used to deduce the analogous results in that context.

math.LO↗

arXiv2010

A categorical approach to internality

Model theoretic internality provides conditions under which the group of automorphisms of a model over a reduct is itself a definable group. In this paper we formulate a categorical analogue of the condition of internality, and prove an analogous result on the categorical level. The model theoretic statement is recovered by considering the category of definable sets.

math.LO↗

arXiv2009

Definable groups of partial automorphisms

The motivation for this paper is to extend the known model theoretic treatment of differential Galois theory to the case of linear difference equations (where the derivative is replaced by an automorphism.) The model theoretic difficulties in this case arise from the fact that the corresponding theory ACFA does not eliminate quantifiers. We therefore study groups of restricted automorphisms, preserving only part of the structure. We give conditions for such a group to be (infinitely) definable, and when these conditions are satisfied we describe the definition of the group and the action explicitly. We then examine the special case when the theory in question is obtained by enriching a stable theory with a generic automorphism. Finally, we interpret the results in the case of ACFA, and explain the connection of our construction with the algebraic theory of Picard-Vessiot extensions. The only model theoretic background assumed is the notion of a definable set.

math.LO↗

arXiv2007

The model completion of the theory of modules over finitely generated commutative algebras

We find the model completion of the theory modules over $A$, where $A$ is a finitely generated commutative algebra over a field $K$. This is done in a context where the field $K$ and the module are represented by sorts in the theory, so that constructible sets associated with a module can be interpreted in this language. The language is expanded by additional sorts for the Grassmanians of all powers of $K^n$, which are necessary to achieve quantifier elimination. The result turns out to be that the model completion is the theory of a certain class of ``big'' injective modules. In particular, it is shown that the class of injective modules is itself elementary. We also obtain an explicit description of the types in this theory.

math.LO↗

arXiv2006

Ind- and Pro- definable sets

We describe the ind- and pro- categories of the category of definable sets, in some first order theory, in terms of points in a sufficiently saturated model.

math.LO↗