Binding groups for algebraic dynamics Permalink


Preprint, 2024

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.

Higher internal covers Permalink

Published in Model Theory, 2023

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

Model theory of fields with free operators in positive characteristic Permalink


Published in Trans. Amer. Math. Soc., 2019

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.

Imaginaries in separably closed valued fields Permalink


Published in Proc. Lond. Math. Soc. (3), 2018

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.

Interpretations and differential Galois extensions Permalink


Published in Int. Math. Res. Not. IMRN, 2016

We give accounts and proofs, using model-theoretic methods among other things, of the following results: Suppose ∂y = Ay is a linear differential equation over a differential field K of characteristic 0, and the field CK of constants of K is existentially closed in K. Then, (i) there exists a Picard–Vessiot extension L of K, namely a differential field extension L of K which is generated by a fundamental system of solutions of the equation, and has no new constants; (ii) if L1 and L2 are two Picard–Vessiot extensions of K which (as fields) have a common embedding over K into an elementary extension of CK, then L1 and L2 are isomorphic over K as differential fields; and (iii) suppose that CK is large in the sense of Pop [21] and also has only finitely many extensions of degree n for all n (Serre’s property (F)). Then, K has a Picard–Vessiot extension L such that CK is existentially closed in L. In fact we state and prove our results in the more general context of logarithmic differential equations over K on (not necessarily linear) algebraic groups over CK, and the corresponding strongly normal extensions of K. We make use of interpretations from model theory as well the Galois groupoid, which are related to the Tannakian theory in [3, 4], but go beyond the linear context. Towards the proof of (iii) we obtain a Galois-cohomological result of possibly independent interest: if k is a field of characteristic 0 with property (F), and G is any algebraic group over k, then H1(k,G) is countable. The current paper replaces the preprint [8] which only dealt with the linear differential equations case and had some mistakes.

Model theory and the Tannakian formalism Permalink

Published in Trans. Amer. Math. Soc., 2015

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.

Tannakian formalism over fields with operators Permalink

Published in Int. Math. Res. Not. IMRN, 2013

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.

A categorical approach to internality Permalink

Published in Models, logics, and higher-dimensional categories, 2011

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.

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

Published in J. Symbolic Logic, 2009

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 Kn, 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.

Definable groups of partial automorphisms Permalink

Published in Selecta Math. (N.S.), 2009

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.

Ind- and Pro- definable sets Permalink

Published in Ann. Pure Appl. Logic, 2007

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.