arXiv subjects

Anand Pillay

Publications and source records attributed to Anand Pillay.

At least 19 recordsLinked to original sources

arXiv2026

Relative stability, relative categoricity and internal covers

Let $T$ be a countable complete theory with a distinguished unary predicate $P$, and let $T^{P}$ be the theory of the $P$-parts of models of $T$ with the induced structure. $T$ is said to be relatively categorical or categorical over $P$ if any isomorphism between the $P$-parts of two models of $T$ lifts to an isomorphism of the models in question. We study the special case of relative categoricity where $T$ is internal to $T^{P}$ (that is, every model $M$ of $T$ is in the definable closure of $P(M)$ together with additional parameters from $M$). We first give a structure theory for such $T$: after passing to $T^{eq}$ and naming a parameter, $T$ is the same thing as a "pure torsor cover" of $T^{P}$, namely simply adjoining to $T^{P}$ a new sort for a torsor $S$ for a $\emptyset$-definable group $G$ in $T^{P}$, with no additional structure. We discuss relative stability, or stability over $P$, and give a characterization of relative stability, superstability, and $\omega$-stability of $T$ in terms of $H$ having the stable chain condition, superstable chain condition, and $\omega$-stable chain condition, respectively.

math.LO↗

arXiv2026

Notes on Lie derivatives, algebraic D-varieties, and Ax's theorem

We discuss the relationship between Lie derivatives and the linear differential equations on cotangent spaces of algebraic D-varieties at sharp points. We also take the liberty to give an account of Ax's theorem (which may be useful as an entry point to the subject for students).

math.AG↗

arXiv2026

Remarks on relative categoricity

The paper is partly a survey with historical background and references, partly provides the opportunity to put in print some unpublished early work, and partly has new results. A special case of relative categoricity is identified (almost internal covers) for which the Gaifman conjecture is proved, full omega-stability over P is introduced, and as counterexample is given to full stability over P.

math.LO↗

arXiv2026

On groups definable in $p$-adically closed fields

This paper is about the $dfg$/$fsg$ decomposition for groups $G$ definable in $p$-adically closed fields. It is proved that for $G$ definably amenable, $G$ has a definable normal $dfg$ subgroup $H$ such that the quotient $G/H$ is a definable $fsg$ group. The result was known for groups definable in $o$-minimal expansions of real closed fields (see \cite{C-P-o-mini}). We also give a version for arbitrary (not necessarily definably amenable) groups $G$ definable in $p$-adically closed fields: there is a definable $dfg$ subgroup $H$ of $G$ such that the homogeneous space $G/H$ is definable and definably compact. (In the $o$-minimal case this is Fact 3.25 of \cite{Peterzil-Starchenko-mutypes}). Finally, we also isolate the definably amenable part of $G$, which we call the definably amenable component. Note that $dfg$ stands for ``has a definable $f$-generic type", and $fsg$ for ``has finitely satisfiable generics", which will be discussed together with various equivalences. We will need to understand something about groups of the form $G(k)$ where $k$ is a $p$-adically closed field and $G$ a semisimple algebraic group over $k$, and as part of the analysis we will prove the Kneser-Tits conjecture over $p$-adically closed fields.

math.LO↗

arXiv2026

Galois theory, automorphism groups of prime models, and the Picard-Vessiot closure

We work in the context of a complete totally transcendental theory $T = T^{eq}$. We consider the prime model $M_{A}$ over a set $A$. For intermediate sets $B$ with $A\subseteq B \subseteq M_{A}$ which are normal ($Aut(M_{A}/A)$-invariant) and ``minimal" we give a full Galois correspondence between intermediate definably closed sets $A\subseteq B \subseteq M_{A}$ and ``closed" subgroups of $Aut(B/A)$ (the group of $A$-elementary permutations of $B$). The unique greatest such minimal normal $B$ coincides with Poizat's ``minimal closure" $A_{min}$, so our paper extends (from $acl(A)$ to $A_{min}$) the well-known Galois correspondence between closed subgroups of the profinite group $Aut(acl(A)/A)$ and intermediate definably closed sets. The main result applies to the ``Picard-Vessiot closure" $K^{PV_{\infty}}$ of a differential field $K$ of char $0$ with algebraically closed field $C_{K}$ of constants. We also show that normal differential subfields of $K^{PV_{\infty}}$ containing $K$ are ``iterated $PV$-extensions" of $K$, and the Galois correspondence above holds for these extensions. This fills in some missing parts of Magid's paper [5]. We also discuss exact sequences $1 \to N \to G \to H \to 1$, where $G = Aut(K_2/K)$, $N = Aut(K_2/K_1)$ and $H = Aut(K_1/K)$, $K_1$ is a (maybe infinite type) $PV$ extension of $K$, $K_2$ is a (maybe infinite type) $PV$ extension of $K_1$ and $K_2$ is normal over $K$ and again $C_K$ is algebraically closed. Both $N$ and $H$ have the structure of proalgebraic groups over $C_K$. We show that conjugation by any given element of $G$ is a proalgebraic automorphism of $N$. Moreover if $G$ splits as a semidirect product $N\rtimes H$, then left multiplication by any fixed element of $G$ is a morphism of proalgebraic varieties $N\times H \to N\times H$. This improves and extends observations in Section 4 of [5] which dealt with one example.

math.LO↗

arXiv2025

Definable Galois theory for bimeromorphic geometry

The outlines of a "Galois theory" for bimeromorphic geometry is here developed, via the study of model-theoretic definable binding groups in the theory CCM of compact complex spaces. As an application, a structure theorem about principal meromorphic bundles with algebraic structure group, and admitting no horizontal subvarieties, is deduced. Examples of algebraic groups arising as binding groups are provided, as is a characterisation of when they are linear. Using binding groups in CCM it is shown that, in contrast to the situation in differentially closed fields, there are many algebraic groups which admit nontrivial definable torsors over acl-closed sets in the theory DCCM of existentially closed differential CCM-structures. A self-contained exposition of the binding group theorem in totally transcendental theories, that emphasises the bitorsorial nature of the construction, is also included.

math.LO↗

arXiv2025

The number of countable models of first-order theories

Throughout, $T$ denotes a complete first-order theory in a countable language $L$ that has infinite models and $I(\aleph_0,T)$ denotes the number of countable models of $T$, up to an isomorphism. To determine $I(\aleph_0,T)$, it suffices to consider only countable models of $T$ with domain $\omega$; since there are at most continuum many $L$-structures with domain $\omega$, $I(\aleph_0,T)\leqslant 2^{\aleph_0}$ holds. Theories with $I(\aleph_0,T)=1$ are the $\aleph_0$-categorical theories. These include the theory of an infinite set, theories of infinite-dimensional vector spaces over a finite field, and the theory of dense linear orders. Theories with $I(\aleph_0,T)<2^{\aleph_0}$ are said to have few countable models. In this paper we discuss and survey work done on Vaught's conjecture, Martin's conjecture, and Ehhrenfeuch theories (theories with more than one but only finitely many, countable models).

math.LO↗

arXiv2025

On groups definable in geometric fields with generic derivations

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.

math.LO↗

arXiv2025

Bass modules and embeddings into free modules

We show that the free module of infinite rank $R^{(\kappa)}$ purely embeds every $\kappa$-generated flat left $R$-module iff $R$ is left perfect. Using a Bass module corresponding to a descending chain of principal right ideals, we construct a model of the theory $T$ of $R^{(\kappa)}$ whose projectivity is equivalent to left perfectness, which allows to add a `stronger' equivalent condition: $R^{(\kappa)}$ purely embeds every $\kappa$-generated flat left $R$-module which is a model of $T$. We extend the model-theoretic construction of this Bass module to arbitrary descending chains of pp formulas, resulting in a `Bass theory' of pure-projective modules. We put this new theory to use by, among other things, reproving an old result of Daniel Simson about pure-semisimple rings and Mittag-Leffler modules. This paper is a condensed version, solely about modules, of our larger work arXiv:2407.15864, with two new results added about cyclically presented modules (Cor.14) and finitely presented cyclic modules (Rem.15).

math.RA↗

arXiv2024

An arithmetic algebraic regularity lemma

We give an 'arithmetic regularity lemma' for groups definable in finite fields, analogous to Tao's 'algebraic regularity lemma' for graphs definable in finite fields. More specifically, we show that, for any $M>0$, any finite field $\mathbf{F}$, and any definable group $(G,\cdot)$ in $\mathbf{F}$ and definable subset $D\subseteq G$, each of complexity at most $M$, there is a normal definable subgroup $H\leqslant G$, of index and complexity $O_M(1)$, such that the following holds: for any cosets $V,W$ of $H$, the bipartite graph $(V,W,xy^{-1}\in D)$ is $O_M(|\mathbf{F}|^{-1/2})$-quasirandom. Various analogous regularity conditions follow; for example, for any $g\in G$, the Fourier coefficient $||\widehat{1}_{H\cap Dg}(\pi)||_{\mathrm{op}}$ is $O_M(|\mathbf{F}|^{-1/8})$ for every non-trivial irreducible representation $\pi$ of $H$.

math.LO↗

arXiv2024

Free algebras, universal models and Bass modules

We investigate the question of when free structures of infinite rank (in a variety) possess model-theoretic properties like categoricity in higher power, saturation, or universality. Concentrating on left $R$-modules we show, among other things, that the free module of infinite rank $R^{(\kappa)}$ purely embeds every $\kappa$-generated flat left $R$-module iff $R$ is left perfect. Using a Bass module corresponding to a descending chain of principal right ideals, we construct a model of the theory $T$ of $R^{(\kappa)}$ whose projectivity is equivalent to left perfectness, which allows to add a "stronger" equivalent condition: $R^{(\kappa)}$ purely (equivalently, elementarily) embeds every $\kappa$-generated flat left $R$-module which is a model of $T$. In addition, we extend the model-theoretic construction of this Bass module to arbitrary descending chains of pp formulas, resulting in a `Bass theory' of pure-projective modules. We put this new theory to use by reproving an old result of Daniel Simson about pure-semisimple rings and Mittag-Leffler modules.

math.RA↗

arXiv2024

An analytic version of stable arithmetic regularity

We prove a structure theorem for stable functions on amenable groups, which extends the arithmetic regularity lemma for stable subsets of finite groups. Given a group $G$, a function $f\colon G\to [-1,1]$ is called stable if the binary function $f(x\cdot y)$ is stable in the sense of continuous logic. Roughly speaking, our main result says that if $G$ is amenable, then any stable function on $G$ is almost constant on all translates of a unitary Bohr neighborhood in $G$ of bounded complexity. The proof uses ingredients from topological dynamics and continuous model theory. We also prove several applications which generalize results in arithmetic combinatorics to nonabelian groups.

math.LO↗

arXiv2024

Automorphism groups of prime models, and invariant measures

We adapt the notion of a (relatively) definable subset of Aut(M) when M is a saturated model to the case Aut(M/A) when M is atomic and strongly omega-homogeneous over A. We discuss the existence and uniqueness of invariant measures on the Boolean algebra of definable subsets of Aut(M/A). For example when Th(M) is stable we have existence and uniqueness. We also discuss the compatibility of our definability notions with definable Galois cohomology and differential Galois theory.

math.LO↗

arXiv2023

Tighter Bounds on the Expressivity of Transformer Encoders

Characterizing neural networks in terms of better-understood formal systems has the potential to yield new insights into the power and limitations of these networks. Doing so for transformers remains an active area of research. Bhattamishra and others have shown that transformer encoders are at least as expressive as a certain kind of counter machine, while Merrill and Sabharwal have shown that fixed-precision transformer encoders recognize only languages in uniform $TC^0$. We connect and strengthen these results by identifying a variant of first-order logic with counting quantifiers that is simultaneously an upper bound for fixed-precision transformer encoders and a lower bound for transformer encoders. This brings us much closer than before to an exact characterization of the languages that transformer encoders recognize.

cs.LG↗

arXiv2023

Generalized locally compact models for approximate groups

We give a proof of the existence of generalized definable locally compact models for arbitrary approximate subgroups via an application of topological dynamics in model theory. Our construction is simpler and shorter than the original one obtained by Hrushovski in ``Beyond the Lascar group'', and it uses only basic model theory (mostly spaces of types and realizations of types). The main tools are Ellis groups from topological dynamics considered for suitable spaces of types. However, we need to redevelop some basic theory of topological dynamics for suitable ``locally compact flows'' in place of (compact) flows. We also prove that the generalized definable locally compact model which we constructed is universal in an appropriate category. We note that the main result yields structural information on definable generic subsets of definable groups, with a more precise structural result for generics in the universal cover of $\textrm{SL}_2(\mathbb{R})$.

math.LO↗

arXiv2023

Identifiable specializations for ODE models

The parameter identifiability problem for a dynamical system is to determine whether the parameters of the system can be found from data for the outputs of the system. Verifying whether the parameters are identifiable is a necessary first step before a meaningful parameter estimation can take place. Non-identifiability occurs in practical models. To reparametrize a model to achieve identifiability is a challenge. The existing approaches have been shown to be useful for many important examples. However, these approaches are either limited to linear models and scaling parametrizations or are not guaranteed to find a reparametrization even if it exists. In the present paper, we prove that there always exists a locally identifiable model with the same input-output behaviour as the original one obtained from a given one by a partial specialization of the parameters. As an extra feature of our approach, the resulting (at least) locally identifiable reparameterization has the same shape: the monomials in the new state variables in the new model are formed in the same way as in the original model. Furthermore, we give a sufficient observability condition for the existence of a state space transformation from the original model to the new one. Our proof is constructive and can be translated to an algorithm, which we illustrate by several examples.

eess.SY↗

arXiv2023

Compactifications of pseudofinite and pseudo-amenable groups

We first give simplified and corrected accounts of some results in \cite{PiRCP} on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing \cite{Turing} to give a simplified proof that any definable compactification of a pseudofinite group has an abelian connected component. We then discuss the relationship between Turing's work, the Jordan-Schur Theorem, and a (relatively) more recent result of Kazhdan \cite{Kazh} on approximate homomorphisms, and we use this to widen our scope from finite groups to amenable groups. In particular, we develop a suitable continuous logic framework for dealing with definable homomorphisms from pseudo-amenable groups to compact Lie groups. Together with the stabilizer theorems of \cite{HruAG,MOS}, we obtain a uniform (but non-quantitative) analogue of Bogolyubov's Lemma for sets of positive measure in discrete amenable groups. We conclude with brief remarks on the case of amenable topological groups.

math.LO↗

arXiv2023

Picard-Vessiot extensions, linear differential algebraic groups and their torsors

Let K be differential field with algebraically closed field of constants. Let K^diff be a differential closure of K, and L the (iterated) Picard-Vessiot closure of K inside K^diff. Let G be a linear differential algebraic group over K and X a differential algebraic torsor for G over K. We prove that X(L) is Kolchin-dense in X. When G is finite-dimensional we prove that X(L) = X(K^diff). We give close relationships between Picard-Vessiot extensions of K and torsors for suitable finite-dimensional linear differential algebraic groups over K. We suggest some differential field analogues of the notion of boundedness for fields (Serre's property F).

math.AG↗