Stable Domination and Independence in Algebraically Closed Valued Fields

Authors

Deirdre Haskell, Ehud Hrushovski, Dugald Macpherson

Type: Book

Publication Date: 2007-12-10

Citations: 119

DOI: https://doi.org/10.1017/cbo9780511546471

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Abstract

This book addresses a gap in the model-theoretic understanding of valued fields that had limited the interactions of model theory with geometry. It contains significant developments in both pure and applied model theory. Part I of the book is a study of stably dominated types. These form a subset of the type space of a theory that behaves in many ways like the space of types in a stable theory. This part begins with an introduction to the key ideas of stability theory for stably dominated types. Part II continues with an outline of some classical results in the model theory of valued fields and explores the application of stable domination to algebraically closed valued fields. The research presented here is made accessible to the general model theorist by the inclusion of the introductory sections of each part.

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Summary

Stable domination and independence in algebraically closed valued fields

2005-01-01

Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson

We seek to create tools for a model-theoretic analysis of types in algebraically closed valued fields (ACVF). We give evidence to show that a notion of 'domination by stable part' … We seek to create tools for a model-theoretic analysis of types in algebraically closed valued fields (ACVF). We give evidence to show that a notion of 'domination by stable part' plays a key role. In Part A, we develop a general theory of stably dominated types, showing they enjoy an excellent independence theory, as well as a theory of definable types and germs of definable functions. In Part B, we show that the general theory applies to ACVF. Over a sufficiently rich base, we show that every type is stably dominated over its image in the value group. For invariant types over any base, stable domination coincides with a natural notion of `orthogonality to the value group'. We also investigate other notions of independence, and show that they all agree, and are well-behaved, for stably dominated types. One of these is used to show that every type extends to an invariant type; definable types are dense. Much of this work requires the use of imaginary elements. We also show existence of prime models over reasonable bases, possibly including imaginaries.

Model Theory of Fields

2017-03-02

David R. Marker , Margit Messmer , Anand Pillay

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been … Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the fifth publication in the Lecture Notes in Logic series, the authors give an insightful introduction to the fascinating subject of the model theory of fields, concentrating on its connections to stability theory. In the first two chapters David Marker gives an overview of the model theory of algebraically closed, real closed and differential fields. In the third chapter Anand Pillay gives a proof that there are 2א non-isomorphic countable differential closed fields. Finally, Margit Messmer gives a survey of the model theory of separably closed fields of characteristic p > 0.

<i>Stable domination and independence in algebraically closed valued fields</i> (Lecture Notes in Logic 30)

2010-01-22

Boris Zilber

By D. Haskell, E. Hrushovski and D. Macpherson: 182 pp., £36.00, US$66.00, isbn 978-0-521-88981-0 (Cambridge University Press, Cambridge, 2008). By D. Haskell, E. Hrushovski and D. Macpherson: 182 pp., £36.00, US$66.00, isbn 978-0-521-88981-0 (Cambridge University Press, Cambridge, 2008).

Strongly stably dominated points

2017-10-19

Ehud Hrushovski , François Loeser

This chapter focuses on the properties of strongly stably dominated types over valued fields bases. In this setting, strong stability corresponds to a strong form of the Abhyankar property for … This chapter focuses on the properties of strongly stably dominated types over valued fields bases. In this setting, strong stability corresponds to a strong form of the Abhyankar property for valuations: the transcendence degrees of the extension coincide with those of the residue field extension. The chapter proves a Bertini type result and shows that the strongly stable points form a strict ind-definable subset <italic>V</italic>superscript Number Sign of unit vector V. It then proves a rigidity statement for iso-definable Γ‎-internal subsets of maximal o-minimal dimension of unit vector V, namely that they cannot be deformed by any homotopy leaving appropriate functions invariant. The chapter also describes the closure of iso-definable Γ‎-internal sets in <italic>V</italic>superscript Number Sign and proves that <italic>V</italic>superscript Number Sign is exactly the union of all skeleta.

Homogeneous Model Theory as a Framework for Algebraic Stability Theory

2019-01-01

Felix Weitkaemper

In this thesis we aim to persuade the readers to reconsider homogeneous model theory as a framework in which to study Algebraic Stability Theory. We argue that the tight link … In this thesis we aim to persuade the readers to reconsider homogeneous model theory as a framework in which to study Algebraic Stability Theory. We argue that the tight link between homogeneous model theory and categorical amalgamation constructions gives it a flexibility beyond what the ordinary first-order context can offer while simultaneously it allows the development of a stability-theoretic machinery very close to classical first-order stability. Thus we first introduce homogeneous model theory and categorical amalgamation theory separately (Chapters 1 and 2), laying out our own contributions alongside the known theory, before we utilise their interactions to prove related results pertinent to several different questions in algebraic stability theory. In Chapter 3 we give a very general result on omitting (possibly isolated) types from homogeneous models, generalising an analogous theorem by Hrushovski and Itai [18] regarding differential fields. Our method makes it possible to omit any family of types of trivial geometry from a superstable (and simple) homogeneous model without impacting the dividing structure and thus the stability class of the model. In Chapter 4 we apply homogeneous model theory to the study of Hrushovski constructions, enabling us to give a general treatment of the fundamental stabilitytheoretic properties beyond the combinatorial case to which such general accounts are usually confined. We obtain a general characterisation of dividing and by extension of U-Rank in the amalgamation limits of those constructions, and since we are not limited to finitary cases we can use our results for studying pseudo-exponentiation in the next two chapters. In Chapters 5 and 6 we thus narrow our focus even more and apply the tools developed above to the single class of exponential fields, illustrating the power of our general results through application to a topic which has been under close investigation ever since Boris Zilber proposed pseudo-exponentiation as a model-theoretic approach to complex exponentiation in the 2000s [42]. In passing we will also construct a non-almost-exponentially-algebraically closed quasiminimal exponential field, answering a question asked by Bays and Kirby in their recent [7].

Residue Field Domination in Real Closed Valued Fields

2019-07-02

Clifton Ealy , Deirdre Haskell , Jana Maříková

We define a notion of residue field domination for valued fields which generalizes stable domination in algebraically closed valued fields. We prove that a real closed valued field is dominated … We define a notion of residue field domination for valued fields which generalizes stable domination in algebraically closed valued fields. We prove that a real closed valued field is dominated by the sorts internal to the residue field, over the value group, both in the pure field and in the geometric sorts. These results characterize forking and þ-forking in real closed valued fields (and also algebraically closed valued fields). We lay some groundwork for extending these results to a power-bounded T-convex theory.

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Non-Archimedean Tame Topology and Stably Dominated Types

2016-03-18

Ehud Hrushovski , François Loeser

Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model … Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools. For non-archimedean fields, such as the p -adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry. This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness. Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods. No previous knowledge of non-archimedean geometry is assumed. Model-theoretic prerequisites are reviewed in the first sections.

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Model Theory of Fields

2005-12-15

David Marker , Margit Messmer , Anand Pillay

The model theory of fields is a fascinating subject stretching from Tarski's work on the decidability of the theories of the real and complex fields to Hrushovksi's recent proof of … The model theory of fields is a fascinating subject stretching from Tarski's work on the decidability of the theories of the real and complex fields to Hrushovksi's recent proof of the Mordell-Lang conjecture for function fields. This volume provides an insightful introduction to this active area, concentrating on connections to stability theory.

Algebraically Closed Fields

2019-04-04

Jonathan Kirby

Model theory begins with an audacious idea: to consider statements about mathematical structures as mathematical objects of study in their own right. While inherently important as a tool of mathematical … Model theory begins with an audacious idea: to consider statements about mathematical structures as mathematical objects of study in their own right. While inherently important as a tool of mathematical logic, it also enjoys connections to and applications in diverse branches of mathematics, including algebra, number theory and analysis. Despite this, traditional introductions to model theory assume a graduate-level background of the reader. In this innovative textbook, Jonathan Kirby brings model theory to an undergraduate audience. The highlights of basic model theory are illustrated through examples from specific structures familiar from undergraduate mathematics, paying particular attention to definable sets throughout. With numerous exercises of varying difficulty, this is an accessible introduction to model theory and its place in mathematics.

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Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)

2017-10-19

Ehud Hrushovski , François Loeser

Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model … Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools. For non-archimedean fields, such as the p-adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry. This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness. Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods. No previous knowledge of non-archimedean geometry is assumed and model-theoretic prerequisites are reviewed in the first sections.

Introduction to stability theory and Morley rank

2009-03-13

Martin Ziegler

Model theory and stability theory, with applications in differential algebra and algebraic geometry

2008-05-22

Anand Pillay

This article is based around parts of the tutorial given by E. Bouscaren and A. Pillay at the training workshop at the Isaac Newton Institute, March 29 — April 8, … This article is based around parts of the tutorial given by E. Bouscaren and A. Pillay at the training workshop at the Isaac Newton Institute, March 29 — April 8, 2005. The material is treated in an informal and free-ranging manner. We begin at an elementary level with an introduction to model theory for the non logician, but the level increases throughout, and towards the end of the article some familiarity with algebraic geometry is assumed. We will give some general references now rather than in the body of the article. For model theory, the beginnings of stability theory, and even material on differential fields, we recommend [5] and [8]. For more advanced stability theory, we recommend [6]. For the elements of algebraic geometry see [10], and for differential algebra see [2] and [9]. The material in section 5 is in the style of [7]. The volume [1] also has a self-contained exhaustive treatment of many of the topics discussed in the present article, such as stability, ω-stable groups, differential fields in all characteristics, algebraic geometry, and abelian varieties. Model theory From one point of view model theory operates at a somewhat naive level: that of point-sets, namely (definable) subsets X of a fixed universe M and its Cartesian powers M × … × M . But some subtlety is introduced by the fact that the universe M is “movable”, namely can be replaced by an elementary extension M ′, so a definable set should be thought of more as a functor.

Model theory of fields

1996-01-01

David Marker , Margit Messmer , Anand Pillay

The model theory of fields is an area for important interactions between mathematical, logical and classical mathematics. Recently, there have been major applications of model theoretic ideas to real analytic … The model theory of fields is an area for important interactions between mathematical, logical and classical mathematics. Recently, there have been major applications of model theoretic ideas to real analytic geometry and diophantine geometry. This book provides an introduction to this fascinating subject. In addition to introducing the basic model theory of the fields of real and complex numbers, we concentrate on differential fields and separably closed fields, the two theories used in Hrushovski's proof of the Mordell-Lang conjecture for function fields. This book is of interest to graduate students in either logic or in related areas of mathematics such as differential algebra or real algebraic geometry.

Preliminaries

2017-10-19

Ehud Hrushovski , François Loeser

This chapter provides some background material on definable sets, definable types, orthogonality to a definable set, and stable domination, especially in the valued field context. It considers more specifically these … This chapter provides some background material on definable sets, definable types, orthogonality to a definable set, and stable domination, especially in the valued field context. It considers more specifically these concepts in the framework of the theory ACVF of algebraically closed valued fields and describes the definable types concentrating on a stable definable <italic>V</italic> as an ind-definable set. It also proves a key result that demonstrates definable types as integrals of stably dominated types along some definable type on the value group sort. Finally, it discusses the notion of pseudo-Galois coverings. Every nonempty definable set over an algebraically closed substructure of a model of ACVF extends to a definable type.

A SURVEY ON SOME RESULTS OF VALUED FIELDS IN RECENT MODEL THEORY (Model theoretic aspects of the notion of independence and dimension)

2015-04-01

郁生米田

On orthogonal types to the value group and residual domination

2024-11-25

Pablo Cubides Kovacsics , Silvain Rideau , Mariana Vicaría

In this paper we present a unifying framework of residual domination for henselian valued fields of equicharacteristic zero, encompassing some valued fields with operators. We introduce a general definition for … In this paper we present a unifying framework of residual domination for henselian valued fields of equicharacteristic zero, encompassing some valued fields with operators. We introduce a general definition for residual domination and show that it is well behaved. For instance, we prove a change of base theorem for residual domination over algebraically closed sets. We show that the class of residually dominated types coincides with the types that are orthogonal to the value group, and with the class of types whose reduct to $\ACVF$ (the theory of algebraically closed valued fields with a non-trivial valuation) are generically stable. If the residue field is stable, we show that an $\A$-invariant type concentrated on the field sort is orthogonal to the value group if and only if it is generically stable. If the residue field is simple, the theory of the valued field is $\NTP_{2}$ and algebraically closed sets of imaginary parameters are extension basis, we show that an $\A$-invariant type concentrated on the field sort is orthogonal to the value group if and only if it is generically simple. Examples of the simple case, among others, include the limit theory $\VFA_{0}$ of the Frobenius automorphism acting on an algebraically closed valued field of characteristic $\p$ (where $\p$ tends to infinity), as well as non-principal ultraproducts of the $\p$-adics.

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Model Theory of Fields With Operators – a Survey

2015-01-19

Zoé Chatzidakis

The model theory of fields with operators has proven to be very useful in applications of model theory to problems outside logic. Differentially closed fields and separably closed fields were … The model theory of fields with operators has proven to be very useful in applications of model theory to problems outside logic. Differentially closed fields and separably closed fields were instrumental in Hrushovski’s proof of the conjecture of Mordell-Lang ([39]), and in work of Hrushovski and Pillay on counting the number of transcendental points on certain varieties ([44]). Let me also mention Ax’s work ([1]) on connections between transcendence theory and differential algebra, which has been recently elaborated on by Bertrand and Pillay ([6]) and Kowalski ([51]) in positive characteristic. The model theory of difference fields also provided various achievements: Hrushovski’s new proof of the conjecture of Manin-Mumford ([40]), Scanlon’s approach to p-adic abc conjectures and proximity questions ([89], [90], [91]), and his solution to Denis’ conjecture ([92]); more recently, applications to algebraic dynamics by Hrushovski and the author ([21], [22], [16]), and by Medvedev and Scanlon ([61]). Model theory also provides some insight on the Galois theory of systems of differential (or difference) equations and of Picard-Vessiot extensions, or of strongly normal extensions of Kolchin, see e.g. [74]. The aim of this article is not to present these applications, but to give a survey of what is known of the model theory of these enriched fields. We will discuss such issues as existence of model companions and their various axiomatisations, elimination of quantifiers and decidability; we will also investigate stability theoretic properties and mention some open problems. Another notable omission of this survey is that of exponential fields, which have seen many extraordinary developments in the past twenty years: in the context of the field of real numbers, or in the context of the field of complex numbers. I do not consider myself an expert in this subject, parts of which are still in full progress.

Model theory: an introduction

2003-06-01

David Marker

This book is a modern introduction to model theory which stresses applications to algebra throughout the text. The first half of the book includes classical material on model construction techniques, … This book is a modern introduction to model theory which stresses applications to algebra throughout the text. The first half of the book includes classical material on model construction techniques, type spaces, prime models, saturated models, countable models, and indiscernibles and their applications. The author also includes an introduction to stability theory beginning with Morley's Categoricity Theorem and concentrating on omega-stable theories. One significant aspect of this text is the inclusion of chapters on important topics not covered in other introductory texts, such as omega-stable groups and the geometry of strongly minimal sets. The author then goes on to illustrate how these ingredients are used in Hrushovski's applications to diophantine geometry. David Marker is Professor of Mathematics at the University of Illinois at Chicago. His main area of research involves mathematical logic and model theory, and their applications to algebra and geometry. This book was developed from a series of lectures given by the author at the Mathematical Sciences Research Institute in 1998.

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Fundamentals of Stability Theory

2017-03-02

John T. Baldwin

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been … Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the twelfth publication in the Perspectives in Logic series, John T. Baldwin presents an introduction to first order stability theory, organized around the spectrum problem: calculate the number of models a first order theory T has in each uncountable cardinal. The author first lays the groundwork and then moves on to three sections: independence, dependence and prime models, and local dimension theory. The final section returns to the spectrum problem, presenting complete proofs of the Vaught conjecture for ω-stable theories for the first time in book form. The book provides much-needed examples, and emphasizes the connections between abstract stability theory and module theory.

Non-archimedean tame topology and stably dominated types

2010-01-01

Ehud Hrushovski , François Loeser

Let $V$ be a quasi-projective algebraic variety over a non-archimedean valued field. We introduce topological methods into the model theory of valued fields, define an analogue $\hat {V}$ of the … Let $V$ be a quasi-projective algebraic variety over a non-archimedean valued field. We introduce topological methods into the model theory of valued fields, define an analogue $\hat {V}$ of the Berkovich analytification $V^{an}$ of $V$, and deduce several new results on Berkovich spaces from it. In particular we show that $V^{an}$ retracts to a finite simplicial complex and is locally contractible, without any smoothness assumption on $V$. When $V$ varies in an algebraic family, we show that the homotopy type of $V^{an}$ takes only a finite number of values. The space $\hat {V}$ is obtained by defining a topology on the pro-definable set of stably dominated types on $V$. The key result is the construction of a pro-definable strong retraction of $\hat {V}$ to an o-minimal subspace, the skeleton, definably homeomorphic to a space definable over the value group with its piecewise linear structure.

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Radiality of definable sets

2020-06-12

John Welliaveetil

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Imaginaries and invariant types in existentially closed valued differential fields

2016-09-18

Silvain Rideau

Abstract We answer three related open questions about the model theory of valued differential fields introduced by Scanlon. We show that they eliminate imaginaries in the geometric language introduced by … Abstract We answer three related open questions about the model theory of valued differential fields introduced by Scanlon. We show that they eliminate imaginaries in the geometric language introduced by Haskell, Hrushovski and Macpherson and that they have the invariant extension property. These two results follow from an abstract criterion for the density of definable types in enrichments of algebraically closed valued fields. Finally, we show that this theory is metastable.

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A note on $μ$-stabilizers in ACVF

2019-01-01

Jinhe Ye

We study $\mu$-stabilizers for groups definable in ACVF in the valued field sort. We prove that $\mathrm{Stab}^\mu(p)$ is an infinite unbounded definable subgroup of $G$ when $p$ is standard and … We study $\mu$-stabilizers for groups definable in ACVF in the valued field sort. We prove that $\mathrm{Stab}^\mu(p)$ is an infinite unbounded definable subgroup of $G$ when $p$ is standard and unbounded. In the particular case when $G$ is linear algebraic, we show that $\mathrm{Stab}^\mu(p)$ is a solvable algebraic subgroup of $G$, with $\mathrm{dim}(\mathrm{Stab}^\mu(p))=\mathrm{dim}(p)$ when $p$ is $\mu$-reduced and unbounded.

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Trace Formulas for Motivic Volumes

2016-05-27

François Loeser

Generic stability and stability

2012-10-22

Hans H. Adler , Enrique Casanovas , Anand Pillay

We prove two results about generically stable types $p$ in arbitrary theories. The first, on existence of strong germs, generalizes results from D. Haskell, E. Hrushovski and D. Macpherson on … We prove two results about generically stable types $p$ in arbitrary theories. The first, on existence of strong germs, generalizes results from D. Haskell, E. Hrushovski and D. Macpherson on stably dominated types. The second is an equivalence of forking and dividing, assuming generic stability of $p^{(m)}$ for all $m$. We use the latter result to answer in full generality a question posed by Hasson and Onshuus: If $p(x)\in S(B)$ is stable and does not fork over $A$ then $p\restriction A$ is stable. (They had solved some special cases.)

Imaginaries in equicharacteristic zero henselian fields

2023-01-01

Silvain Rideau , Mariana Vicaría

We prove an elimination of imaginaires results for (almost all) henselian valued fields of equicharacteristic zero. To do so, we consider a mix of sorts introduced in earlier works of … We prove an elimination of imaginaires results for (almost all) henselian valued fields of equicharacteristic zero. To do so, we consider a mix of sorts introduced in earlier works of the two authors and define a generalized version of the k-linear imaginaries. For a large class of value groups containing all subgroups of (\mathbb{R}^n) for some (n), we prove that the imaginaries of such a valued field can be elimininated in the field, the k-linear imaginaries and the imaginaries of the value group.

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Introduction to theories without the independence property

2008-01-01

Hans H. Adler

We present an updated exposition of the classical theory of complete first order theories without the independence property (also called NIP theories or dependent theories). We present an updated exposition of the classical theory of complete first order theories without the independence property (also called NIP theories or dependent theories).

Definable equivalence relations and zeta functions of groups (with an appendix by Raf Cluckers)

2018-07-18

Ehud Hrushovski , Benjamin Martin , Silvain Rideau

We prove that the theory of the p -adics {\mathbb Q}_p admits elimination of imaginaries provided we add a sort for {\mathrm GL}_n({\mathbb Q}_p)/{\mathrm GL}_n({\mathbb Z}_p) for each n . … We prove that the theory of the p -adics {\mathbb Q}_p admits elimination of imaginaries provided we add a sort for {\mathrm GL}_n({\mathbb Q}_p)/{\mathrm GL}_n({\mathbb Z}_p) for each n . We also prove that the elimination of imaginaries is uniform in p . Using p -adic and motivic integration, we deduce the uniform rationality of certain formal zeta functions arising from definable equivalence relations. This also yields analogous results for definable equivalence relations over local fields of positive characteristic. The appendix contains an alternative proof, using cell decomposition, of the rationality (for fixed p ) of these formal zeta functions that extends to the subanalytic context. As an application, we prove rationality and uniformity results for zeta functions obtained by counting twist isomorphism classes of irreducible representations of finitely generated nilpotent groups; these are analogous to similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald for subgroup zeta functions of finitely generated nilpotent groups.

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A Guide to NIP Theories

2015-07-05

Pierre Simon

The study of NIP theories has received much attention from model theorists in the last decade, fuelled by applications to o-minimal structures and valued fields. This book, the first to … The study of NIP theories has received much attention from model theorists in the last decade, fuelled by applications to o-minimal structures and valued fields. This book, the first to be written on NIP theories, is an introduction to the subject that will appeal to anyone interested in model theory: graduate students and researchers in the field, as well as those in nearby areas such as combinatorics and algebraic geometry. Without dwelling on any one particular topic, it covers all of the basic notions and gives the reader the tools needed to pursue research in this area. An effort has been made in each chapter to give a concise and elegant path to the main results and to stress the most useful ideas. Particular emphasis is put on honest definitions, handling of indiscernible sequences and measures. The relevant material from other fields of mathematics is made accessible to the logician.

Interpretable fields in various valued fields

2022-04-28

Yatir Halevi , Assaf Hasson , Ya’acov Peterzil

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Cohomology of algebraic varieties over non-archimedean fields

2022-01-01

Pablo Cubides Kovacsics , Mário J. Edmundo , Jinhe Ye

Abstract We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed nontrivially valued nonarchimedean field K based on Hrushovski-Loeser’s stable completion. In parallel, we develop a sheaf … Abstract We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed nontrivially valued nonarchimedean field K based on Hrushovski-Loeser’s stable completion. In parallel, we develop a sheaf cohomology of definable subsets in o-minimal expansions of the tropical semi-group $\Gamma _{\infty }$ , where $\Gamma $ denotes the value group of K . For quasi-projective varieties, both cohomologies are strongly related by a deformation retraction of the stable completion homeomorphic to a definable subset of $\Gamma _{\infty }$ . In both contexts, we show that the corresponding cohomology theory satisfies the Eilenberg-Steenrod axioms, finiteness and invariance, and we provide natural bounds of cohomological dimension in each case. As an application, we show that there are finitely many isomorphism types of cohomology groups in definable families. Moreover, due to the strong relation between the stable completion of an algebraic variety and its analytification in the sense of V. Berkovich, we recover and extend results on the singular cohomology of the analytification of algebraic varieties concerning finiteness and invariance.

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Independence in computable algebra

2015-06-23

Matthew Harrison‐Trainor , Alexander Melnikov , Antonio Montalbán

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A study of skeleta in non-Archimedean geometry

2015-06-30

John Welliaveetil

This thesis is a reflection of the interaction between Berkovich geometry and model theory. Using the results of Hrushovski and Loeser, we show that several interesting topological phenomena that concern … This thesis is a reflection of the interaction between Berkovich geometry and model theory. Using the results of Hrushovski and Loeser, we show that several interesting topological phenomena that concern the analytifications of varieties are governed by certain finite simplicial complexes embedded in them. Our work consists of the following two sets of results. Let k be an algebraically closed non-Archimedean non trivially real valued field which is complete with respect to its valuation. 1) Let $\phi : C' \to C$ be a finite morphism between smooth projective irreducible $k$-curves.The morphism $\phi$ induces a morphism $\phi^{an} : C'^{an} \to C^{an}$ between the Berkovich analytifications of the curves. We construct a pair of deformation retractions of $C'^{an}$ and $C^{an}$ which are compatible with the morphism $\phi^{\mathrm{an}}$ andwhose images $\Upsilon_{C'^{an}}$, $\Upsilon_{C^{an}}$ are closed subspaces of $C'^{an}$, $C^{an}$ that are homeomorphic to finite metric graphs. We refer to such closed subspaces as skeleta.In addition, the subspaces $\Upsilon_{C'^{an}}$ and $\Upsilon_{C^{an}}$ are such that their complements in their respective analytifications decompose into the disjoint union of isomorphic copies of Berkovich open balls. The skeleta can be seen as the union of vertices and edges, thus allowing us to define their genus. The genus of a skeleton in a curve $C$ is in fact an invariant of the curve which we call $g^{an}(C)$. The pair of compatible deformation retractions forces the morphism $\phi^{an}$ to restrict to a map $\Upsilon_{C'^{an}} \to \Upsilon_{C^{an}}$. We study how the genus of $\Upsilon_{C'^{an}}$ can be calculated using the morphism $\phi^{an}_{|\Upsilon_{C'^{an}}$ and invariants defined on $\Upsilon_{C^{an}}$. 2) Let $\phi$ be a finite endomorphism of $\mathbb{P}^1_k$. Given a closed point $x \in \mathbb{P}^1_k$, we are interested in the radius $f(x)$ of the largest Berkovich open ball centered at $x$ over which the morphism $\phi^{\mathrm{an}}$ is a topological fibration. Interestingly, the function $f : \mathbb{P}_k^1(k) \to \mathbb{R}_{\geq 0}$ admits a strong tameness property in that it is controlled by a non-empty finite graph contained in $\mathbb{P}^{1,an}_k$. We show that this result can be generalized to the case of finite morphisms $\phi : V' \to V$ between integral projective $k$-varieties where $V$ is normal.

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On Stably Pointed Varieties and Generically Stable Groups in ACVF

2018-09-21

Yatir Halevi

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Relative decidability and definability in henselian valued fields

2011-10-11

Joseph Flenner

Abstract Let ( K, v ) be a henselian valued field of characteristic 0. Then K admits a definable partition on each piece of which the leading term of a … Abstract Let ( K, v ) be a henselian valued field of characteristic 0. Then K admits a definable partition on each piece of which the leading term of a polynomial in one variable can be computed as a definable function of the leading term of a linear map. The main step in obtaining this partition is an answer to the question, given a polynomial f ( x ) ∈ K [ x ], what is v ( f ( x ))? Two applications are given: first, a constructive quantifier elimination relative to the leading terms, suggesting a relative decision procedure; second, a presentation of every definable subset of K as the pullback of a definable set in the leading terms subjected to a linear translation.

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Some supplements to Feferman–Vaught related to the model theory of adeles

2014-07-02

Jamshid Derakhshan , Angus Macintyre

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About Hrushovski and Loeser’s Work on the Homotopy Type of Berkovich Spaces

2016-01-01

Antoine Ducros

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Invariant Types in Model Theory

2020-12-01

Rosario Mennuni

An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the 'Save PDF' … An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the 'Save PDF' action button.

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Un principe d’Ax–Kochen–Ershov imaginaire

2024-07-04

Martin Hils , Silvain Rideau

We study interpretable sets in henselian and \sigma -henselian valued fields with value group elementarily equivalent to \mathbb{Q} or \mathbb{Z} . Our first result is an Ax–Kochen–Ershov type principle for … We study interpretable sets in henselian and \sigma -henselian valued fields with value group elementarily equivalent to \mathbb{Q} or \mathbb{Z} . Our first result is an Ax–Kochen–Ershov type principle for weak elimination of imaginaries in finitely ramified characteristic zero henselian fields – relative to value group imaginaries and residual linear imaginaries. We extend this result to the valued difference context and show, in particular, that existentially closed equicharacteristic zero multiplicative difference valued fields eliminate imaginaries in the geometric sorts; the \omega -increasing case corresponds to the theory of the non-standard Frobenius automorphism acting on an algebraically closed valued field. On the way, we establish some auxiliary results on separated pairs of characteristic zero henselian fields and on imaginaries in linear structures, which are also of independent interest.

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Theories without the tree property of the second kind

2013-10-28

Artem Chernikov

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Distal and non-distal NIP theories

2012-11-01

Pierre Simon

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Changing Models

2019-04-04

Jonathan Kirby

The Amalgamation Property for automorphisms of ordered abelian groups

2022-01-01

Jan Dobrowolski , Rosario Mennuni

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, … 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.

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Hilbert’s Nullstellensatz

2019-04-04

Jonathan Kirby

A note on μ-stabilizers in ACVF

2022-11-24

Jinhe Ye

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Residue field domination in some henselian valued fields

2023-10-21

Clifton Ealy , Deirdre Haskell , Pierre Simon

We generalize previous results about stable domination and residue field domination to henselian valued fields of equicharacteristic 0 with bounded Galois group, and we provide an alternate characterization of stable … We generalize previous results about stable domination and residue field domination to henselian valued fields of equicharacteristic 0 with bounded Galois group, and we provide an alternate characterization of stable domination in algebraically closed valued fields for types over parameters in the field sort.

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A Course in Model Theory

2012-03-08

Katrin Tent , Martin Ziegler

This concise introduction to model theory begins with standard notions and takes the reader through to more advanced topics such as stability, simplicity and Hrushovski constructions. The authors introduce the … This concise introduction to model theory begins with standard notions and takes the reader through to more advanced topics such as stability, simplicity and Hrushovski constructions. The authors introduce the classic results, as well as more recent developments in this vibrant area of mathematical logic. Concrete mathematical examples are included throughout to make the concepts easier to follow. The book also contains over 200 exercises, many with solutions, making the book a useful resource for graduate students as well as researchers.

An Invitation to Model Theory

2019-04-04

Jonathan Kirby

Introductory notes on the model theory of valued fields

2011-09-22

Zoé Chatzidakis

These notes will give some very basic definitions and results from model theory. They contain many examples, and in particular discuss extensively the various languages used to study valued fields. … These notes will give some very basic definitions and results from model theory. They contain many examples, and in particular discuss extensively the various languages used to study valued fields. They are intended as giving the necessary background to read the papers by Cluckers-Loeser, Delon, Halupczok and Kowalski in this volume. We also mention a few recent results or directions of research in the model theory of valued fields, but omit completely those themes which will be discussed elsewhere in this volume. So for instance, we do not even mention motivic integration.

Elementary Extensions

2019-04-04

Jonathan Kirby

Imaginaries in separably closed valued fields

2018-02-01

Martin Hils , Moshe Kamensky , Silvain Rideau

We show that separably closed valued fields of finite imperfection degree (either with λ-functions or commuting Hasse derivations) eliminate imaginaries in the geometric language. We then use this classification of … We show that separably closed valued fields of finite imperfection degree (either with λ-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.

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Cohomology of algebraic varieties over non-archimedean fields

2020-01-01

Pablo Cubides Kovacsics , Mário J. Edmundo , Jinhe Ye

We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed non-trivially valued non-archimedean field $K$ based on Hrushovski-Loeser's stable completion. In parallel, we develop a sheaf cohomology … We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed non-trivially valued non-archimedean field $K$ based on Hrushovski-Loeser's stable completion. In parallel, we develop a sheaf cohomology of definable subsets in o-minimal expansions of the tropical semi-group $Γ_\infty$, where $Γ$ denotes the value group of $K$. For quasi-projective varieties, both cohomologies are strongly related by a deformation retraction of the stable completion homeomorphic to a definable subset of $Γ_\infty$. In both contexts, we show that the corresponding cohomology theory satisfies the Eilenberg-Steenrod axioms, finiteness and invariance, and we provide natural bounds of cohomological dimension in each case. As an application, we show that there are finitely many isomorphism types of cohomology groups in definable families. Moreover, due to the strong relation between the stable completion of an algebraic variety and its analytification in the sense of V. Berkovich, we recover and extend results on the topological cohomology of the analytification of algebraic varieties concerning finiteness and invariance.

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Cardinality Considerations

2019-04-04

Jonathan Kirby

LOCALLY CONSTANT FUNCTIONS IN <i>C</i>-MINIMAL STRUCTURES

2015-03-01

Pablo Cubides Kovacsics

Abstract Let M be a C -minimal structure and T its canonical tree (which corresponds in an ultrametric space to the set of closed balls with radius different than ∞ … Abstract Let M be a C -minimal structure and T its canonical tree (which corresponds in an ultrametric space to the set of closed balls with radius different than ∞ ordered by inclusion). We present a description of definable locally constant functions f : M → T in C -minimal structures having a canonical tree with infinitely many branches at each node and densely ordered branches. This provides both a description of definable subsets of T in one variable and analogues of known results in algebraically closed valued fields.

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Polynomial approximation of Berkovich spaces and definable types

2013-10-26

Jérôme Poineau

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An Invitation to Extension Domination

2023-08-01

Kyle Gannon , Jinhe Ye

Motivated by the theory of domination for types, we introduce a notion of domination for Keisler measures called extension domination. We argue that this variant of domination behaves similarly to … Motivated by the theory of domination for types, we introduce a notion of domination for Keisler measures called extension domination. We argue that this variant of domination behaves similarly to its typesetting counterpart. We prove that extension domination extends domination for types and that it forms a preorder on the space of global Keisler measures. We then explore some basic properties related to this notion (e.g., approximations by formulas, closure under localizations, convex combinations). We also prove a few preservation theorems and provide some explicit examples.

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Valued difference fields and NTP2

2014-06-24

Artem Chernikov , Martin Hils

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ON THE NUMBER OF DEDEKIND CUTS AND TWO-CARDINAL MODELS OF DEPENDENT THEORIES

2015-01-23

Artem Chernikov , Saharon Shelah

For an infinite cardinal ${\it\kappa}$ , let $\text{ded}\,{\it\kappa}$ denote the supremum of the number of Dedekind cuts in linear orders of size ${\it\kappa}$ . It is known that ${\it\kappa}<\text{ded}\,{\it\kappa}\leqslant 2^{{\it\kappa}}$ … For an infinite cardinal ${\it\kappa}$ , let $\text{ded}\,{\it\kappa}$ denote the supremum of the number of Dedekind cuts in linear orders of size ${\it\kappa}$ . It is known that ${\it\kappa}<\text{ded}\,{\it\kappa}\leqslant 2^{{\it\kappa}}$ for all ${\it\kappa}$ and that $\text{ded}\,{\it\kappa}<2^{{\it\kappa}}$ is consistent for any ${\it\kappa}$ of uncountable cofinality. We prove however that $2^{{\it\kappa}}\leqslant \text{ded}(\text{ded}(\text{ded}(\text{ded}\,{\it\kappa})))$ always holds. Using this result we calculate the Hanf numbers for the existence of two-cardinal models with arbitrarily large gaps and for the existence of arbitrarily large models omitting a type in the class of countable dependent first-order theories. Specifically, we show that these bounds are as large as in the class of all countable theories.

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STRONGLY MINIMAL REDUCTS OF VALUED FIELDS

2016-06-01

Piotr Kowalski , Serge Randriambololona

Abstract We prove that if a strongly minimal nonlocally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying … Abstract We prove that if a strongly minimal nonlocally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying field.

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Definable functions in tame expansions of algebraically closed valued fields

2020-03-01

Pablo Cubides Kovacsics , Françoise Delon

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The Theory of Classical Valuations

1999-01-01

Paulo Ribenboim

On the category of models of a complete theory

1982-06-01

Daniel Lascar

Let T be a countable complete theory and C ( T ) the category whose objects are the models of T and morphisms are the elementary maps. The main object … Let T be a countable complete theory and C ( T ) the category whose objects are the models of T and morphisms are the elementary maps. The main object of this paper will be the study of C ( T ). The idea that a better understanding of the category may give us model theoretic information about T is quite natural: The (semi) group of automorphisms (endomorphisms) of a given structure is often a powerful tool for studying this structure. But certainly, one of the very first questions to be answered is: “to what extent does this category C ( T ) determine T ?” There is some obvious limitation: for example let T 0 be the theory of infinite sets (in a language containing only =) and T 1 the theory, in the language ( =, U ( ν 0 ), f ( ν 0 )) stating that: (1) U is infinite. (2) f is a bijective map from U onto its complement. It is quite easy to see that C ( T 0 ) is equivalent to C ( T 1 ). But, in this case, T 0 and T 1 can be “interpreted” each in the other. To make this notion of interpretation precise, we shall associate with each theory T a category, loosely denoted by T , defined as follows: (1) The objects are the formulas in the given language. (2) The morphisms from into are the formulas such that (i.e. f defines a map from ϕ into ϕ ; two morphisms defining the same map in all models of T should be identified).

The notion of independence in categories of algebraic structures, part I: Basic properties

1988-05-01

Gabriel Srour

Closed sets and chain conditions in stable theories

1984-12-01

Anand Pillay , Gabriel Srour

An impressive theory has been developed, largely by Shelah, around the notion of a stable theory. This includes detailed structure theorems for the models of such theories as well as … An impressive theory has been developed, largely by Shelah, around the notion of a stable theory. This includes detailed structure theorems for the models of such theories as well as a generalized notion of independence. The various stability properties can be defined in terms of the numbers of types over sets, or in terms of the complexity of definable sets. In the concrete examples of stable theories, however, one finds an important distinction between “positive” and “negative” information, such a distinction not being an a priori consequence of the general definitions. In the naive examples this may take the form of distinguishing between say a class of a definable equivalence relation and the complement of a class. In the more algebraic examples, this distinction may have a “topological” significance, for example with the Zariski topology on (the set of n -tuples of) an algebraically closed field, the “closed” sets being those given by sets of polynomial equalities. Note that in the latter case, every definable set is a Boolean combination of such closed sets (the definable sets are precisely the constructible sets). Similarly, stability conditions in practice reduce to chain conditions on certain “special” definable sets (e.g. in modules, stable groups). The aim here is to develop and present such notions in the general (model-theoretic) context. The basic notion is that of an “equation”. Given a complete theory T in a language L , an L -formula φ ( x̄, ȳ ) is said to be an equation (in x̄ ) if any collection Φ of instances of φ (i.e. of formulae φ ( x̄, ā )) is equivalent to a finite subset Φ′ ⊂ Φ .

Rigid Subanalytic Sets

1993-02-01

Leonard Lipshitz