Deirdre Haskell

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Stable Domination and Independence in Algebraically Closed Valued Fields

2007-12-10

Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson

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 … 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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Definable sets in algebraically closed valued fields: elimination of imaginaries

2006-01-01

Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson

It is shown that if K is an algebraically closed valued field with valuation ring R, then Th(K) has elimination of imaginaries if sorts are added whose elements are certain … It is shown that if K is an algebraically closed valued field with valuation ring R, then Th(K) has elimination of imaginaries if sorts are added whose elements are certain cosets in Kn of certain definable R-submodules of Kn (for all ). The proof involves the development of a theory of independence for unary types, which play the role of 1-types, followed by an analysis of germs of definable functions from unary sets to the sorts.

A version of o-minimality for the p-adics

1997-12-01

Deirdre Haskell , Dugald Macpherson

In this paper we formulate a notion similar to o -minimality but appropriate for the p -adics. The paper is in a sense a sequel to [11] and [5]. In … In this paper we formulate a notion similar to o -minimality but appropriate for the p -adics. The paper is in a sense a sequel to [11] and [5]. In [11] a notion of minimality was formulated, as follows. Suppose that L, L + are first-order languages and + is an L + -structure whose reduct to L is . Then + is said to be -minimal if, for every N + elementarily equivalent to + , every parameterdefinable subset of its domain N + is definable with parameters by a quantifier-free L -formula. Observe that if L has a single binary relation which in is interpreted by a total order on M , then we have just the notion of strong o-minimality , from [13]; and by a theorem from [6], strong o -minimality is equivalent to o -minimality. If L has no relations, functions, or constants (other than equality) then the notion is just strong minimality . In [11], -minimality is investigated for a number of structures . In particular, the C-relation of [1] was considered, in place of the total order in the definition of strong o -minimality. The C -relation is essentially the ternary relation which naturally holds on the maximal chains of a sufficiently nice tree; see [1], [11] or [5] for more detail, and for axioms. Much of the motivation came from the observation that a C -relation on a field F which is preserved by the affine group AGL(1, F ) (consisting of permutations ( a,b ) : x ↦ ax + b , where a ∈ F \ {0} and b ∈ F ) is the same as a non-trivial valuation: to get a C -relation from a valuation ν, put C ( x;y,z ) if and only if ν( y − x ) < ν( y − z ).

Vapnik-Chervonenkis density in some theories without the independence property, I

2015-12-22

Matthias Aschenbrenner , Alf Dolich , Deirdre Haskell +2 more

We recast the problem of calculating Vapnik-Chervonenkis (VC) density into one of counting types, and thereby calculate bounds (often optimal) on the VC density for some weakly o-minimal, weakly quasi-o-minimal, … We recast the problem of calculating Vapnik-Chervonenkis (VC) density into one of counting types, and thereby calculate bounds (often optimal) on the VC density for some weakly o-minimal, weakly quasi-o-minimal, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal theories.

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One-Dimensional p -Adic Subanalytic Sets

1999-02-01

Lou van den Dries , Deirdre Haskell , Dugald Macpherson

In this paper we extend two theorems from [2] on p-adic subanalytic sets, where p is a fixed prime number, Qp is the field of p-adic numbers and Zp is … In this paper we extend two theorems from [2] on p-adic subanalytic sets, where p is a fixed prime number, Qp is the field of p-adic numbers and Zp is the ring of p-adic integers. One of these theorems [2, 3.32] says that each subanalytic subset of Zp is semialgebraic. This is extended here as follows.

Cell decompositions of C-minimal structures

1994-03-01

Deirdre Haskell , Dugald Macpherson

Vapnik–Chervonenkis Density in Some Theories without the Independence Property, II

2013-01-01

Matthias Aschenbrenner , Alf Dolich , Deirdre Haskell +2 more

We study the Vapnik–Chervonenkis (VC) density of definable families in certain stable first-order theories. In particular, we obtain uniform bounds on the VC density of definable families in finite U-rank … We study the Vapnik–Chervonenkis (VC) density of definable families in certain stable first-order theories. In particular, we obtain uniform bounds on the VC density of definable families in finite U-rank theories without the finite cover property, and we characterize those abelian groups for which there exist uniform bounds on the VC density of definable families.

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Grothendieck Rings of ℤ-Valued Fields

2001-06-01

Raf Cluckers , Deirdre Haskell

Abstract We prove the triviality of the Grothendieck ring of a ℤ-valued field K under slight conditions on the logical language and on K . We construct a definable bijection … Abstract We prove the triviality of the Grothendieck ring of a ℤ-valued field K under slight conditions on the logical language and on K . We construct a definable bijection from the plane K 2 to itself minus a point. When we specialize to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point.

Unexpected imaginaries in valued fields with analytic structure

2013-05-15

Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson

Abstract We give an example of an imaginary defined in certain valued fields with analytic structure which cannot be coded in the ‘geometric’ sorts which suffice to code all imaginaries … Abstract We give an example of an imaginary defined in certain valued fields with analytic structure which cannot be coded in the ‘geometric’ sorts which suffice to code all imaginaries in the corresponding algebraic setting.

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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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GANZSTELLENSÄTZE IN THEORIES OF VALUED FIELDS

2008-06-01

Deirdre Haskell , Yoav Yaffe

The purpose of this paper is to study an analogue of Hilbert's seventeenth problem for functions over a valued field which are integral definite on some definable set; that is, … The purpose of this paper is to study an analogue of Hilbert's seventeenth problem for functions over a valued field which are integral definite on some definable set; that is, that map the given set into the valuation ring. We use model theory to exhibit a uniform method, on various theories of valued fields, for deriving an algebraic characterization of such functions. As part of this method we refine the concept of a function being integral at a point, and make it dependent on the relevant class of valued fields. We apply our framework to algebraically closed valued fields, model complete theories of difference and differential valued fields, and real closed valued fields.

A transfer theorem in constructive p-adic algebra

1992-07-01

Deirdre Haskell

Vapnik-Chervonenkis density in some theories without the independence property, I

2011-01-01

M. Aschenbrenner , A. Dolich , Deirdre Haskell +2 more

Real closed valued fields with analytic structure

2019-12-05

Pablo Cubides Kovacsics , Deirdre Haskell

Abstract We show quantifier elimination theorems for real closed valued fields with separated analytic structure and overconvergent analytic structure in their natural one-sorted languages and deduce that such structures are … Abstract We show quantifier elimination theorems for real closed valued fields with separated analytic structure and overconvergent analytic structure in their natural one-sorted languages and deduce that such structures are weakly o-minimal. We also provide a short proof that algebraically closed valued fields with separated analytic structure (in any rank) are C -minimal.

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A note on valuation definable expansions of fields

1998-06-01

Deirdre Haskell , Dugald Macpherson

In this note, we consider models of the theories of valued algebraically closed fields and convexly valued real closed fields, their reducts to the pure field or ordered field language … In this note, we consider models of the theories of valued algebraically closed fields and convexly valued real closed fields, their reducts to the pure field or ordered field language respectively, and expansions of these by predicates which are definable in the valued field. We show that, in terms of definability, there is no structure properly between the pure (ordered) field and the valued field. Our results are analogous to several other definability results for reducts of algebraically closed and real closed fields; see [9], [10], [11] and [12]. Throughout this paper, definable will mean definable with parameters. T heorem A. Let ℱ = ( F , +, ×, V ) be a valued, algebraically closed field, where V denotes the valuation ring. Let A be a subset of F n definable in ℱ v . Then either A is definable in ℱ = ( F , +, ×) or V is definable in . T heorem B. Let ℛ v = ( R , <, +, ×, V ) be a convexly valued real closed field, where V denotes the valuation ring. Let Abe a subset of R n definable in ℛ v . Then either A is definable in ℛ = ( R , <, +, ×) or V is definable in . The proofs of Theorems A and B are quite similar. Both ℱ v and ℛ v admit quantifier elimination if we adjoin a definable binary predicate Div (interpreted by Div( x, y ) if and only if v ( x ) ≤ v ( y )). This is proved in [14] (extending [13]) in the algebraically closed case, and in [4] in the real closed case. We show by direct combinatorial arguments that if the valuation is not definable then the expanded structure is strongly minimal or o-minimal respectively. Then we call on known results about strongly minimal and o-minimal fields to show that the expansion is not proper.

Dependence on parameters in analogues of hilberts seventeenth problem

1996-01-01

Luc Bélair , Deirdre Haskell

Model Theory of Analytic Functions: Some Historical Comments

2012-08-13

Deirdre Haskell

Abstract Model theorists have been studying analytic functions since the late 1970s. Highlights include the seminal work of Denef and van den Dries on the theory of the p -adics … Abstract Model theorists have been studying analytic functions since the late 1970s. Highlights include the seminal work of Denef and van den Dries on the theory of the p -adics with restricted analytic functions, Wilkie's proof of o-minimality of the theory of the reals with the exponential function, and the formulation of Zilber's conjecture for the complex exponential. My goal in this talk is to survey these main developments and to reflect on today's open problems, in particular for theories of valued fields.

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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SOME BACKGROUND ON ALGEBRAICALLY CLOSED VALUED FIELDS

2007-12-10

Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

J. Denef and L. van den Dries. p-Adic and real subanalytic sets. Annals of mathematics, ser. 2 vol. 128 (1988), pp. 79–138.

1997-12-01

Deirdre Haskell

An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Valued fields and elimination of imaginaries

2017-03-29

Deirdre Haskell

Unexpected imaginaries in valued fields with analytic structure

2011-12-21

Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson

We give an example of an imaginary defined in certain valued fields with analytic structure which cannot be coded in the `geometric' sorts which suffice to code all imaginaries in … We give an example of an imaginary defined in certain valued fields with analytic structure which cannot be coded in the `geometric' sorts which suffice to code all imaginaries in the corresponding algebraic setting.

Grothendieck rings of \mathbb{Z}-valued fields

2003-01-01

Raf Cluckers , Deirdre Haskell

We prove the triviality of the Grothendieck ring of a integer-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the … We prove the triviality of the Grothendieck ring of a integer-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the plane K^2 to itself minus a point. When we specialize to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point.

Residue field domination in some henselian valued fields

2022-01-01

Clifton Ealy , Deirdre Haskell , Pierre Simon

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Unexpected imaginaries in valued fields with analytic structure

2011-01-01

Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson

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.

Residue field domination in some henselian valued fields

2023-10-21

Clifton Ealy , Deirdre Haskell , Pierre Simon

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Residue field domination in some henselian valued fields

2022-01-01

Clifton Ealy , Deirdre Haskell , Pierre Simon

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Real closed valued fields with analytic structure

2019-12-05

Pablo Cubides Kovacsics , Deirdre Haskell

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Residue Field Domination in Real Closed Valued Fields

2019-07-02

Clifton Ealy , Deirdre Haskell , Jana Maříková

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Valued fields and elimination of imaginaries

2017-03-29

Deirdre Haskell

Vapnik-Chervonenkis density in some theories without the independence property, I

2015-12-22

Matthias Aschenbrenner , Alf Dolich , Deirdre Haskell +2 more

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Unexpected imaginaries in valued fields with analytic structure

2013-05-15

Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson

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Vapnik–Chervonenkis Density in Some Theories without the Independence Property, II

2013-01-01

Matthias Aschenbrenner , Alf Dolich , Deirdre Haskell +2 more

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Model Theory of Analytic Functions: Some Historical Comments

2012-08-13

Deirdre Haskell

Unexpected imaginaries in valued fields with analytic structure

2011-12-21

Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson

Vapnik-Chervonenkis density in some theories without the independence property, I

2011-01-01

M. Aschenbrenner , A. Dolich , Deirdre Haskell +2 more

Unexpected imaginaries in valued fields with analytic structure

2011-01-01

Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson

GANZSTELLENSÄTZE IN THEORIES OF VALUED FIELDS

2008-06-01

Deirdre Haskell , Yoav Yaffe

Stable Domination and Independence in Algebraically Closed Valued Fields

2007-12-10

Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson

View PDF Chat

SOME BACKGROUND ON ALGEBRAICALLY CLOSED VALUED FIELDS

2007-12-10

Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson

Definable sets in algebraically closed valued fields: elimination of imaginaries

2006-01-01

Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson

Stable domination and independence in algebraically closed valued fields

2005-01-01

Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson

Grothendieck rings of \mathbb{Z}-valued fields

2003-01-01

Raf Cluckers , Deirdre Haskell

Grothendieck Rings of ℤ-Valued Fields

2001-06-01

Raf Cluckers , Deirdre Haskell

One-Dimensional p -Adic Subanalytic Sets

1999-02-01

Lou van den Dries , Deirdre Haskell , Dugald Macpherson

A note on valuation definable expansions of fields

1998-06-01

Deirdre Haskell , Dugald Macpherson

A version of o-minimality for the p-adics

1997-12-01

Deirdre Haskell , Dugald Macpherson

J. Denef and L. van den Dries. p-Adic and real subanalytic sets. Annals of mathematics, ser. 2 vol. 128 (1988), pp. 79–138.

1997-12-01

Deirdre Haskell

Dependence on parameters in analogues of hilberts seventeenth problem

1996-01-01

Luc Bélair , Deirdre Haskell

Cell decompositions of C-minimal structures

1994-03-01

Deirdre Haskell , Dugald Macpherson

A transfer theorem in constructive p-adic algebra

1992-07-01

Deirdre Haskell

Coauthor	Papers Together
Dugald Macpherson	13
Ehud Hrushovski	7
Clifton Ealy	3
Pierre Simon	2
Sergei Starchenko	2
Matthias Aschenbrenner	2
Alf Dolich	2
Raf Cluckers	2
Jana Maříková	1
Lou van den Dries	1
Pablo Cubides Kovacsics	1
S. Starchenko	1
D. Macpherson	1
A. Dolich	1
M. Aschenbrenner	1
Yoav Yaffe	1
Luc Bélair	1

Rigid Subanalytic Sets

1993-02-01

Leonard Lipshitz

p-adic and Real Subanalytic Sets

1988-07-01

Jan Denef , Lou van den Dries

On variants of o-minimality

1996-06-01

Dugald Macpherson , Charles Steinhorn

One-Dimensional p -Adic Subanalytic Sets

1999-02-01

Lou van den Dries , Deirdre Haskell , Dugald Macpherson

Real closed rings II. model theory

1983-12-01

Gregory Cherlin , Max Dickmann

Cell decompositions of C-minimal structures

1994-03-01

Deirdre Haskell , Dugald Macpherson

Definable sets in algebraically closed valued fields: elimination of imaginaries

2006-01-01

Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson

Formally p-adic Fields

1984-01-01

Alexander Prestel , Peter Roquette

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On definable subsets of p-adic fields

1976-09-01

Angus Macintyre

The brilliant work of Ax-Kochen [1], [2], [3] and Ersov [6], and later work by Kochen [7] have made clear very striking resemblances between real closed fields and p -adically … The brilliant work of Ax-Kochen [1], [2], [3] and Ersov [6], and later work by Kochen [7] have made clear very striking resemblances between real closed fields and p -adically closed fields, from the model-theoretical point of view. Cohen [5], from a standpoint less model-theoretic, also contributed much to this analogy. In this paper we shall point out a feature of all the above treatments which obscures one important resemblance between real and p -adic fields. We shall outline a new treatment of the p -adic case (not far removed from the classical treatments cited above), and establish an new analogy between real closed and p -adically closed fields. We want to describe the definable subsets of p -adically closed fields. Tarski [9] in his pioneering work described the first-order definable subsets of real closed fields. Namely, if K is a real-closed field and X is a subset of K first-order definable on K using parameters from K then X is a finite union of nonoverlapping intervals (open, closed, half-open, empty or all of K ). In particular, if X is infinite, X has nonempty interior. Now, there is an analogous question for p -adically closed fields. If K is p -adically closed, what are the definable subsets of K ? To the best of our knowledge, this question has not been answered until now. What is the difference between the two cases? Tarski's analysis rests on elimination of quantifiers for real closed fields. Elimination of quantifiers for p -adically closed fields has been achieved [3], but only when we take a cross-section π as part of our basic data. The problem is that in the presence of π it becomes very difficult to figure out what sort of set is definable by a quantifier free formula. We shall see later that use of the cross-section increases the class of definable sets.

One-dimensional fibers of rigid subanalytic sets

1998-03-01

Leonard Lipshitz , Zachary Robinson

Let K be an algebraically closed field of any characteristic, complete with respect to the non-trivial ultrametric absolute value ∣·∣: K → ℝ + . By R denote the valuation … Let K be an algebraically closed field of any characteristic, complete with respect to the non-trivial ultrametric absolute value ∣·∣: K → ℝ + . By R denote the valuation ring of K , and by ℘ its maximal ideal. We work within the class of subanalytic sets defined in [5], but our results here also hold for the strongly subanalytic sets introduced in [11] as well as for those subanalytic sets considered in [6]. Let X ⊂ R 1 be subanalytic. In [8], we showed that there is a decomposition of X as a union of a finite number of special sets U ⊂ R 1 (see below). In this note, in Theorem 1.6, we obtain a version of this result which is uniform in parameters, thereby answering a question brought to our attention by Angus Macintyre. It follows immediately from Theorem 1.6 that the theory of K in the language (see [5] and [6]) is C -minimal in the sense of [3] and [9]. The analogous uniformity result in the p -adic case was recently proved in [12]. D efinition 1.1. (i) A disc in R 1 is a set of one of the two following forms: A special set in R 1 is a disc minus a finite union of discs. (ii) R -domains u ⊂ R m , and their associated rings of analytic functions, , are defined inductively as follows. R m is an R -domain and , the ring of strictly convergent power series in X 1 ,…, X m over K . If u is an R -domain with associated ring , (where K 〈 X, Y 〉〚ρ〛 S is a ring of separated power series, see [5, §2] and [1, §1]) and f , have no common zero on u and ◸ ϵ {<, ≤}, then is an R -domain and where J is the ideal generated by I and f − gZ ( Z is a new variable) if ◸ is ≤, and where J is the ideal generated by I and f − g τ (τ a new variable) if ◸ is <. (See [8, Definition 2.2].) R -domains generalize the rational domains of [2, §7.2.3]. It is true, but not easy to prove, that only depends on u as a point set, and is independent of the particular representation of u .

Canonical forms for definable subsets of algebraically closed and real closed valued fields

1995-09-01

Jan E. Holly

Abstract We present a canonical form for definable subsets of algebraically closed valued fields by means of decompositions into sets of a simple form, and do the same for definable … Abstract We present a canonical form for definable subsets of algebraically closed valued fields by means of decompositions into sets of a simple form, and do the same for definable subsets of real closed valued fields. Both cases involve discs, forming “Swiss cheeses” in the algebraically closed case, and cuts in the real closed case. As a step in the development, we give a proof for the fact that in “most” valued fields F , if f ( x ), g ( x ) ∈ F [ x ] and v is the valuation map, then the set { x : v ( f ( x )) ≤ v ( g ( x ))} is a Boolean combination of discs; in fact, it is a finite union of Swiss cheeses. The development also depends on the introduction of “valued trees”, which we define formally.

A version of o-minimality for the p-adics

1997-12-01

Deirdre Haskell , Dugald Macpherson

T-convexity and tame extensions

1995-03-01

Lou van den Dries , Adam H. Lewenberg

Abstract Let T be a complete o-minimal extension of the theory of real closed fields. We characterize the convex hulls of elementary substructures of models of T and show that … Abstract Let T be a complete o-minimal extension of the theory of real closed fields. We characterize the convex hulls of elementary substructures of models of T and show that the residue field of such a convex hull has a natural expansion to a model of T . We give a quantifier elimination relative to T for the theory of pairs (ℛ, V ) where ℛ ⊨ T and V ≠ ℛ is the convex hull of an elementary substructure of ℛ. We deduce that the theory of such pairs is complete and weakly o-minimal. We also give a quantifier elimination relative to T for the theory of pairs with ℛ a model of T and a proper elementary substructure that is Dedekind complete in ℛ. We deduce that the theory of such “tame” pairs is complete.

The Theory of Classical Valuations

1999-01-01

Paulo Ribenboim

Groups, measures, and the NIP

2007-02-02

Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay

We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating … We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$ in saturated $o$-minimal structures to compact Lie groups. We also prove some other structural results about such $G$, for example the existence of a left invariant finitely additive probability measure on definable subsets of $G$. We finally introduce the new notion of "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.

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Definable sets in ordered structures. II

1986-01-01

Julia F. Knight , Anand Pillay , Charles Steinhorn

It is proved that any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> … It is proved that any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (in which the underlying order is dense) is strongly <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal (namely, every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> elementarily equivalent to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal). It is simultaneously proved that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal, then every definable set of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-tuples of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has finitely many "definably connected components."

Zeta functions from definable equivalence relations

2006-12-31

Ehud Hrushovski , Benjamin Martin

We prove that the theory of the $p$-adics ${\mathbb Q}_p$ admits elimination of imaginaries provided we add a sort for ${\rm GL}_n({\mathbb Q}_p)/{\rm GL}_n({\mathbb Z}_p)$ for each $n$. We also … We prove that the theory of the $p$-adics ${\mathbb Q}_p$ admits elimination of imaginaries provided we add a sort for ${\rm GL}_n({\mathbb Q}_p)/{\rm GL}_n({\mathbb Z}_p)$ 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.

Fields with analytic structure

2011-05-13

Raf Cluckers , Leonard Lipshitz

We present a unifying theory of fields with certain classes of analytic functions, called fields with analytic structure. Both real closed fields and Henselian valued fields are considered. For real … We present a unifying theory of fields with certain classes of analytic functions, called fields with analytic structure. Both real closed fields and Henselian valued fields are considered. For real closed fields with analytic structure, o -minimality is shown. For Henselian valued fields, both the model theory and the analytic theory are developed. We give a list of examples that comprises, to our knowledge, all principal, previously studied, analytic structures on Henselian valued fields, as well as new ones. The b -minimality is shown, as well as other properties useful for motivic integration on valued fields. The paper is reminiscent of [Denef, van den Dries, p-adic and real subanalytic sets . Ann. of Math. (2) 128 (1988) 79–138], of [Cohen, Paul J. Decision procedures for real and p-adic fields . Comm. Pure Appl. Math. 22 (1969)131–151], and of [Fresnel, van der Put, Rigid analytic geometry and its applications . Progress in Mathematics, 218 Birkhäuser (2004)], and unifies work by van den Dries, Haskell, Macintyre, Macpherson, Marker, Robinson, and the authors.

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Elimination of quantifiers in algebraic structures

1983-01-01

Angus Macintyre , K.F. McKenna , Lou van den Dries

Imaginaries in real closed valued fields

2005-06-14

T. Mellor

Vapnik-Chervonenkis Classes of Definable Sets

1992-04-01

M. Laskowski

We show that a class of subsets of a structure uniformly definable by a first-order formula is a Vapnik-Chervonenkis class if and only if the formula does not have the … We show that a class of subsets of a structure uniformly definable by a first-order formula is a Vapnik-Chervonenkis class if and only if the formula does not have the independence property. Via this connection we obtain several new examples of Vapnik-Chervonenkis classes, including sets of positivity of finitely subanalytic functions.

Maximum number of edges joining vertices on a cube

2003-05-19

Khaled Abdel-Ghaffar

Analytic $p$-adic cell decomposition and integrals

2003-10-29

Raf Cluckers

We prove a conjecture of Denef on parameterized $p$-adic analytic integrals using an analytic cell decomposition theorem, which we also prove in this paper. This cell decomposition theorem describes piecewise … We prove a conjecture of Denef on parameterized $p$-adic analytic integrals using an analytic cell decomposition theorem, which we also prove in this paper. This cell decomposition theorem describes piecewise the valuation of analytic functions (and more generally of subanalytic functions), the pieces being geometrically simple sets, called cells. We also classify subanalytic sets up to subanalytic bijection.

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Geometric Set Systems

1998-01-01

Jiřı́ Matoušek

Weakly Normal Groups

1987-01-01

U. Hrushovski , Anthony L. Pillay

Unavoidable Traces Of Set Systems

2005-12-01

József Balogh , Béla Bollobás

Prototypes for definable subsets of algebraically closed valued fields

1997-12-01

Jan E. Holly

Abstract Elimination of imaginaries for 1-variable definable equivalence relations is proved for a theory of algebraically closed valued fields with new sorts for the disc spaces. The proof is constructive, … Abstract Elimination of imaginaries for 1-variable definable equivalence relations is proved for a theory of algebraically closed valued fields with new sorts for the disc spaces. The proof is constructive, and is based upon a new framework for proving elimination of imaginaries, in terms of prototypes which form a canonical family of formulas for defining each set that is definable with parameters. The proof also depends upon the formal development of the tree-like structure of valued fields, in terms of valued trees , and a decomposition of valued trees which is used in the coding of certain sets of discs.

A course on empirical processes

1984-01-01

R. M. Dudley

Linear Equations in Variables which Lie in a Multiplicative Group

2002-05-01

Jan‐Hendrik Evertse , Hans Peter Schlickewei , Winfried Schmidt

Let K be a field of characteristic 0 and let n be a natural number.Let Γ be a subgroup of the multiplicative group (K * ) n of finite rank … Let K be a field of characteristic 0 and let n be a natural number.Let Γ be a subgroup of the multiplicative group (K * ) n of finite rank r.Given a 1 , . . ., a n ∈ K * write A(a 1 , . . ., a n , Γ) for the number of solutionsWe derive an explicit upper bound for A(a 1 , . . ., a n , Γ) which depends only on the dimension n and on the rank r.

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The rationality of the Poincar� series associated to thep-adic points on a variety

1984-02-01

Jan Denef

On Goldie and dual Goldie dimensions

1984-03-01

Piotr Grzeszczuk , E. R. Puczyłowski

Types dans les corps valués munis d'applications coefficients

1999-06-01

Luc Bélair

We transpose Delon's analysis of types in valued fields to unramified henselian valued fields of mixed characteristic, by using coefficient maps of order $n$. This yields the Ax-Kochen-Ershov transfer principle … We transpose Delon's analysis of types in valued fields to unramified henselian valued fields of mixed characteristic, by using coefficient maps of order $n$. This yields the Ax-Kochen-Ershov transfer principle for the independence property in this class of valued fields.

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A combinatorial problem; stability and order for models and theories in infinitary languages

1972-04-01

Saharon Shelah

DEFINITION 1.3.P3(λ, μ, a) holds if |S| DEFINITION 1.3.P3(λ, μ, a) holds if |S|

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Les groupes 𝜔-stables de rang fini

1985-01-01

Daniel Lascar

We prove that a group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi>ω</mml:mi> <mml:annotation … We prove that a group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi>ω</mml:mi> <mml:annotation encoding="application/x-tex">\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-stable of finite Morley rank is nonmultidimensional. If moreover it is connected and does not have any infinite normal abelian definable subgroup, then it is isomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Pi upper H Subscript i slash upper K"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Π</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\Pi {H_i}/K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript i"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{H_i}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\omega _1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-categorical groups and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a finite group.

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Types dans les corps valués

1996-01-01

Luc Bélair , Michel Bousquet

RESUME. L’analyse de Delon des types dans les corps values devient particulierement transparente dans le langage des corps values auquel on ajoute une ≪ application coeﬃcient ≫. On obtient ainsi … RESUME. L’analyse de Delon des types dans les corps values devient particulierement transparente dans le langage des corps values auquel on ajoute une ≪ application coeﬃcient ≫. On obtient ainsi une preuve du transfert a la Ax-Kochen-Ershov de la propriete d’independance en egale caracteristique zero. / ABSTRACT. Delon’s analysis of types in valued

The theory of ordered abelian groups does not have the independence property

1984-01-01

Yuri Gurevich , Peter H. Schmitt

We prove that no complete theory of ordered abelian groups has the independence property, thus answering a question by B. Poizat. The main tool is a result contained in the … We prove that no complete theory of ordered abelian groups has the independence property, thus answering a question by B. Poizat. The main tool is a result contained in the doctoral dissertation of Yuri Gurevich and also in P. H. Schmitt’s <italic>Elementary properties of ordered abelian groups</italic>, which basically transforms statements on ordered abelian groups into statements on coloured chains. We also prove that every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-type in the theory of coloured chains has at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{2^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> coheirs, thereby strengthening a result by B. Poizat.

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The model-theoretic content of Lang’s conjecture

2009-03-13

Anand Pillay

Rings of finite representation type and modules of finite Morley rank

1984-06-01

Mike Prest

Quasi-o-minimal structures

2000-09-01

Oleg Belegradek , Ya’acov Peterzil , Frank Olaf Wagner

Abstract A structure ( M , <, …) is called quasi-o-minimal if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0-definable subsets … Abstract A structure ( M , <, …) is called quasi-o-minimal if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0-definable subsets and intervals. We give a series of natural examples of quasi-o-minimal structures which are not o-minimal; one of them is the ordered group of integers. We develop a technique to investigate quasi-o-minimality and use it to study quasi-o-minimal ordered groups (possibly with extra structure). Main results: any quasi-o-minimal ordered group is abelian; any quasi-o-minimal ordered ring is a real closed field, or has zero multiplication; every quasi-o-minimal divisible ordered group is o-minimal; every quasi-o-minimal archimedian densely ordered group is divisible. We show that a counterpart of quasi-o-minimality in stability theory is the notion of theory of U -rank 1.

On $ω_1$-categorical theories of abelian groups

1971-01-01

Angus Macintyre

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On the density of families of sets

1972-07-01

N. Sauer

Essentially periodic ordered groups

2000-11-01

Françoise Point , Frank Olaf Wagner

On $ω_1$-categorical theories of fields

1971-01-01

Angus Macintyre

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Stability, the f.c.p., and superstability; model theoretic properties of formulas in first order theory

1971-10-01

Saharon Shelah

Generically stable and smooth measures in NIP theories

2012-12-13

Ehud Hrushovski , Anand Pillay , Pierre Simon

We formulate the measure analogue of generically stable types in first order theories with $NIP$ (without the independence property), giving several characterizations, answering some questions from an earlier paper by … We formulate the measure analogue of generically stable types in first order theories with $NIP$ (without the independence property), giving several characterizations, answering some questions from an earlier paper by Hrushovski and Pillay, and giving another treatment of uniqueness results from the same paper. We introduce a notion of "generic compact domination", relating it to stationarity of the Keisler measures, and also giving definable group versions. We also prove the "approximate definability" of arbitrary Borel probability measures on definable sets in the real and $p$-adic fields.

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Dp-Minimality: Basic Facts and Examples

2011-07-01

Alfred Dolich , John Goodrick , David Lippel

We study the notion of dp-minimality, beginning by providing several essential facts about dp-minimality, establishing several equivalent definitions for dp-minimality, and comparing dp-minimality to other minimality notions. The majority of … We study the notion of dp-minimality, beginning by providing several essential facts about dp-minimality, establishing several equivalent definitions for dp-minimality, and comparing dp-minimality to other minimality notions. The majority of the rest of the paper is dedicated to examples. We establish via a simple proof that any weakly o-minimal theory is dp-minimal and then give an example of a weakly o-minimal group not obtained by adding traces of externally definable sets. Next we give an example of a divisible ordered Abelian group which is dp-minimal and not weakly o-minimal. Finally we establish that the field of p-adic numbers is dp-minimal.

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Lectures on Formally Real Fields

1984-01-01

Alexander Prestel

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Strongly dependent theories

2014-07-28

Saharon Shelah

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Magidor-Malitz quantifiers in modules

1984-03-01

Andreas Baudisch

Abstract We prove the elimination of Magidor-Malitz quantifiers for R -modules relative to certain -core sentences and positive primitive formulas. For complete extensions of the elementary theory of R-modules it … Abstract We prove the elimination of Magidor-Malitz quantifiers for R -modules relative to certain -core sentences and positive primitive formulas. For complete extensions of the elementary theory of R-modules it follows that all Ramsey quantifiers (ℵ 0 -interpretation) are eliminable. By a result of Baldwin and Kueker [1] this implies that there is no R -module having the finite cover property.

Model theory of modules over a serial ring

1995-03-01

Paul C. Eklof , Ivo Herzog