Definable sets in algebraically closed valued fields: elimination of imaginaries

Haskell, Deirdre; Hrushovski, Ehud; Macpherson, Dugald

doi:10.1515/crelle.2006.066

Cited by 69 publications

(217 citation statements)

References 9 publications

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“…In this section, we wish to make precise the idea that, in C-minimal theories, n + 1-types can be viewed as generic types of balls parametrised by realisations of an n-type. This is an obvious higher dimensional generalisation of the unary notion of genericity in a ball (see [HHM06,Definition 2.3.4]). To do so, we introduce a class of ∆-types (see Definition (5.11)) for ∆ a finite set of L-formulas that will play a central role in the rest of this text.…”

Section: Types and Uniform Families Of Ballsmentioning

confidence: 99%

Imaginaries and invariant types in existentially closed valued differential fields

Rideau

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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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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Section: Types and Uniform Families Of Ballsmentioning

confidence: 99%

Imaginaries and invariant types in existentially closed valued differential fields

Rideau

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…This was a theme in the modeltheoretic analysis of algebraically closed valued fields [10], [11], [14]. It was also an aspect of the paper [16], although there the main motivation was to resolve some conjectures about definably compact groups in o-minimal structures.…”

Section: Introductionmentioning

confidence: 99%

On NIP and invariant measures

Hrushovski¹,

Pillay²

2011

J. Eur. Math. Soc.

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Abstract. We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = tp(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over bdd(A), (ii) analogous statements for Keisler measures and definable groups, including the fact that G 000 = G 00 for G definably amenable, (iii) definitions, characterizations and properties of "generically stable" types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in o-minimal expansions of real closed fields.

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“…Thus J( V ) = D; this gives (2). Since P (J(p)) generates p, we have (1). With this definition of J, (4) is clear:…”

Section: Proof It Suffices To Prove This Formentioning

confidence: 99%

“…We will not repeat the proofs of (1,2) from [1]; (1) is rather straightforward, see 2.6.6; (2) For (3), let A n be the O-algebra of strict 3 upper triangular matrices. Let J be the subalgebra defined by: i≤j≤n val(a ij ) ⋄ ij α ij .…”

Section: Imaginaries In Acvfmentioning

confidence: 99%

Imaginaries and definable types in algebraically closed valued fields

Hrushovski¹

2014

EMS Series of Congress Reports

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This manuscript is largely an exposition of material from [1], [2] and [3], regarding definable types in the model completion of the theory of valued fields, and the classification of imaginary sorts. The proof of the latter is based here on definable types rather than invariant types, and on the notion of generic reparametrization; it allows a more conceptual view than we had when [1] was written.

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Definable sets in algebraically closed valued fields: elimination of imaginaries

Cited by 69 publications

References 9 publications

Imaginaries and invariant types in existentially closed valued differential fields

Imaginaries and invariant types in existentially closed valued differential fields

On NIP and invariant measures

Imaginaries and definable types in algebraically closed valued fields

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