Distality in valued fields and related structures

Aschenbrenner, Matthias; Chernikov, Artem; Gehret, Allen; Ziegler, Martin

doi:10.48550/arxiv.2008.09889

Cited by 3 publications

(14 citation statements)

References 40 publications

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“…Part of the proof consists in showing a similar reduction result for short exact sequences in the framework recently investigated by M. Aschenbrenner, A. Chernikov, A. Gehret and M. Ziegler [1] (see Theorem 5.2.5).…”

mentioning

confidence: 79%

“…We may L bp -embed A ′ |L in M over (M 0 ) |L by beauty of M and thus L eq bp -embed A ′ in M * over M 0 since they are generated by real elements. We may assume by (1), that the real part of B generates all of B. Take A ′′ realizing tp(A ′ /A)|B and set B ′ = ( A ′′ ∪ B , P (B)).…”

Section: 38mentioning

confidence: 99%

“…Any two beautiful pairs are then backand-forth equivalent, and one may define beauty transfer as before. Assuming K-beautiful pairs exist and T bp ( K) has beauty transfer, it then follows that the definable types in F ‹ K are uniformly definable, so one may a posteriori expand the L P -theory T bp ( K) to an L bp -theory and get back to the context of (1).…”

Section: 416mentioning

confidence: 99%

“…Let T now denote a complete theory of dense regular non-divisible ordered abelian groups. Due to the failure of uniform definability of types in T , in the following theorem, we need to work in the more general framework of SE-structures in the sense of Remark 2.4.17 (1). We denote by K the class of L bp -substructures of SE-pairs A = (A, P (A)) such that P (A) is a model of T , P (A) is convex in A and for every prime q, the embedding P (A)/qP (A) into A/qA is a group isomorphism.…”

Section: 34mentioning

confidence: 99%

“…Let T abc = Th L abc (M ) and T * abc = Th L * abc (M ). In [1] the authors work in a definitional expansion L abcq of L abc , adding new sorts for A/nA for all n ∈ N (identifying A/0A with A), as well as, for every natural number n, unary functions π n : A → A/nA (denoting the canonical projection) and ρ n : B → A/nA which, on ν −1 (nC), is the composition of the group homomorphisms…”

Section: Beautiful Pairs Of Pure Short Exact Sequencesmentioning

confidence: 99%

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Beautiful pairs

Kovacsics¹,

Hils²,

Ye³

2021

Preprint

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We introduce an abstract framework to study certain classes of stably embedded pairs of models of a complete L-theory T , called beautiful pairs, which comprises Poizat's belles paires of stable structures and van den Dries-Lewenberg's tame pairs of o-minimal structures. Using an amalgamation construction, we relate several properties of beautiful pairs with classical Fraïssé properties.After characterizing beautiful pairs of various theories of ordered abelian groups and valued fields, including the theories of algebraically, p-adically and real closed valued fields, we show an Ax-Kochen-Ershov type result for beautiful pairs of henselian valued fields. As an application, we derive strict pro-definability of particular classes of definable types. When T is one of the theories of valued fields mentioned above, the corresponding classes of types are related to classical geometric spaces such as Berkovich and Huber's analytifications. In particular, we recover a result of Hrushovski-Loeser on the strict pro-definability of stably dominated types in algebraically closed valued fields.

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mentioning

confidence: 79%

Section: 38mentioning

confidence: 99%

Section: 416mentioning

confidence: 99%

Section: 34mentioning

confidence: 99%

Section: Beautiful Pairs Of Pure Short Exact Sequencesmentioning

confidence: 99%

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Beautiful pairs

Kovacsics¹,

Hils²,

Ye³

2021

Preprint

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Model-theoretic Elekes-Szabó for stable and o-minimal hypergraphs

Chernikov¹,

Peterzil²,

Starchenko³

2021

Preprint

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A theorem of Elekes and Szabó recognizes algebraic groups among certain complex algebraic varieties with maximal size intersections with finite grids. We establish a generalization to relations of any arity and dimension, definable in: 1) stable structures with distal expansions (includes algebraically and differentially closed fields of characteristic 0); and 2) o-minimal expansions of groups. Our methods provide explicit bounds on the power saving exponent in the non-group case. Ingredients of the proof include: a higher arity generalization of the abelian group configuration theorem in stable structures, along with a purely combinatorial variant characterizing Latin hypercubes that arise from abelian groups; and Zarankiewicz-style bounds for hypergraphs definable in distal structures. ContentsDate: 2021-04-07 00:43:29Z. 1 2 ARTEM CHERNIKOV, YA'ACOV PETERZIL, AND SERGEI STARCHENKO 5.1. On the notion of p-dimension 42 5.2. Fiber-algebraic relations and p-irreducibility 45 5.3. On general position 46 5.4. Main theorem: the statement and some reductions 49 5.5. Proof of Theorem 5.31: the two cases 54 5.6. Case 1: dim p (Z) < s − 2. 56 5.7. Case 2: dim p (Z) = s − 2. 57 5.8. Proof of Theorem 5.24 for ternary Q 58 5.9. Discussion and some applications 61 6. Main theorem in the o-minimal case 64 6.1. Main theorem and some reductions 64 6.2. The proof of Theorem 6.8 66 6.3. Obtaining a nice Q-relation 70 6.4. Discussion and some applications 72 References 75

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Hensel minimality II: Mixed characteristic and a diophantine application

Cluckers,

Halupczok,

Rideau-Kikuchi

et al. 2021

Preprint

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We extend the results of [8] on Hensel minimality to include also the mixed characteristic case. This completes our axiomatic framework for tame non-archimedean geometry over Henselian valued fields of characteristic zero. We prove a Jacobian property, a strong form of Taylor approximations of definable functions, resplendency results and cell decomposition, all under 1-h-minimality. We obtain a diophantine application of counting rational points of bounded height on Hensel minimal curves.

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Distality in valued fields and related structures

Cited by 3 publications

References 40 publications

Beautiful pairs

Beautiful pairs

Model-theoretic Elekes-Szabó for stable and o-minimal hypergraphs

Hensel minimality II: Mixed characteristic and a diophantine application

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