Eliminating field quantifiers in strongly dependent henselian fields

Halevi, Yatir; Hasson, Assaf

doi:10.1090/proc/14203

Cited by 18 publications

(26 citation statements)

References 21 publications

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“…It follows from [11,Corollary 4.4] that, combined with the main result of [12], Conjecture 3.3 determines all possible first order theories of strongly dependent fields. The following is a special case of an unpublished result of S. Anscombe and the third author: Theorem 3.11.…”

Section: Strongly Dependent Fieldsmentioning

confidence: 97%

“…The general case follows from, essentially, the same argument. Here are the details: Since (K, v) is strongly dependent and Kv is infinite, (K, v, ac) eliminates field quantifiers (see [11,Theorem 1], it is a model of T 1 in the notation there) and it follows that Kv and vK are stably embedded. So their respective dp-ranks (as pure structures) are the same as their dp-ranks with the structure induced from (K, v, ac).…”

Section: Strongly Dependent Fieldsmentioning

confidence: 99%

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A Conjectural Classification of Strongly Dependent Fields

Halevi¹,

Hasson²,

Jahnke³

2019

Bull. symb. log

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Section: Strongly Dependent Fieldsmentioning

confidence: 97%

Section: Strongly Dependent Fieldsmentioning

confidence: 99%

A Conjectural Classification of Strongly Dependent Fields

Halevi¹,

Hasson²,

Jahnke³

2019

Bull. symb. log

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“…We may finally drop the ac-map, the valued field remains strongly dependent. For a direct proof of this fact see also a subsequent paper [14].…”

Section: (Claim)mentioning

confidence: 83%

“…Remark. Theorem 5.14 can also be deduced from elimination of field quantifiers and [33, Claim 1.17 (2)], see [14].…”

Section: (Claim)mentioning

confidence: 99%

Strongly dependent ordered abelian groups and Henselian fields

Halevi

Hasson

2019

Isr. J. Math.

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Strongly dependent ordered abelian groups have finite dp-rank. They are precisely those groups with finite spines and |{p prime : [G : pG] = ∞}| < ∞. We apply this to show that if K is a strongly dependent field, then (K, v) is strongly dependent for any henselian valuation v. 1 2. Preliminaries and notation Throughout the text G will denote a group, usually abelian and often ordered, C will denote a sufficiently saturated model of Th(G). By definable we will mean definable with parameters. We will need a few results from [30]. Since this text is not readily available, we try to keep the present work as self contained as possible, referring to more accessible sources whenever we are aware of such. In particular, for the study of ordered abelian groups we chose the language of [5], rather than the language used by Schmitt. The next sub-section is dedicated to a quick overview of (parts) of the language we are using, and to the basic properties of definable sets.2.1. Ordered abelian groups. Recall that an abelian group (G; +) is orderd if it is equipped with a linear ordering < such that a < b implies a + g < b + g for all a, b, g ∈ G. An ordered abelian group is discrete if it has a minimal positive element, and dense otherwise. It is archimedean if for all a, b ∈ G there exists n ∈ Z such that

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“…Though in [1] Bélair does not claim Fact 3.12 for algebraically maximal Kaplansky fields in mixed characteristic his proof seems to work equally well in that setting. A more self contained proof is available in [12]. Combined with [10, Proposition 5.9] we get that for the last sentence in the above corollary to hold (for algebraically closed Kaplansky fields of any characteristics) we do not need the value group and the residue field to be pure.…”

Section: 14mentioning

confidence: 76%

Definable valuations induced by multiplicative subgroups and NIP fields

2019

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We study the algebraic implications of the non-independence property (NIP) and variants thereof (dp-minimality) on infinite fields, motivated by the conjecture that all such fields which are neither real closed nor separably closed admit a (definable) henselian valuation. Our results mainly focus on Hahn fields and build up on Will Johnson's preprint "dp-minimal fields", arXiv:to find the right analogue of super-stability in the context of NIP theories, Shelah introduced the notion of strong NIP. As part of establishing this analogy, Shelah showed [32, Claim 5.40] that the theory of a separably closed field that is not algebraically closed is not strongly NIP. In fact Shelah's proof actually shows that strongly NIP fields are perfect 1 . Shelah conjectured [32, Conjecture 5.34] that (interpreting its somewhat vague formulation) strongly NIP fields are real closed, algebraically closed or support a definable non-trivial (henselian) valuation. Recently, this conjecture was proved 2 by Johnson [20] in the special case of dp-minimal fields (and, independently, assuming the definability of the valuation, henselianity is proved in [19]).The two main open problems in the field are:(1) Let K be an infinite (strongly) NIP field that is neither separably closed nor real closed. Does K support a non-trivial definable valuation? (2) Are all (strongly) NIP fields henselian (i.e., admit some non-trivial henselian valuation) or, at least, t-henselian (i.e., elementarily equivalent in the language of rings, to a henselian field)?A positive answer to Questions (2) would imply, for example, that strongly 3 dependent fields are elementarily equivalent to Hahn fields over well understood base fields [11, Theorem 3.11]:Equi-characteristic: R((t Γ )), C((t Γ )) or F p ((t Γ )). Finite residue field: Q((t Γ )) where Q is a p-adically closed field if the field admits a henselian valuation with finite residue field. Kaplansky: L((t Γ )) where L is a rank 1 Kaplansky field with residue field as in (1) above.where in all cases Γ is a strongly dependent ordered abelian group (see [10] for the classification of such groups).In view of the above, a natural strategy for studying Shelah's conjecture would be to, on the one hand, study the conjecture for Hahn fields (with dependent residue fields), as the key example and -on the other handusing the information gained in the study of Hahn fields, try to generalise Johnson's results from dp-minimal fields to the strongly dependent setting.The simplest extension of Johnson's proof of Shelah's conjecture for dpminimal fields would be to finite extensions of dp-minimal fields. Section 2 is dedicated to showing that this extension is vacuous, namely we prove that a finite extension of a dp-minimal field is again dp-minimal (see Theorem 1 Shelah's proof only uses the simple fact that if char(K) = p > 0 then either K is perfect or [K × : (K × ) p ] is infinite. See, e.g., [27, Remark 2.5] 2

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Eliminating field quantifiers in strongly dependent henselian fields

Cited by 18 publications

References 21 publications

A Conjectural Classification of Strongly Dependent Fields

A Conjectural Classification of Strongly Dependent Fields

Strongly dependent ordered abelian groups and Henselian fields

Definable valuations induced by multiplicative subgroups and NIP fields

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