Pseudo real closed fields, pseudo p-adically closed fields and NTP2

Montenegro, Samaria

doi:10.1016/j.apal.2016.09.004

Cited by 20 publications

(29 citation statements)

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“…Other algebraic examples of NTP 2 structures were identified recently, including bounded pseudo real closed and pseudo p-adically closed fields [15], certain model complete multivalued fields [13] and certain valued difference fields, e.g., the theory VFA 0 of a nonstandard Frobenius on an algebraically closed valued field of characteristic zero [6]. See also [7] and [12] for some general results about groups and fields definable in NTP 2 structures.…”

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confidence: 99%

HENSELIAN VALUED FIELDS AND inp-MINIMALITY

Chernikov¹,

Simon²

2019

J. symb. log.

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confidence: 99%

HENSELIAN VALUED FIELDS AND inp-MINIMALITY

Chernikov¹,

Simon²

2019

J. symb. log.

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“…We conclude by showing that Shelah's conjecture is equivalent to the statement that any NIP field not contained in the algebraic closure of a finite field is t-henselian.Theorem (Proposition 6.3). Shelah's conjecture for NIP fields implies the henselianity conjecture.The proof of the main result generalizes methods used by Johson [20, Chapter 11], Montenegro [31] and Duret [7] by proving the following strong approximation theorem on curves.Theorem (Proposition 4.2). Let K be a perfect field and τ 1 , τ 2 two distinct V-topologies on K with τ 1 t-henselian.…”

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confidence: 65%

Definable V-topologies, Henselianity and NIP

2019

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We initiate the study of definable V-topolgies and show that there is at most one such V-topology on a t-henselian NIP field. Equivalently, we show that if (K, v 1 , v 2 ) is a bi-valued NIP field with v 1 henselian (resp. t-henselian) then v 1 and v 2 are comparable (resp. dependent). As a consequence Shelah's conjecture for NIP fields implies the henselianity conjecture for NIP fields. Furthermore, the latter conjecture is proved for any field admitting a henselian valuation with a dp-minimal residue field. We conclude by showing that Shelah's conjecture is equivalent to the statement that any NIP field not contained in the algebraic closure of a finite field is t-henselian.Theorem (Proposition 6.3). Shelah's conjecture for NIP fields implies the henselianity conjecture.The proof of the main result generalizes methods used by Johson [20, Chapter 11], Montenegro [31] and Duret [7] by proving the following strong approximation theorem on curves.Theorem (Proposition 4.2). Let K be a perfect field and τ 1 , τ 2 two distinct V-topologies on K with τ 1 t-henselian. Let C be a geometrically integral separated curve over K, X ∈ τ 1 and Y ∈ τ 2 infinite subsets of C(K). Then X ∩ Y is nonempty.The topological theme is recapitulated in the the final section. We show that a well behaved interaction between definable sets and open sets is equivalent, without further model theoretic assumptions, to t-henselianity. This allows us to deduce the following new (to the best of our knowledge) general characterization of henselianity in terms of t-henselianity.Theorem (Theorem 7.5, Proposition 7.7). Let (K, v) be a valued field. Then (K, v) is henselian if and only if (Kv ∆ ,v) is t-henselian for every coarsening v ∆ of v.Moreover, if K is not real closed then v has a non-trivial henselian coarsening if and only if for every algebraic extension L/K, any extension of v to L is t-henselian.As a final result, we use all of the above to show the following reformulations of Shelah's Conjecture.Theorem (Corollary 7.6). The following are equivalent:(1) Every infinite NIP field is either separably closed, real closed or admits a non-trivial henselian valuation.

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“…Another, somewhat overkill, approach would be to adapt the general outline of the proof presented in this paper. The only result on bounded pseudo p-adically closed fields which is proved in this paper and whose pseudo real closed equivalent is not already proved in [Mon17b] is Proposition (2.41). But the bounded pseudo real closed equivalent is an easy consequence of Proposition (2.37).…”

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confidence: 72%

“…As interest in less restrictive tameness notions grew, for example notions like NTP 2 that do not preclude the existence of any definable order, it also became important to find algebraic examples, in particular enriched fields, that would provide us with study cases. In [Mon17a] and [Mon17b], the first author thus started a neostability flavored study of two classes of large fields -see [Pop96] -extending the class of pseudo algebraically closed fields: pseudo p-adically closed fields and pseudo real closed fields. Those two classes consist of the fields over which any absolutely irreducible variety with a simple point over every p-adically closed (respectively real closed) extension has a rational point.…”

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confidence: 99%

Imaginaries, invariant types and pseudo \(p\)-adically closed fields

Montenegro¹,

Rideau-Kikuchi²

2020

Trans. Amer. Math. Soc.

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In this paper, we give a general criterion for elimination of imaginaries using an abstract independence relation. We also study germs of definable functions at certain well-behaved invariant types. Finally we apply these results to prove the elimination of imaginaries in bounded pseudo p-adically closed fields.so-called density theorems [Mon17b, Theorem 3.17 and 6.11] -which generalizes cellular decomposition to the multi-valued (respectively multi-ordered) setting. She also proves an amalgamation theorem [Mon17b, Theorem 3.21 and 6.13] which is a weaker version of the independence theorem of simple theories which holds in this non-simple setting. She then uses those new tools to prove new classification results, the main one [Mon17b, Theorem 4.24 and 8.5] being that bounded pseudo p-adically closed and pseudo real closed fields are NTP 2 of finite burden. One particularity that stands out in the first author's work is that, although most results are proved for both classes, elimination of imaginaries was only proved for pseudo real closed fields [Mon17a]. The initial motivation for this paper was to repair that asymmetry. As it turns out, we were also able to repair a small gap in the proof of the pseudo real closed case, cf. §2.8.1. Since bounded pseudo p-adically closed fields that are not pseudo algebraically closed fields come with finitely many definable valuations, one cannot expect elimination of imaginaries in a language with just one sort for the field, contrary to what happens with pseudo real closed fields. But since the work of [HHM06], we know how to circumvent that particular issue: we have to add, for each valuation, codes for certain definable modules over the valuation ring; that is, work in the so-called geometric language. The main question regarding the imaginaries in bounded pseudo p-adically closed fields then becomes to prove that there are no imaginaries arising from the interaction between the various valuations and, therefore, that it suffices to add the geometric sorts for each of the valuations. The main result of this paper, Theorem (2.58), is a positive answer to this question. We deduce it from our abstract criterion, Proposition (1.17), applied to quantifier free invariant independence, cf Definition (1.18). The results of Section 1.2 play a fundamental role by allowing us to deduce n-ary extension from unary extension for that particular independence relation. Our first step towards this elimination result, and a core ingredient of the rest of the paper, is Proposition (2.26) which states that not only are the geometric sorts for each valuation orthogonal but also that the structure of any given geometric sort is the one induced by the relevant p-adic closure. We then proceed to deduce, from this strong statement on the independence of the valuations, a result on the structure of definable subsets of the valued field, where, at first sight, the valuations do interact. The key ingredient of the first author's proof that pseudo real closed fields eliminate imaginaries is [Mon17b, ...

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Pseudo real closed fields, pseudo p-adically closed fields and NTP2

Cited by 20 publications

References 32 publications

HENSELIAN VALUED FIELDS AND inp-MINIMALITY

HENSELIAN VALUED FIELDS AND inp-MINIMALITY

Definable V-topologies, Henselianity and NIP

Imaginaries, invariant types and pseudo \(p\)-adically closed fields

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