The canonical topology on dp-minimal fields

Johnson, Will

doi:10.1142/s0219061318500071

Cited by 22 publications

(17 citation statements)

References 19 publications

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“…The answer is positive for a dp-minimal field by the results of Johnson [14] (so under the assumptions of Theorem 1.4, we have thatK is dp-minimal if and only if both k and Γ are dp-minimal), but the proof relies on the construction of a valuation which doesn't seem to be available in the general inp-minimal case.…”

Section: Remarks and Questionsmentioning

confidence: 99%

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HENSELIAN VALUED FIELDS AND inp-MINIMALITY

Chernikov¹,

Simon²

2019

J. symb. log.

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Section: Remarks and Questionsmentioning

confidence: 99%

“…Johnson [14] shows that a dp-minimal not strongly minimal field admits a definable Henselian valuation. It follows that if K is dp-minimal, then K × /(K × ) p is finite for all prime p (a Fact which Johnson states and uses).…”

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confidence: 99%

HENSELIAN VALUED FIELDS AND inp-MINIMALITY

Chernikov¹,

Simon²

2019

J. symb. log.

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“…In the last two sections of the paper we turn to the problem of constructing definable valuations on (strongly) NIP fields. As Johnson's methods of [20] do not seem to generalise easily even to the finite dp-rank case, we study a more general construction due to Koenigsmann. We give, provided the field K is neither real closed nor separably closed (without further model theoretic assumptions), an explicit first order sentence ψ K in the language of rings such that K |= ψ K implies the existence of a non-trivial valuation ring definable (over the same parameters appearing in ψ K ) in the language of rings. As we will show (see the discussion following Corollary 1.4) if K is t-henselian then K |= ψ K .…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

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confidence: 96%

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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“…In its full generality Shelah's conjecture is considered our of reach of present techniques. The only instance of the conjecture known is due to Johnson, [19], in the special case of NIP fields of dp-rank one (i.e. dp-minimal fields).…”

Section: Introductionmentioning

confidence: 99%

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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The canonical topology on dp-minimal fields

Cited by 22 publications

References 19 publications

HENSELIAN VALUED FIELDS AND inp-MINIMALITY

HENSELIAN VALUED FIELDS AND inp-MINIMALITY

Definable valuations induced by multiplicative subgroups and NIP fields

Definable V-topologies, Henselianity and NIP

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