Artin-Schreier extensions in NIP and simple fields

Kaplan, Itay; Scanlon, Thomas; Wagner, Frank Olaf

doi:10.1007/s11856-011-0104-7

Cited by 51 publications

(95 citation statements)

References 22 publications

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“…Proof. By [23,Corollary 4.5] every infinite NIP field of characteristic p > 0 contains F alg p . But then M cannot satisfy the strong Erdős-Hajnal property by the proposition above, contradicting distality.…”

Section: 3mentioning

confidence: 99%

Regularity lemma for distal structures

Chernikov

Starchenko

2018

J. Eur. Math. Soc.

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It is known that families of graphs with a semialgebraic edge relation of bounded complexity satisfy much stronger regularity properties than arbitrary graphs, and that they can be decomposed into very homogeneous semialgebraic pieces up to a small error (e.g., see [33,2,16,18]). We show that similar results can be obtained for families of graphs with the edge relation uniformly definable in a structure satisfying a certain model theoretic property called distality, with respect to a large class of generically stable measures. Moreover, distality characterizes these strong regularity properties. This applies in particular to graphs definable in arbitrary o-minimal structures and in p-adics.1 2 ARTEM CHERNIKOV AND SERGEI STARCHENKO Theorem 1.2 (Erdős, Hajnal and Pach [14]). For any finite graph H there is a constant δ = δ(H) > 0 such that every H-free graph on n vertices has a homogeneous pairThe following definition is taken from [17].Remark 1.4. Is is shown in [2] that if a family of finite graphs G has the strong Erdős-Hajnal property and is closed under taking induced subgraphs then it has the Erdős-Hajnal property.In this paper we consider families of graphs whose edge relations are given by a fixed definable relation in a first-order structure.Definition 1.5. Let M be a first-order structure and R ⊆ M k × M k be a definable relation. Consider the family G R of all finite graphs V = (G, E) where G ⊆ M k is a finite subset and E = (V ×V )∩R. We say that R satisfies the (strong) Erdős-Hajnal property if the family G R does.We extend this notion to the bi-partite case. Definition 1.6. Let M be a first-order structure and R ⊆ M m × M n a definable relation. (1) A pair of subsets A ⊆ M m , B ⊆ M n is called R-homogeneous if either A × B ⊆ R or (A × B) ∩ R = ∅. (2) We say that the relation R satisfies the strong Erdős-Hajnal property if there is a constant δ = δ(R) > 0 such that for any finite subsets A ⊆ M m , B ⊆ M n there are A 0 ⊆ A, B 0 ⊆ B with |A 0 | ≥ δ|A|, |B 0 | ≥ δ|B|, and the pair A 0 , B 0 is R-homogeneous.Our motivation for this work comes from the following remarkable theorem by Alon et al.Theorem 1.7 ([2, Theorem 1.1]). If R ⊆ R n × R m is a semialgebraic relation then R has the strong Erdős-Hajnal property.Remark 1.8.(i) Although it is not stated explicitly in [2], but can be easily derived from the proof, homogeneous pairs in the above theorem can be chosen to be relatively uniformly definable. (ii) The above theorem was generalized by Basu (see [5]) to (topologically closed) relations definable in arbitrary o-minimal expansions of real closed fields.Besides the Erdős-Hajnal property for semialgebraic graphs, the above theorem has many other applications including unit distance problems [32], improved bounds in higher dimensional semialgebraic Ramsey theorem [11], [2, Theorem 1.2], algorithmic property testing [18], and can also be used to obtain a strong Szemerédi-type regularity lemma for semialgebraic graphs [16,18] (see also Section 5).The aim of this article is to demonstrate that the above r...

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Section: 3mentioning

confidence: 99%

Regularity lemma for distal structures

Chernikov

Starchenko

2018

J. Eur. Math. Soc.

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“…Since char(K) = p it is perfect and so vK is p-divisible. Moreover, as K is dependent it follows from the proof of [24,Proposition 5.3] that Kv is Artin-Schreier closed, and therefore infinite.…”

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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“…It will suffice, therefore, to find a definable non-trivial valuation on some finite extension L ≥ K (since if O is a non-trivial valuation ring on L then O ∩ K is a non-trivial valuation ring in K). It is therefore, harmless to assume that √ −1 ∈ K. By [22,Theorem 4.4] K is Artin-Schreier closed. So the same is true of any finite extension L ≥ K. This implies (e.g., [27, Lemma 2.4] 6 ) that either K is separably closed, or there exists some finite separable extension L ≥ K and q = char(K) such that (L × ) q = L × (in fact, by [22,Corollary 4.5] K has no finite separable extensions of degree divisible by p).…”

Section: N ′mentioning

confidence: 99%

“…It is therefore, harmless to assume that √ −1 ∈ K. By [22,Theorem 4.4] K is Artin-Schreier closed. So the same is true of any finite extension L ≥ K. This implies (e.g., [27, Lemma 2.4] 6 ) that either K is separably closed, or there exists some finite separable extension L ≥ K and q = char(K) such that (L × ) q = L × (in fact, by [22,Corollary 4.5] K has no finite separable extensions of degree divisible by p). Since √ −1 ∈ L it follows that, letting L(q) denote the q-closure of L, we have [L(q) : L] = ∞ ([8, Theorem 4.3.5]).…”

Section: N ′mentioning

confidence: 99%

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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Artin-Schreier extensions in NIP and simple fields

Abstract: We show that NIP fields have no Artin-Schreier extension, and that simple fields have only a finite number of them.

Cited by 51 publications

References 22 publications

Regularity lemma for distal structures

Regularity lemma for distal structures

Strongly dependent ordered abelian groups and Henselian fields

Definable valuations induced by multiplicative subgroups and NIP fields

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