Distality for the Asymptotic Couple of the Field of Logarithmic Transseries

Gehret, Allen; Kaplan, Elliot

doi:10.1215/00294527-2020-0010

Cited by 4 publications

(6 citation statements)

References 13 publications

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“…It is well-known that if T is distal, then T is NIP; for instance, see [34,Proposition 2.8]. Distality can be thought of as a notion of pure instability among NIP theories.…”

Section: Preliminaries On Distalitymentioning

confidence: 99%

Distality in valued fields and related structures

Aschenbrenner¹,

Chernikov²,

Gehret³

et al. 2020

Preprint

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We investigate distality and existence of distal expansions in valued fields and related structures. In particular, we characterize distality in a large class of ordered abelian groups, provide an AKE-style characterization for henselian valued fields, and demonstrate that certain expansions of fields, e.g., the differential field of logarithmic-exponential transseries, are distal. As a new tool for analyzing valued fields we employ a relative quantifier elimination for pure short exact sequences of abelian groups.

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“…It is well-known that if T is distal, then T is NIP; for instance, see [34,Proposition 2.8]. Distality can be thought of as a notion of pure instability among NIP theories.…”

Section: Preliminaries On Distalitymentioning

confidence: 99%

Distality in valued fields and related structures

Aschenbrenner¹,

Chernikov²,

Gehret³

et al. 2020

Preprint

Self Cite

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“…Proof. It is well known that distality implies NIP; e.g., see [7,Proposition 2.8] for a proof. Proof.…”

Section: Model Theory Of Hamel Spaces Now Let Lmentioning

confidence: 99%

“…Proof. This is an exercise in simplification and bookkeeping which mimics the proof of [7,Lemma 4.4], except that it uses the fact that the value set v(G = ) is a C -linearly independent subset of G (Lemma 4.3).…”

Section: Indiscernible Lemmasmentioning

confidence: 99%

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Hamel Spaces and Distal Expansions

Gehret¹,

Nell²

2019

J. symb. log.

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“…Hieronymi and Nell developed criteria for determining when certain expansions of o-minimal theories remain distal [13], and Nell continued this work by studying distal behavior in dense pairs of o-minimal structures [18]. In 2018, the asymptotic couple of the field of logarithmic transseries was shown to be distal by Gehret and Kaplan [10], and in 2020, Aschenbrenner, Chernikov, Gehret, and Ziegler explored distality in valued fields and, among other things, proved that the differential field of the logarithmic-exponential transseries is distal [1].…”

Section: Introductionmentioning

confidence: 99%

Distality Rank

WALKER¹

2022

J. symb. log.

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Building on Pierre Simon’s notion of distality, we introduce distality rank as a property of first-order theories and give examples for each rank m such that $1\leq m \leq \omega $ . For NIP theories, we show that distality rank is invariant under base change. We also define a generalization of type orthogonality called m-determinacy and show that theories of distality rank m require certain products to be m-determined. Furthermore, for NIP theories, this behavior characterizes m-distality. If we narrow the scope to stable theories, we observe that m-distality can be characterized by the maximum cycle size found in the forking “geometry,” so it coincides with $(m-1)$ -triviality. On a broader scale, we see that m-distality is a strengthening of Saharon Shelah’s notion of m-dependence.

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Distality for the Asymptotic Couple of the Field of Logarithmic Transseries

Cited by 4 publications

References 13 publications

Distality in valued fields and related structures

Distality in valued fields and related structures

Hamel Spaces and Distal Expansions

Distality Rank

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