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Abstract

We construct a nontrivial definable type V field topology on any dp-minimal field (Formula presented.) that is not strongly minimal, and prove that definable subsets of (Formula presented.) have small boundary. Using this topology and its properties, we show that in any dp-minimal field (Formula presented.), dp-rank of definable sets varies definably in families, dp-rank of complete types is characterized in terms of algebraic closure, and (Formula presented.) is finite for all (Formula presented.). Additionally, by combining the existence of the topology with results of Jahnke, Simon and Walsberg [Dp-minimal valued fields, J. Symbolic Logic 82(1) (2017) 151–165], it follows that dp-minimal fields that are neither algebraically closed nor real closed admit nontrivial definable Henselian valuations. These results are a key stepping stone toward the classification of dp-minimal fields in [Fun with fields, Ph.D. thesis, University of California, Berkeley (2016)].
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... Various specialisations of this conjecture were considered in [11], [6] and [7]. In [11] the investigation restricts to strongly NIP, as in the original conjecture by Shelah. ...
... Strongly NIP imposes a boundary on the dp-rank (see [13,Definition 4.12]), an important measure of complexity for NIP structures. In [6] and [7] the query is further narrowed to fields which are dp-minimal (see [13,Definition 4.27] for the definition). Dp-minimality is a further refinement of strongly NIP, limiting the dp-rank as much as possible. ...
... In [6,Corollary 6.6] it is established that every dp-minimal ordered field is real closed or admits a non-trivial definable henselian valuation. Shelah's conjecture is verified for the dp-minimal case in [7,Theorem 1.6], which establishes that every infinite dp-minimal field is real closed, algebraically closed or admits a non-trivial definable henselian valuation. ...
Preprint
Full-text available
We give an explicit algebraic characterisation of all definable henselian valuations on a dp-minimal real field. Additionally we characterise all dp-minimal real fields that admit a definable henselian valuation with real closed residue field. We do so by first proving this for the more general setting of almost real closed fields.
... Johnson [Joh18] 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). ...
... The answer is positive for a dp-minimal field by the results of Johnson [Joh18] (so under the assumptions of Theorem 4, we have thatK is dp-minimal if and only if both k and Γ are dpminimal), but the proof relies on the construction of a valuation which doesn't seem to be available in the general inp-minimal case. ...
Preprint
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Preprint
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... But R V does not interpret any 1-dimensional algebraically closed fields [13,Theorem 4.21], a contradiction. Let us elaborate 2 : by [20] any dp-minimal unstable field is an SW-uniformity (in the sense of [30]). Moreover, as we have seen, T -convex expansions of o-minimal fields have generic differentiability (this is necessary to apply the facts we now use). ...
Preprint
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... Without getting into technical details, let us give a brief overview of the strategy of the proof and through it also some of the intuition underlying these new definitions. As in [11], our method is local, in the sense that the subgroup ν in the conclusion of Theorem 1.1 appears as a generalised infinitesimal group (as in, e.g., [25] or [18]). ...
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