We survey the history of Shelah’s conjecture on strongly dependent fields, give an equivalent formulation in terms of a classification of strongly dependent fields and prove that the conjecture implies that every strongly dependent field has finite dp-rank.

In this article, we develop tame topology over dp-minimal structures equipped with definable uniformities satisfying certain assumptions. Our assumptions are enough to ensure that definable sets are tame: there is a good notion of dimension on definable sets, definable functions are almost everywhere continuous, and definable sets are finite unions of graphs of definable continuous “multivalued functions.” This generalizes known statements about weakly o-minimal, C-minimal, and P-minimal theories.

Let K be an NIP field and let v be a Henselian valuation on K. We ask whether [Formula: see text] is NIP as a valued field. By a result of Shelah, we know that if v is externally definable, then [Formula: see text] is NIP. Using the definability of the canonical p-Henselian valuation, we show that whenever the residue field of v is not separably closed, then v is externally definable. In the case of separably closed residue field, we (...) show that [Formula: see text] is NIP as a pure valued field. (shrink)

Given a Henselian valuation, we study its definability (with and without parameters) by examining conditions on the value group. We show that any Henselian valuation whose value group is not closed in its divisible hull is definable in the language of rings, using one parameter. Thereby we strengthen known definability results. Moreover, we show that in this case, one parameter is optimal in the sense that one cannot obtain definability without parameters. To this end, we present a construction method for (...) a t-Henselian non-Henselian ordered field elementarily equivalent to a Henselian field with a specified value group. (shrink)

We prove that NIP valued fields of positive characteristic are henselian, and we begin to generalize the known results on dp-minimal fields to dp-finite fields. On any unstable dp-finite field K, we define a type-definable group of “infinitesimals,” corresponding to a canonical group topology on (K, +). We reduce the classification of positive characteristic dp-finite fields to the construction of non-trivial Aut(K/A)-invariant valuation rings.

We study the algebraic implications of the non-independence property and variants thereof on infinite fields, motivated by the conjecture that all such fields which are neither real closed nor separably closed admit a henselian valuation. Our results mainly focus on Hahn fields and build up on Will Johnson’s “The canonical topology on dp-minimal fields” :1850007, 2018).

We initiate the study of definable [Formula: see text]-topologies and show that there is at most one such [Formula: see text]-topology on a [Formula: see text]-henselian NIP field. Equivalently, we show that if [Formula: see text] is a bi-valued NIP field with [Formula: see text] henselian, then [Formula: see text] and [Formula: see text] are comparable. 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 [Formula: see text]-henselian. (shrink)

The following conjecture is due to Shelah–Hasson: Any infinite strongly NIP field is either real closed, algebraically closed, or admits a non‐trivial definable henselian valuation, in the language of rings. We specialise this conjecture to ordered fields in the language of ordered rings, which leads towards a systematic study of the class of strongly NIP almost real closed fields. As a result, we obtain a complete characterisation of this class.