In the spirit of the Ax-Kochen-Ershov principle, we show that in certain cases the burden of a Henselian valued field can be computed in terms of the burden of its residue field and that of its value group. To do so, we first see that the burden of such a field is equal to the burden of its RV-sort. These results are generalizations of Chernikov and Simon's work in [CS16].In [She90], Shelah started to classify first order theories by elaborating a hierarchy of combinatorial properties of families of definable sets. The study of these properties leads to a better understanding of algebraic structure. Among these properties, one may mention stability, NIP (No Independence Property), simplicity, NTP 2 (No Tree Property of the 2 nd kind), and their rank-one versions, namely strong minimality, dp-minimality, SU-rank 1, and inp-minimality respectively. From this classification arises a complex map of tameness properties, and locating a given concrete first-order theory in this hierarchy is often interesting and challenging. In the particular context of Henselian valued fields, a now common approach is to find an Ax-Kochen-Ershov-like principle. The theorem of Ax-Kochen-Ershov states that any Henselian valued field of equicharacteristic 0 is model complete relative to the residue field and value group. More generally, the question is the following: let P be a certain combinatorial property and assume both the theory of the residue field and that of the value group satisfy P . Then, does the theory of the valued field satisfy P ? Let us state the theorem of Delon [Del81] as an example: a Henselian valued field of equicharacteristic 0 is NIP if and only if both of its residue field and its value group are NIP 1 .This approach is based on the faith that the study of the residue group and the valued field might be enough to classify the valued field. But as we will see, it might be reasonable in some cases to consider also another interpretable sort. The first hint of this fact might be seen in the theorem of Pas: we know that a Henselian valued field (K, Γ, k) of equicharacteristic zero equipped with an angular component (also called ac-map), eliminates K-quantifiers relative to the value group and the residue field. But it is important to notice that the ac-map is needed here, as the theorem fails otherwise. This has the disadvantage of adding new definable sets to the structure. For instance, any ultraproduct of p-adic fields over a non-principal utrafilter on prime numbers is inp-minimal (see [CS16]), but it is of burden 2 (i.e. NTP 2 of dimension 2) when one adds an ac-map. Basarab and Kuhlmann's different approach was to use another natural sort, capturing both information from the value group and the residue field: the leading term structure (see subsection 2.2), also called the RV-sort. Unlike the ac-map, it is always interpretable, and adding it to the language does not add definable sets. In [Bas91], Basarab was the first to give quantifier elimination results relative to this sort. Later in [Fle11], Flenner gave ...

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In this thesis, we study transfer principles in the context of certain Henselian valued fields, namely Henselian valued fields of equicharacteristic $0$ , algebraically closed valued fields, algebraically maximal Kaplansky valued fields, and unramified mixed characteristic Henselian valued fields with perfect residue field. First, we compute the burden of such a valued field in terms of the burden of its value group and its residue field. The burden is a cardinal related to the model theoretic complexity and a notion of dimension associated to $\text {NTP}_2$ theories. We show, for instance, that the Hahn field $\mathbb {F}_p^{\text {alg}}((\mathbb {Z}[1/p]))$ is inp-minimal (of burden 1), and that the ring of Witt vectors $W(\mathbb {F}_p^{\text {alg}})$ over $\mathbb {F}_p^{\text {alg}}$ is not strong (of burden $\omega $ ). This result extends previous work by Chernikov and Simon and realizes an important step toward the classification of Henselian valued fields of finite burden. Second, we show a transfer principle for the property that all types realized in a given elementary extension are definable. It can be written as follows: a valued field as above is stably embedded in an elementary extension if and only if its value group is stably embedded in the corresponding extension of value groups, its residue field is stably embedded in the corresponding extension of residue fields, and the extension of valued fields satisfies a certain algebraic condition. We show, for instance, that all types over the power series field $\mathbb {R}((t))$ are definable. Similarly, all types over the quotient field of $W(\mathbb {F}_p^{\text {alg}})$ are definable. This extends previous work of Cubides and Delon and of Cubides and Ye.These distinct results use a common approach, which has been developed recently. It consists of establishing first a reduction to an intermediate structure called the leading term structure, or $\operatorname {\mathrm {RV}}$ -sort, and then of reducing to the value group and residue field. This leads us to develop similar reduction principles in the context of pure short exact sequences of abelian groups.Abstract prepared by Pierre Touchard.E-mail: pierre.pa.touchard@gmail.comURL: https://miami.uni-muenster.de/Record/a612cf73-0a2f-42c4-b1e4-7d28934138a9

One can associate to a valued field an inverse system of valued hyperfields (H i ) i∈I in a natural way. We investigate when, conversely, such a system arise from a valued field. First, we extend a result of Krasner by showing that the inverse limit of certain systems are stringent valued hyperfields. Secondly, we describe a Hahn-like construction which yields a henselian valued field from a stringent valued hyperfield. In addition, we provide an axiomatisation of the theory of stringent valued hyperfields in a language consisting of two binary function symbols ⊕ and • and two constant symbols 0 and 1.

In analogy to valued fields, we study model-theoretic properties of valued vector spaces with variable base field by proving transfer principles down to the skeleton and down to the value set and base field. For instance, we give a formula which computes its burden in terms of the burden of its base field and its value set. To do this, we study these transfer principles in the context of lexicographic products of structures.