Burden in Henselian Valued Fields

Abstract

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 ...

Residue field domination in henselian valued fields of equicharacteristic zero

“…In [19,Theorem 7.6] Chernikov proved that a henselian valued field of equicharacteristic zero in the L ac language is N T P 2 if and only if its residue field is N T P 2 . Later in [20,Theorem 3.11] P. Touchard proved that if K = (K, k, Γ, ac, res, v) is a henselian valued field of equicharacteristic zero then bdn(K ac ) = bdn(k) + bdn(Γ), where bdn(X) is the burden of the definable set X as defined in [20,Definition 1.12]. He also showed that if a valued field of equicharacteristic zero is considered in the language L then bdn(K) = max n≥0 (bdn(k × (k × ) n ) + bdn(nΓ)), therefore a henselian valued field of equicharacteristic zero is N T P 2 if and only if its residue field is N T P 2 .…”

On Model Theory of Valued Vector Spaces

“…When we do, this will be indicated to the reader with a superscript ⋆: Card ⋆ , bdn ⋆ , sup ⋆ etc. For more details, notably about the arithmetic of Card ⋆ , see [22,Section 1.1.3]). We will also use Rideau-Kikushi's terminology in [18,Annex A].…”

“…We may assume the array of parameters to be mutually indiscernible. By Proposition 1.3 and elimination of the disjunction in inp-patterns (see for instance [22]), we may assume that every formula φ(x, b, c) with variable x ∈ V and parameters b ∈ V , c ∈ K is a conjunction of:…”

“…The authors of [2] use first a result of Flenner [6] -which eliminates 2020Mathematics Subject Classification: Primary: 03C10 03C45; Secondary: 03C60, 15A03 * The author was partially supported through the "Oberwolfach Leibniz Fellows" program -Project-ID2043r, by Mathematisches Forschungsinstitut Oberwolfach in 2020 and supported by a grant of the University of Campania 'Luigi Vanvitelli' in the framework of V:ALERE 2019 (GoAL project) 1 for instance, the ultraproduct U Qp has burden 1 in the language of valued fields, and 2 in the language of Pas. See [4] and [22]).…”

“…For this second step, they observe that it is enough to see the RV-sort as an enriched pure exact sequence of abelian groups. This two step reduction method has inspired many transfer principles in the context of (enriched) henselian valued fields (see for instance [8], [9], [22], [23]). In this paper, we study simpler objects: vector spaces and valued vector spaces (with value preserving scalar multiplication).…”