We define a notion of residue field domination for valued fields which generalizes stable domination in algebraically closed valued fields. We prove that a real closed valued field is dominated by the sorts internal to the residue field, over the value group, both in the pure field sort and in the geometric sorts. These results characterize forking and þ-forking in real closed valued fields (and also algebraically closed valued fields). We lay some groundwork for extending these results to a power-bounded $T$-convex theory.
In this paper we study domination in an Ax-Kochen/Ershov style results for henselian valued fields of equicharacteristic zero for elements in the home sort.
We show that any theory of tame henselian valued fields is NIP if and only if the theory of its residue field and the theory of its value group are NIP. Moreover, we show that if $(K,v)$ is a henselian valued field of residue characteristic $\mathrm{char}(Kv)=p$ for which $K^\times/(K^\times)^p$ is finite in case $p>0$, then $(K,v)$ is NIP iff $Kv$ is NIP and $v$ is roughly tame.
Motivated by the Ax-Kochen/Ershov principle, a large number of questions about henselian valued fields have been shown to reduce to analogous questions about the value group and residue field. In this paper, we investigate the burden of henselian valued fields in the three-sorted Denef-Pas language. If $T$ is a theory of henselian valued fields admitting relative quantifier elimination (in any characteristic), we show that the burden of $T$ is equal to the sum of the burdens of its value group and residue field. As a consequence, $T$ is NTP${}_2$ if and only if its residue field and value grou…
Ax-Kochen-Ershov principles for finitely ramified henselian fields
Université Paris Cité（巴黎西岱大学）；Sorbonne Université（索邦大学）；CNRS, IMJ-PRG（法国国家科学研究中心，IMJ-PRG）；Technische Universität Dresden（德累斯顿工业大学）；University of Manchester（曼彻斯特大学）；University of Münster（明斯特大学）；Institute for Logic, Language and Computation (ILLC), University of Amsterdam（阿姆斯特丹大学逻辑、语言与计算研究所）
We show that, for a certain large class of power-bounded $o$-minimal $\mathcal{L}_T$-theories $T$ whose field of exponents is infinite-dimensional as a vector space over the rationals, any definable set in a $T$-convex valued field $(\mathcal{R}, \mathfrak{O})$ is in a precise sense the limit of a family of $\mathcal{L}_T$-definable sets indexed over the residue field. Alternatively, in the mainstream model-theoretic language, this says that if $(\mathcal{R}', \mathfrak{O}')$ is an elementary substructure of $(\mathcal{R}, \mathfrak{O})$ and if the residue field of $\mathfrak{O}$ contains an e…
In this article, we study the structure of finitely ramified mixed characteristic valued fields. For any two complete discrete valued fields $K_1$ and $K_2$ of mixed characteristic with perfect residue fields, we show that if the $n$-th residue rings are isomorphic for each $n\ge 1$, then $K_1$ and $K_2$ are isometric and isomorphic. More generally, for $n_1\ge 1$, there is $n_2$ depending only on the ramification indices of $K_1$ and $K_2$ such that any homomorphism from the $n_1$-th residue ring of $K_1$ to the $n_2$-th residue ring of $K_2$ can be lifted to a homomorphism between the valuat…
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 generalisations of a work of Chernikov and Simon.
Let $K$ be an NIP field and let $v$ be a henselian valuation on $K$. We ask whether $(K,v)$ is NIP as a valued field. By a result of Shelah, we know that if $v$ is externally definable, then $(K,v)$ 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 $(K,v)$ is NIP as a pure valued field.
Let $\mathcal M=\langle K;O\rangle$ be a real closed valued field and let $k$ be its residue field. We prove that every interpretable field in $\mathcal M$ is definably isomorphic to either $K$, $K(\sqrt{-1})$, $k$, or $k(\sqrt{-1})$. The same result holds when $K$ is a model of $T$, for $T$ an o-minimal power bounded expansion of a real closed field, and $O$ is a $T$-convex subring. The proof is direct and does not make use of known results about elimination of imaginaries in valued fields.
Elimination results for tame fields with finite residue fields
Université Paris Cité, Sorbonne Université, CNRS, IMJ-PRG（巴黎西岱大学、索邦大学、法国国家科学研究中心、IMJ-PRG）；Heinrich Heine University Düsseldorf, Faculty of Mathematics and Natural Sciences（杜塞尔多夫海因里希·海涅大学、数学与自然科学学院）
本文基于 Kuhlmann 和 Lisinski 的工作，研究有限域上的 Hahn 系列场的理论，证明每个公式等价于存在 y 使得多项式 f(x₁,…,xₙ,y)=0。
We show a transfer principle for the property that all types realised in a given elementary extension are definable. It can be written as follows: a Henselian valued fields is stably embedded in an elementary extension if and only if its value group is stably embedded in its corresponding extension, its residue field is stably embedded in its corresponding extension, and the extension of valued fields satisfies a certain algebraic condition. We show for instance that all types over the Hahn field $\mathbb{R}((\mathbb{Z}))$ are definable. Similarly, all types over the quotient field of the Witt…
In this paper we study elimination of imaginaries in some classes of henselian valued fields of equicharacteristic zero and residue field algebraically closed. The results are sensitive to the complexity of the value group. We focus first in the case where the ordered abelian group has finite spines, and then prove a better result for the dp-minimal case. An ordered abelian with finite spines weakly eliminates imaginaries once we add sorts for the quotient groups $Γ/ Δ$ for each definable convex subgroup $Δ$, and sorts for the quotient groups $Γ/ Δ+ lΓ$ where $Δ$ is a definable convex subgroup…
The aim of this article is to give a self-contained account of the algebra and model theory of Cohen rings, a natural generalization of Witt rings. Witt rings are only valuation rings in case the residue field is perfect, and Cohen rings arise as the Witt ring analogon over imperfect residue fields. Just as one studies truncated Witt rings to understand Witt rings, we study Cohen rings of positive characteristic as well as of characteristic zero. Our main results are a relative completeness and a relative model completeness result for Cohen rings, which imply the corresponding Ax-Kochen/Ershov…
For any two complete discrete valued fields $K_1$ and $K_2$ of mixed characteristic with perfect residue fields, we show that if the $n$-th valued hyperfields of $K_1$ and $K_2$ are isomorphic over $p$ for each $n\ge1$, then $K_1$ and $K_2$ are isomorphic. More generally, for $n_1,n_2\ge 1$, if $n_2$ is large enough, then any homomorphism, which is over $p$, from the $n_1$-th valued hyperfield of $K_1$ to the $n_2$-th valued hyperfield of $K_2$ can be lifted to a homomorphism from $K_1$ to $K_2$. We compute such $n_2$ effectively, which depends only on the ramification indices of $K_1$ and $K_…
We continue the effort of grokking the structure of power-bounded $T$-convex valued fields, whose theory is in general referred to as TCVF. In the present paper our focus is on certain expansion of it that is equipped with a tempered exponential function beyond the valuation ring. In order to construct such a tempered exponential function, the signed value group is also converted into a model of $T$ plus exponentiation and is in fact identified with (a section of) the residue field via the composition of a diagonal cross-section and an angular component map. In a sense, the resulting universal…
Existential fragments of theories of henselian valued fields
Université Paris Cité, Sorbonne Université, CNRS, IMJ-PRG（巴黎西岱大学，索邦大学，法国国家科学研究中心，IMJ-PRG）；Institut für Algebra, Technische Universität Dresden（德累斯顿工业大学代数研究所）
We prove that a valued field of positive characteristic $p$ that has only finitely many distinct Artin-Schreier extensions (which is a property of infinite NTP$_2$ fields) is dense in its perfect hull. As a consequence, it is a deeply ramified field and has $p$-divisible value group and perfect residue field. Further, we prove a partial analogue for valued fields of mixed characteristic and observe an open problem about 1-units in this setting. Finally, we fill a gap that occurred in a proof in an earlier paper in which we first introduced a classification of Artin-Schreier defect extensions.
In this article we study definable functions in tame expansions of algebraically closed valued fields. For a given definable function we have two types of results: of type (I), which hold at a neighborhood of infinity, and of type (II), which hold locally for all but finitely many points in the domain of the function. In the first part of the article, we show type (I) and (II) results concerning factorizations of definable functions over the value group. As an application, we show that tame expansions of algebraically closed valued fields having value group $\mathbb{Q}$ (like $\mathbb{C}_p$ an…
We prove a uniform version of non-Archimedean Yomdin-Gromov parametrizations in a definable context with algebraic Skolem functions in the residue field. The parametrization result allows us to bound the number of F_q[t]-points of bounded degrees of algebraic varieties, uniformly in the cardinality q of the finite field F_q and the degree, generalizing work by Sedunova for fixed q. We also deduce a uniform non-Archimedean Pila-Wilkie theorem, generalizing work by Cluckers-Comte-Loeser.
Let $\mathcal{K}=(K,v,\ldots)$ be a dp-minimal expansion of a non-trivially valued field of characteristic $0$ and $\mathcal{F}$ an infinite field interpretable in $\mathcal{K}$. Assume that $\mathcal{K}$ is one of the following: (i) $V$-minimal, (ii) power bounded $T$-convex, or (iii) $P$-minimal (assuming additionally in (iii) generic differentiability of definable functions). Then $\mathcal{F}$ is definably isomorphic to a finite extension $K$ or, in cases (i) and (ii), its residue field. In particular, every infinite field interpretable in $\mathbb{Q}_p$ is definably isomorphic to a finite…
We prove a tight bound on the number of realized $0/1$ patterns (or equivalently on the Vapnik-Chervonenkis codensity) of definable families in models of the theory of algebraically closed valued fields with a non-archimedean valuation. Our result improves the best known result in this direction proved by Aschenbrenner, Dolich, Haskell, Macpherson and Starchenko, who proved a weaker bound in the restricted case where the characteristics of the field $K$ and its residue field are both assumed to be $0$. The bound obtained here is optimal and without any restriction on the characteristics. We ob…
