Henselian Valued Fields Research Articles

Ax–Kochen–Ershov principles for finitely ramified henselian fields

We study the model theory of finitely ramified henselian valued fields of fixed initial ramification, obtaining versions of the Ax–Kochen–Ershov principle as follows. We identify the induced structure on the residue field and show that once the residue field is endowed with this structure, the theory of the valued field is determined by the theories of the enriched residue field and the value group. Similarly, we show that the existential theory of the valued field is determined by the positive existential theory of the enriched residue field. We also prove that an embedding of finitely ramified henselian valued fields is existentially closed as soon as the induced embeddings of value group and residue field are existentially closed. This last result requires no enrichment of the residue field, in analogy to the corresponding result for model completeness, which holds by results of Ershov and Ziegler.

The domination monoid in henselian valued fields

The domination monoid in henselian valued fields

Characterizing NIP henselian fields

AbstractIn this paper, we characterize NIP (Not the Independence Property) henselian valued fields modulo the theory of their residue field, both in an algebraic and in a model‐theoretic way. Assuming the conjecture that every infinite NIP field is either separably closed, real closed, or admits a nontrivial henselian valuation, this allows us to obtain a characterization of all theories of NIP fields.

Residue field domination in some henselian valued fields

Residue field domination in some henselian valued fields

Stably embedded submodels of Henselian valued fields

Stably embedded submodels of Henselian valued fields

A note on saturated distinguished chains of length one

Let [Formula: see text] be a Henselian valued field. In this paper, we investigate algebraic elements [Formula: see text] having saturated distinguished chains of length one over [Formula: see text]. We provide some characterizations for them. In particular, we characterize all irreducible polynomials over [Formula: see text] whose roots have saturated distinguished chains of length one. Moreover, using the properties of such elements we give various necessary, sufficient, or both conditions for the metric invariants of algebraic elements to be equal.

Burden in Henselian valued fields

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 leading term structure. These results are generalisations of Chernikov and Simon's work in [11].

Elimination of imaginaries in C((Γ))$\mathbb {C}((\Gamma ))$

AbstractIn this paper, we study elimination of imaginaries in 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 on the case where the ordered abelian group has finite spines, and then prove a better result for the dp‐minimal case. In previous work the author proved that an ordered abelian with finite spines weakly eliminates imaginaries once one adds sorts for the quotient groups for each definable convex subgroup , and sorts for the quotient groups where is a definable convex subgroup and . We refer to these sorts as the quotient sorts. Jahnke, Simon, and Walsberg (J. Symb. Log. 82 (2017) 151–165) characterized ‐minimal ordered abelian groups as those without singular primes, that is, for every prime number one has .We prove the following two theorems:

Bitangents to plane quartics via tropical geometry: rationality, $$\mathbb {A}^1$$-enumeration, and real signed count

We explore extensions of tropical methods to arithmetic enumerative problems such as $$\mathbb {A}^1$$ -enumeration with values in the Grothendieck–Witt ring and rationality over Henselian valued fields, using bitangents to plane quartics as a test case. We consider quartic curves over valued fields whose tropicalizations are smooth and satisfy a mild genericity condition. We then express obstructions to rationality of bitangents and their points of tangency in terms of twisting of edges of the tropicalization; the latter depends only on the tropicalization and the initial coefficients of the defining equation modulo squares. We also show that the $${{\,\textrm{GW}\,}}$$ -multiplicity of a tropical bitangent, i.e. the multiplicity with which its lifts contribute to the $$\mathbb {A}^1$$ -enumeration of bitangents as defined by Larson and Vogt (Res Math Sci 8(26):1–21, 2021), can be computed from the tropicalization of the quartic together with the initial coefficients of the defining equation. As an application, we show that the four lifts of most tropical bitangent classes contribute $$2\mathbb {H}$$ , twice the class of the hyperbolic plane, to the $$\mathbb {A}^1$$ -enumeration. These results rely on a degeneration theorem relating the Grothendieck–Witt ring of a Henselian valued field to the Grothendieck–Witt ring of its residue field, in residue characteristic not equal to two.

Spherically complete models of Hensel minimal valued fields

AbstractWe prove that Hensel minimal expansions of finitely ramified Henselian valued fields admit spherically complete immediate elementary extensions. More precisely, the version of Hensel minimality we use is 0‐hmix‐minimality (which, in equi‐characteristic 0, amounts to 0‐h‐minimality).

Defect extensions and a characterization of tame fields

We study the relation between two important classes of valued fields: tame fields and defectless fields. We show that in the case of valued fields of equal characteristic or rank one valued fields of mixed characteristic, tame fields are exactly the henselian valued fields for which all algebraic extensions are defectless fields. In general tame fields form a proper subclass of henselian valued fields for which all algebraic extensions are defectless fields. We introduce a wider class of roughly tame fields and show that every algebraic extension of a given valued field is defectless if and only if its henselization is roughly tame. Proving the above results we also present constructions of Galois defect extensions in positive as well as mixed characteristic.

The model theory of Cohen rings

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 type results for unramified henselian valued fields also in case the residue field is imperfect.

Tame fields and distinguished pairs

A henselian valued field is called defectless (resp. tame) if each of its finite extensions is defectless (resp. tame). For a henselian field K in [3], a question was posed: “If every simple algebraic extension of K is defectless, then is it true that K is defectless?” An example showed that the answer is “no” in general. In this paper, we show that the analogue of this question for tame fields has an affirmative answer. Indeed it is proved that if every simple algebraic extension of a henselian field K is tame, then it is a tame field. Moreover, for an algebraic element θ over a tame field K, it is known that all those elements appearing in a saturated distinguished chain for θ stay inside of . This rises the problem of studying conditions under which the next element to θ in a chain does not necessarily stay inside of . Here we will give a sufficient condition for when there is no algebraic element such that is a distinguished pair.

Burden of Henselian Valued Fields in the Denef–Pas Language

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 article, 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 NTP2 if and only if its residue field and value group are; the same is true for the statements “T is strong” and “T has finite burden.”

The classification of dp-minimal and dp-small fields

Continuing the work of Johnson (2018), we classify the pure fields and pure valued fields of dp-rank 1, up to elementary equivalence. In the process, we give a few new examples of dp-minimal fields. Specializing to the case of dp-small fields, we prove that dp-small fields and VC-minimal fields are all algebraically closed or real closed, as had been conjectured by Guingona (2014). This generalizes earlier work of Haskell and Macpherson (1994) and Macpherson et al. (2000), showing that C -minimal fields are algebraically closed, and weakly o-minimal fields are real closed. In fact, we obtain slight strengthenings of these earlier results, since we assume no compatibility between the field structure on the one hand and the C -relation or order on the other hand. We also give a new proof of Jahnke, Simon, and Walsberg’s (2017) theorem that dp-minimal valued fields are henselian. Lastly, we apply the Kaplan–Scanlon–Wagner (2011) theorem (NIP fields are Artin–Schreier closed) to prove some general properties of strongly dependent valued fields. For example, strongly dependent henselian valued fields are defectless.

Abstract key polynomials and distinguished pairs

In this paper, for a henselian valued field [Formula: see text] of arbitrary rank and an extension [Formula: see text] of [Formula: see text] to [Formula: see text] we use abstract key polynomials for [Formula: see text] to obtain distinguished pairs and saturated distinguished chains.

Distality in valued fields and related structures

We investigate distality and existence of distal expansions in valued fields and related structures. In particular, we characterize distality in a large class of ordered abelian groups, provide an Ax-Kochen-Eršov-style characterization for henselian valued fields, and demonstrate that certain expansions of fields, e.g., the differential field of logarithmic-exponential transseries, are distal. As a new tool for analyzing valued fields we employ a relative quantifier elimination for pure short exact sequences of abelian groups.

Hasse invariant for the tame Brauer group of a higher local field

We generalize the Hasse invariant of local class field theory to the tame Brauer group of a higher dimensional local field, and use it to study the arithmetic of central simple algebras, which are given a priori as tensor products of standard cyclic algebras. We also compute the tame Brauer dimension (or period-index bound) and the cyclic length of a general henselian-valued field of finite rank and finite residue field.

Subfields of algebraically maximal Kaplansky fields

Using the ramification theory of tame and Kaplansky fields, we show that maximal Kaplansky fields contain maximal immediate extensions of each of their subfields. Likewise, algebraically maximal Kaplansky fields contain maximal immediate algebraic extensions of each of their subfields. This study is inspired by problems that appear in henselian valued fields of rank higher than 1 when a Hensel root of a polynomial is approximated by the elements generated by a (transfinite) Newton algorithm. The main result of this article has been applied in the theory of separated (also called vector space defectless) extensions of valued fields.

On the non-uniqueness of maximal purely wild extensions

In this paper, we illustrate certain criteria which are sufficient for a henselian valued field to admit non-isomorphic maximal purely wild extensions.