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)

We prove that every ultraproduct of p-adics is inp-minimal. More generally, we prove an Ax-Kochen type result on preservation of inp-minimality for Henselian valued fields of equicharacteristic 0 in the RV language.

This article is a logical continuation of the Henri Lombardi and Franz-Viktor Kuhlmann article [9]. We address some classical points of the theory of valued fields with an elementary and constructive point of view. We deal with Krull valuations, and not simply discrete valuations. First of all, we show how to construct the Henselization of a valued field; we restrict to fields in which one has at one's disposal algorithmic tools to test the nullity or the valuation ring membership. It (...) is therefore a work that differs as much in spirit as in field of application from that of Mines, Richman and Bridges , who address the framework of Heyting fields and discrete valuation. We show then in a constructive way a batch of classical results in Henselian fields, notably factorization criteria and Krasner's Lemma. We conclude by a construction of the inertia field of a valued field. (shrink)

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 is a henselian valued field of residue characteristic \=p\) such that if \, depending on the characteristic of K either the degree of imperfection or the index of the pth powers is finite, then is NIP iff Kv is NIP and v is (...) roughly separably tame. (shrink)

We prove that if an equicharacteristic valued field has a ℤ-group as its value group and admits quantifier elimination in the main sort of the prototypical Denef-Pas style language then it is henselian. In fact the proof of this suggests that a reasonable class of Denef-Pas style languages is natural with respect to henselianity.

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.”. (shrink)

We give an elementary theory of Henselian local rings and construct the Henselisation of a local ring. All our theorems have an algorithmic content.

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)

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 $\mathbb {F}_p^{\text {alg}})$ is inp-minimal, and that the ring of Witt vectors $W$ over $\mathbb {F}_p^{\text {alg}}$ is not strong. 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 $\mathbb {R})$ are definable. Similarly, all types over the quotient field of $W$ 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.prepared by Pierre Touchard.E-mail: [email protected]: https://miami.uni-muenster.de/Record/a612cf73-0a2f-42c4-b1e4-7d28934138a9. (shrink)

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)

Let (K, v) be a henselian valued field of characteristic 0. Then K admits a definable partition on each piece of which the leading term of a polynomial in one variable can be computed as a definable function of the leading term of a linear map. The main step in obtaining this partition is an answer to the question, given a polynomial f(x) ∈ K[x], what is v(f(x))? Two applications are given: first, a constructive quantifier elimination relative to the (...) leading terms, suggesting a relative decision procedure; second, a presentation of every definable subset of K as the pullback of a definable set in the leading terms subjected to a linear translation. (shrink)

In well-known papers ([A-K1], [A-K2], and [E]) J. Ax, S. Kochen, and J. Ershov prove a transfer theorem for henselian valued fields. Here we prove an analogue for henselian valued and ordered fields. The orders for which this result apply are the usual orders and also the higher level orders introduced by E. Becker in [B1] and [B2]. With certain restrictions, two henselian valued and ordered fields are elementarily equivalent if and only if their value groups (with (...) a little bit more structure) and their residually ordered residue fields (a henselian valued and ordered field induces in a natural way an order in its residue field) are elementarily equivalent. Similar results are proved for elementary embeddings and ∀-extensions (extensions where the structure is existentially closed). (shrink)

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 field 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 ring $$W(\mathbb {F}_p^{\text {alg}})$$ are definable. This extends a work of Cubides and Delon and of Cubides and Ye. (shrink)

We give a definition, in the ring language, of Zp inside Qp and of Fp[[t]] inside Fp), which works uniformly for all p and all finite field extensions of these fields, and in many other Henselian valued fields as well. The formula can be taken existential-universal in the ring language, and in fact existential in a modification of the language of Macintyre. Furthermore, we show the negative result that in the language of rings there does not exist a uniform (...) definition by an existential formula and neither by a universal formula for the valuation rings of all the finite extensions of a given Henselian valued field. We also show that there is no existential formula of the ring language defining Zp inside Qp uniformly for all p. For any fixed finite extension of Qp, we give an existential formula and a universal formula in the ring language which define the valuation ring. (shrink)

We study the model theory of fields k carrying a henselian valuation with real closed residue field. We give a criteria for elementary equivalence and elementary inclusion of such fields involving the value group of a not necessarily definable valuation. This allows us to translate theories of such fields to theories of ordered abelian groups, and we study the properties of this translation. We also characterize the first-order definable convex subgroups of a given ordered abelian group and prove that (...) the definable real valuation rings of k are in correspondence with the definable convex subgroups of the value group of a certain real valuation of k. (shrink)

In [12], P. Scowcroft and L. van den Dries proved a cell decomposition theorem for p-adically closed fields. We work here with the notion of P-minimal fields defined by D. Haskell and D. Macpherson in [6]. We prove that a P-minimal field K admits cell decomposition if and only if K has definable selection. A preprint version in French of this result appeared as a prepublication [8].

We initiate a systematic study of the class of theories without the tree property of the second kind — NTP2. Most importantly, we show: the burden is “sub-multiplicative” in arbitrary theories ; NTP2 is equivalent to the generalized Kimʼs lemma and to the boundedness of ist-weight; the dp-rank of a type in an arbitrary theory is witnessed by mutually indiscernible sequences of realizations of the type, after adding some parameters — so the dp-rank of a 1-type in any theory is (...) always witnessed by sequences of singletons; in NTP2 theories, simple types are co-simple, characterized by the co-independence theorem, and forking between the realizations of a simple type and arbitrary elements satisfies full symmetry; a Henselian valued field of characteristic is NTP2 if and only if the residue field is NTP2 , so in particular any ultraproduct of p-adics is NTP2; adding a generic predicate to a geometric NTP2 theory preserves NTP2. (shrink)

In this paper, we investigate a new model theoretical tree property (TP), called the antichain tree property (ATP). We develop combinatorial techniques for ATP. First, we show that ATP is always witnessed by a formula in a single free variable, and for formulas, not having ATP is closed under disjunction. Second, we show the equivalence of ATP and [Formula: see text]-ATP, and provide a criterion for theories to have not ATP (being NATP). Using these combinatorial observations, we find algebraic examples (...) of ATP and NATP, including pure groups, pure fields and valued fields. More precisely, we prove Mekler’s construction for groups, Chatzidakis’ style criterion for pseudo-algebraically closed (PAC) fields, and the AKE-style principle for valued fields preserving NATP. We give a construction of an antichain tree in the Skolem arithmetic and atomless Boolean algebras. (shrink)

Recently, Cluckers et al. developed a new axiomatic framework for tame non-Archimedean geometry, called Hensel minimality. It was extended to mixed characteristic together with the author. Hensel minimality aims to mimic o-minimality in both strong consequences and wide applicability. In this paper, we continue the study of Hensel minimality, in particular focusing on [Formula: see text]-h-minimality and [Formula: see text]-h-minimality, for [Formula: see text] a positive integer. Our main results include an analytic criterion for [Formula: see text]-h-minimality, preservation of [Formula: (...) see text]-h-minimality under coarsening of the valuation and [Formula: see text]-dimensional geometry. (shrink)

We introduce a new notion of tame geometry for structures admitting an abstract notion of balls. The notion is named b-minimality and is based on definable families of points and balls. We develop a dimension theory and prove a cell decomposition theorem for b-minimal structures. We show that b-minimality applies to the theory of Henselian valued fields of characteristic zero, generalizing work by Denef–Pas [25, 26]. Structures which are o-minimal, v-minimal, or p-minimal and which satisfy some slight extra conditions (...) are also b-minimal, but b-minimality leaves more room for nontrivial expansions. The b-minimal setting is intended to be a natural framework for the construction of Euler characteristics and motivic or p-adic integrals. The b-minimal cell decomposition is a generalization of concepts of Cohen [11], Denef [15], and the link between cell decomposition and integration was first made by Denef [13]. (shrink)

We introduce a very weak language L M on p-adic fields K, which is just rich enough to have exactly the same definable subsets of the line K that one has using the ring language. (In our context, definable always means definable with parameters.) We prove that the only definable functions in the language L M are trivial functions. We also give a definitional expansion $L\begin{array}{*{20}{c}} ' \\ M \\ \end{array} $ of L M in which K has quantifier elimination, (...) and we obtain a cell decomposition result for L M -definable sets. Our language L M can serve as a p-adic analogue of the very weak language (<) on the real numbers, to define a notion of minimality on the field of p-adic numbers and on related valued fields. These fields are not necessarily Henselian and may have positive characteristic. (shrink)

A field K in a ring language $\mathcal {L}$ is finitely undecidable if $\mbox {Cons}(T)$ is undecidable for every nonempty finite $T \subseteq {\mathtt{Th}}(K; \mathcal {L})$. We extend a construction of Ziegler and (among other results) use a first-order classification of Anscombe and Jahnke to prove every NIP henselian nontrivially valued field is finitely undecidable. We conclude (assuming the NIP Fields Conjecture) that every NIP field is finitely undecidable. This work is drawn from the author’s PhD thesis [48, Chapter (...) 3]. (shrink)

We 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).

We study first-order properties of the quotient rings C(V)/P by a prime ideal P, where C(V) is the ring of p-adic valued continuous definable functions on some affine p-adic variety V. We show that they are integrally closed Henselian local rings, with a p-adically closed residue field and field of fractions, and they are not valuation rings in general but always satisfy ∀ x, y(x|y 2 ∨ y|x 2 ).

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.

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.

In spite of the analogies between Q p and F p ((t)) which became evident through the work of Ax and Kochen, an adaptation of the complete recursive axiom system given by them for Q p to the case of F p ((t)) does not render a complete axiom system. We show the independence of elementary properties which express the action of additive polynomials as maps on F p ((t)). We formulate an elementary property expressing this action and show that (...) it holds for all maximal valued fields. We also derive an example of a rather simple immediate valued function field over a henselian defectless ground field which is not a henselian rational function field. This example is of special interest in connection with the open problem of local uniformization in positive characteristic. (shrink)

Separated and immediate extensions of valued fields. The notion of separated extension of valued fields was introduced by Baur. He showed that extensions of maximal fields are separated. We prove that, when (K, v) is Henselian with residual characteristic 0, then $(K, v) \subset (L, w)$ is separated iff L is linearly disjoint over K from each immediate extension of K.

Separated and immediate extensions of valued fields. The notion of separated extension of valued fields was introduced by Baur. He showed that extensions of maximal fields are separated. We prove that, when $(K, v)$ is Henselian with residual characteristic 0, then $(K, v) \subset (L, w)$ is separated iff $L$ is linearly disjoint over $K$ from each immediate extension of $K$.

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 examine fields in which model theoretic algebraic closure coincides with relative field theoretic algebraic closure. These are perfect fields with nice model theoretic behaviour. For example, they are exactly the fields in which algebraic independence is an abstract independence relation in the sense of Kim and Pillay. Classes of examples are perfect PAC fields, model complete large fields and henselian valued fields of characteristic 0.

An exponential $\exp $ on an ordered field $$. The structure $$ is then called an ordered exponential field. A linearly ordered structure $$ is called o-minimal if every parametrically definable subset of M is a finite union of points and open intervals of M.The main subject of this thesis is the algebraic and model theoretic examination of o-minimal exponential fields $$ whose exponential satisfies the differential equation $\exp ' = \exp $ with initial condition $\exp = 1$. This study (...) is mainly motivated by the Transfer Conjecture, which states as follows:Any o-minimal exponential field $$ whose exponential satisfies the differential equation $\exp ' = \exp $ with initial condition $\exp =1$ is elementarily equivalent to $\mathbb {R}_{\exp }$.Here, $\mathbb {R}_{\exp }$ denotes the real exponential field $$, where $\exp $ denotes the standard exponential $x \mapsto \mathrm {e}^x$ on $\mathbb {R}$. Moreover, elementary equivalence means that any first-order sentence in the language $\mathcal {L}_{\exp } = \{+,-,\cdot,0,1, <,\exp \}$ holds for $$ if and only if it holds for $\mathbb {R}_{\exp }$.The Transfer Conjecture, and thus the study of o-minimal exponential fields, is of particular interest in the light of the decidability of $\mathbb {R}_{\exp }$. To the date, it is not known if $\mathbb {R}_{\exp }$ is decidable, i.e., whether there exists a procedure determining for a given first-order $\mathcal {L}_{\exp }$ -sentence whether it is true or false in $\mathbb {R}_{\exp }$. However, under the assumption of Schanuel’s Conjecture—a famous open conjecture from Transcendental Number Theory—a decision procedure for $\mathbb {R}_{\exp }$ exists. Also a positive answer to the Transfer Conjecture would result in the decidability of $\mathbb {R}_{\exp }$. Thus, we study o-minimal exponential fields with regard to the Transfer Conjecture, Schanuel’s Conjecture, and the decidability question of $\mathbb {R}_{\exp }$.Overall, we shed light on the valuation theoretic invariants of o-minimal exponential fields—the residue field and the value group—with additional induced structure. Moreover, we explore elementary substructures and extensions of o-minimal exponential fields to the maximal ends—the smallest elementary substructures being prime models and the maximal elementary extensions being contained in the surreal numbers. Further, we draw connections to models of Peano Arithmetic, integer parts, density in real closure, definable Henselian valuations, and strongly NIP ordered fields.Parts of this thesis were published in [2–5].Abstract prepared by Lothar Sebastian KrappE-mail: [email protected]: https://d-nb.info/1202012558/34. (shrink)

A chain-closed field is defined as a chainable field (i.e. a real field such that, for all n ∈ N, Σ K2n+1 ≠ Σ K2n) which does not admit any "faithful" algebraic extension, and can also be seen as a field having a Henselian valuation ν such that the residue field K/ν is real closed and the value group ν K is odd divisible with |ν K/2ν K| = 2. If K admits only one such valuation, we show that (...) f ∈ K(X) is in $\mathbf{\Sigma} K(X)^{2n} \operatorname{iff}$ for any real algebraic extension L of $K, "f(L) \subseteq \mathbf{\Sigma}L^{2n}"$ holds. The conclusion is also true for K = R((t)) (a chainable but not chain-closed field), and in the case n = 1 it holds for several variables and any real field K. (shrink)

The theory of immediate pairs of Henselian valued fields, with a given residual theory (of characteristic zero) and a given theory of valuation group (nonzero), is undecidable and has 2ℵ0 completions.

A chain-closed field is defined as a chainable field which does not admit any "faithful" algebraic extension, and can also be seen as a field having a Henselian valuation $\nu$ such that the residue field $K/\nu$ is real closed and the value group $\nu K$ is odd divisible with $|\nu K/2\nu K| = 2$. If $K$ admits only one such valuation, we show that $f \in K$ is in $\mathbf{\Sigma} K^{2n} \operatorname{iff}$ for any real algebraic extension $L$ of $K, (...) "f \subseteq \mathbf{\Sigma}L^{2n}"$ holds. The conclusion is also true for $K = \mathbf{R})$, and in the case $n = 1$ it holds for several variables and any real field $K$. (shrink)

We study when the property that a field is dense in its real and p-adic closures is elementary in the language of rings and deduce that all models of the theory of algebraic fields have this property.