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Ordered fields dense in their real closure and definable convex valuations
Abstract
In this paper, we undertake a systematic model- and valuation-theoretic study of the class of ordered fields which are dense in their real closure. We apply this study to determine definable henselian valuations on ordered fields, in the language of ordered rings. In light of our results, we re-examine the Shelah–Hasson Conjecture (specialized to ordered fields) and provide an example limiting its valuation-theoretic conclusions.
... While some of these results deal with the existence of nontrivial definable Henselian valuations and others consider the quantifier complexity of defining formulas (cf. [9, Sections 2 and 3]), in this paper we are interested in the definability of a given Henselian valuation with a certain value group. 1 The strongest currently known definability results which only pose conditions on the value group of a given Henselian valuation are exhibited in [14,21]. In the following, we state the results under consideration for this paper and outline how we strengthen these. ...
... While in [14] definability is treated in the language of rings L r , the definability results in [21] are stated for ordered fields in the language of ordered rings L or . Fact 1.3 [21,Theorem 5.3(2)]. ...
... While in [14] definability is treated in the language of rings L r , the definability results in [21] are stated for ordered fields in the language of ordered rings L or . Fact 1.3 [21,Theorem 5.3(2)]. Let (K, <, v) be an ordered Henselian valued field such that vK is not closed in its divisible hull. ...
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.
... While some of these results deal with the existence of non-trivial definable henselian valuations and others consider the quantifier complexity of defining formulas (cf. [9,Sections 2 & 3]), in this paper we are interested in the definability of a given henselian valuation with a certain value group. 1 The strongest currently known definability results which only pose conditions on the value group of a given henselian valuation are exhibited in [14] and [21]. In the following, we state the results under consideration for this paper and outline how we strengthen these. ...
... While in [14] definability is treated in the language of rings L r , the definability results in [21] are stated for ordered fields in the language of ordered rings L or . Fact 1.3. ...
... Fact 1.3. [21,Theorem 5.3 (2)]. Let (K, <, v) be an ordered henselian valued field such that vK is not closed in its divisible hull. ...
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.
... Further let w be a definable henselian valuation with real closed residue field. Then w = v 0 follows from [10,Proposition 5.9]. Also by [1,Theorem 4.4] this implies that G 0 is definable in G. Also note that whenever G 0 is non-trivial and definable in (G, +, <), then it follows from Fact 1.6 that G 0 = G p for some prime p. ...
... As a result it is C 2 /p n C 2 = B n /p n B n = Z/p n Z and therefore finite. As was pointed out in [10], C 2 has no nontrivial divisible convex subgroup, but for every prime p it has p-divisible convex subgroups. Hence an almost real closed field K with v K (K × ) = C 2 fails to have the property (2) from Theorem 1.8, and therefore also the others. ...
We give an explicit algebraic characterisation of all definable henselian valuations on a dp-minimal real field. Additionally we characterise all dp-minimal real fields that admit a definable henselian valuation with real closed residue field. We do so by first proving this for the more general setting of almost real closed fields.
... It is shown in [19] that any almost real closed field which is not real closed cannot be dense in its real closure. Thus, any dp-minimal ordered field which is dense in its real closure is real closed. ...
... A preliminary version of this work is contained in our arXiv preprint[18], which contains also a systematic study of L or -definable henselian valuations in ordered fields as well as of the class of ordered fields which are dense in their real closure. This systematic study, of independent interest, will be the subject of a separate publication[19].© 2021 Wiley-VCH GmbH www.mlq-journal.org ...
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.
... It is shown in [20] that any almost real closed field which is not real closed cannot be dense in its real closure. Thus, any dp-minimal ordered field which is dense in its real closure is real closed. ...
... A preliminary version of this work is contained in our arXiv preprint[19], which contains also a systematic study of Lor-definable henselian valuations in ordered fields as well as of the class of ordered fields which are dense in their real closure. This systematic study, of independent interest, will be the subject of a separate publication[20]. ...
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.
... Moreover, the majority of Chapter 8 and Section 7.1 are the main part of the preprint Krapp, S. Kuhlmann and Lehéricy [60]. This was slightly extended and split into two parts [61,62], both of which are submitted for publication. ...
... Corollary 8. 62. Let (K, <) be an almost real closed field and let G be an ordered abelian group. ...
An exponential exp on an ordered field (K, +, −, ·, 0, 1, <) is an order-preserving isomorphism from the ordered additive group (K, +, 0, <) to the ordered multiplicative group of positive elements(K^(>0), ·, 1, <). The structure (K, +, −, ·, 0, 1, <, exp) is then called an ordered exponential field. A linearly ordered structure (M, <, ...) is called o-minimal if every parametrically definable subset ofMis a finite union of points and open intervals of M.The main subject of this thesis is the algebraic and model theoretic examinationof o-minimal exponential fields (K, +, −, ·, 0, 1, <, exp) whose exponential satisfiesthe differential equation exp′ = exp with initial condition exp(0) = 1. This study is mainly motivated by the Transfer Conjecture, which states as follows: Any o-minimal exponential field (K, +, −, ·, 0, 1, <, exp) whose exponential satisfies the differential equation exp′ = exp with initial condition exp(0) = 1is elementarily equivalent to R_exp. Here, R_exp denotes the real exponential field (R, +, −, ·, 0, 1, <, exp), where exp denotes the standard exponential on R. Moreover, elementary equivalence means that any first-order sentence in the language L_exp = {+, −, ·, 0, 1, <, exp} holds for (K, +, −, ·, 0, 1, <, exp) if and only if it holds for R_exp. The Transfer Conjecture, and thus the study of o-minimal exponentialfields, is of particular interest in the light of the decidability of R_exp. To the date, it is not known if R_exp is decidable, i.e. whether there exists a procedure determining for agiven first-order L_exp-sentence whether it is true or false in R_exp. However, underthe assumption of Schanuel’s Conjecture – a famous open conjecture from Transcendental Number Theory – a decision procedure for R_exp exists. Also a positive answerto the Transfer Conjecture would result in the decidability of R_exp. Thus, we study o-minimal exponential fields with regard to the Transfer Conjecture, Schanuel’s Conjecture and the decidability question of 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 thesurreal numbers. Further, we draw connections to models of Peano Arithmetic, integer parts, density in real closure, definable henselian valuations and strongly NIP ordered fields.
... (i) The cases (i) and (ii) of Theorem 3.1 are optimal in the sense that, in general, one cannot obtain parameter-free definability. More precisely, in [Krapp et al. 2022, Examples 4.9 and 4.10] two ordered valued fields (L 1 , <, v 1 ) and (L 2 , <, v 2 ) are presented such that the following hold: ...
... Parts of this thesis were published in [2][3][4][5]. ...
An exponential on an ordered field (K,+,0, to the ordered multiplicative group of positive elements (K,+,-,\cdot ,0,1, is then called an ordered exponential field (cf. [6]). A linearly ordered structure (K,+,-,\cdot ,0,1, whose exponential satisfies the differential equation with initial condition . This study is mainly motivated by the Transfer Conjecture, which states as follows:
Any o-minimal exponential field \exp ' = \exp \exp (0)=1\mathbb {R}_{\exp }\mathbb {R}_{\exp }(\mathbb {R},+,-,\cdot ,0,1, , where denotes the standard exponential on . Moreover, elementary equivalence means that any first-order sentence in the language (K,+,-,\cdot ,0,1, if and only if it holds for .
The Transfer Conjecture, and thus the study of o-minimal exponential fields, is of particular interest in the light of the decidability of . To the date, it is not known if is decidable, i.e., whether there exists a procedure determining for a given first-order -sentence whether it is true or false in . However, under the assumption of Schanuel’s Conjecture—a famous open conjecture from Transcendental Number Theory—a decision procedure for exists (cf. [7]). Also a positive answer to the Transfer Conjecture would result in the decidability of (cf. [1]). Thus, we study o-minimal exponential fields with regard to the Transfer Conjecture, Schanuel’s Conjecture, and the decidability question of .
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 Krapp
E-mail : sebastian.krapp@uni-konstanz.de
URL : https://d-nb.info/1202012558/34
We study metric valued fields in continuous logic, following Ben Yaacov's approach, thus working in the metric space given by the projective line. As our main result, we obtain an approximate Ax-Kochen-Ershov principle in this framework, completely describing elementary equivalence in equicharacteristic 0 in terms of the residue field and value group. Moreover, we show that, in any characteristic, the theory of metric valued difference fields does not admit a model-companion. This answers a question of Ben Yaacov.
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.
We construct a nontrivial definable type V field topology on any dp-minimal field (Formula presented.) that is not strongly minimal, and prove that definable subsets of (Formula presented.) have small boundary. Using this topology and its properties, we show that in any dp-minimal field (Formula presented.), dp-rank of definable sets varies definably in families, dp-rank of complete types is characterized in terms of algebraic closure, and (Formula presented.) is finite for all (Formula presented.). Additionally, by combining the existence of the topology with results of Jahnke, Simon and Walsberg [Dp-minimal valued fields, J. Symbolic Logic 82(1) (2017) 151–165], it follows that dp-minimal fields that are neither algebraically closed nor real closed admit nontrivial definable Henselian valuations. These results are a key stepping stone toward the classification of dp-minimal fields in [Fun with fields, Ph.D. thesis, University of California, Berkeley (2016)].
Strongly dependent ordered abelian groups have finite dp-rank. They are precisely those groups with finite spines and . We conclude that, if K is a strongly dependent field, then (K,v) is strongly dependent for any henselian valuation v and the value group and residue field are stably embedded as pure structures.
We study the algebraic implications of the non-independence property (NIP) and variants thereof (dp-minimality) on infinite fields, motivated by the conjecture that all such fields which are neither real closed nor separably closed admit a definable henselian valuation. Our results mainly focus on Hahn fields and build up on Will Johnson's preprint "dp-minimal fields", arXiv: 1507.02745v1, July 2015.
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.
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).
The study of NIP theories has received much attention from model theorists in the last decade, fuelled by applications to o-minimal structures and valued fields. This book, the first to be written on NIP theories, is an introduction to the subject that will appeal to anyone interested in model theory: graduate students and researchers in the field, as well as those in nearby areas such as combinatorics and algebraic geometry. Without dwelling on any one particular topic, it covers all of the basic notions and gives the reader the tools needed to pursue research in this area. An effort has been made in each chapter to give a concise and elegant path to the main results and to stress the most useful ideas. Particular emphasis is put on honest definitions, handling of indiscernible sequences and measures. The relevant material from other fields of mathematics is made accessible to the logician.
Although the study of the definability of henselian valuations has a long history starting with J. Robinson, most of the results in this area were proven during the last few years. We survey these results which address the definability of concrete henselian valuations, the existence of definable henselian valuations on a given field, and questions of uniformity and quantifier complexity.
We show that dp-minimal valued fields are henselian and that a dp-minimal
field admitting a definable type V topology is either real closed,
algebraically closed or admits a non-trivial definable henselian valuation. We
give classifications of dp-minimal ordered abelian groups and dp-minimal
ordered fields without additional structure.
We study the question which henselian fields admit definable henselian
valuations (with or without parameters). We show that every field which admits
a henselian valuation with non-divisible value group admits a
parameter-definable (non-trivial) henselian valuation. In equicharacteristic
0, we give a complete characterization of henselian fields admitting a
parameter-definable (non-trivial) henselian valuation. We also obtain partial
characterization results of fields admitting 0-definable (non-trivial)
henselian valuations. We then draw some Galois-theoretic conclusions from our
results.
On a Henselian valued field (K,V), where V is the valuation ring, if the value group contains a convex p-regular subgroup that is not p-divisible, then V is definable in the language of rings. A Henselian valuation ring with a regular non-divisible value group is always 0-definable. In particular, some results of Ax and of Konenigmann are generalized.
We give model theoretic criteria for and -
formulas in the ring language to define uniformly the valuation rings
of models of an elementary theory of
henselian valued fields. As one of the applications we obtain the existence of
an -formula defining uniformly the valuation rings
of valued henselian fields whose residue class
field k is finite, pseudo-finite, or hilbertian. We also obtain -formulas and such that defines
uniformly k[[t]] in k((t)) whenever k is finite or the function field of
a real or complex curve, and does the job if k is any number
field.
In this note we investigate the question whether a henselian valued field
carries a non-trivial 0-definable henselian valuation (in the language of
rings). It follows from the work of Prestel and Ziegler that there are
henselian valued fields which do not admit a 0-definable non-trivial henselian
valuation. We give conditions on the residue field which ensure the existence
of a parameter-free definiton. In particular, we show that a henselian valued
field admits a non-trivial 0-definable valuation when the residue field is
separably closed or sufficiently non-henselian, or when the absolute Galois
group of the (residue) field is non-universal.
1ordered fields for which Hilbert's conjecture holds ? This paper deals with this and related questions. In particular, the principal result in this direction will be a proof that Hilbert' s conjecture holds on an ordered field, K, if and only if K is dense in its real closure and K is uniquely orderable. The fact that all ordered fields are not dense in their real closures can easily be seen by considering the field Q(t), where t is a transcendental which is placed greater than all the rationals. There is nothing from Q(t) in the interval (~-, Z ~-). In fact, if A is the field of real 2 algebraic numbers, no non-rational element of A is a limit point of Q(t).
Let us call an integer part of an ordered field any subring such that every element of the field lies at distance less than 1 from a unique element of the ring. We show that every real closed field has an integer part.
In this paper, we are concerned with the arithmetical definability of certain notions of integers and rationals in terms of other notions. The results derived will be applied to obtain a negative solution of corresponding decision problems.
In Section 1, we show that addition of positive integers can be defined arithmetically in terms of multiplication and the unary operation of successor S (where Sa = a + 1). Also, it is shown that both addition and multiplication can be defined arithmetically in terms of successor and the relation of divisibility | (where x|y means x divides y ).
We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing [Sh:715], [Sh:783] and related works. Those are properties (= classes) somewhat parallel to superstability among stable theory, though are different from it even for stable theories. We show equivalence of some of their definitions, investigate relevant ranks and give some examples, e.g., the first order theory of the p-adics is strongly dependent. The most notable result is: if |A| + |T| ≤ µ, I ⊆ ℭ and |I|≥ℶ|T|+(µ), then some J ⊆ I of cardinality µ+ is an indiscernible sequence over A.
Elementary properties of ordered abelian groups
- A Robinson
- E Zakon