In this paper, we develop three different subjects. We study and prove alternative versions of Hrushovski’s ‘stabilizer theorem’, we generalize part of the basic theory of definably amenable NIP groups …
In this paper, we develop three different subjects. We study and prove alternative versions of Hrushovski’s ‘stabilizer theorem’, we generalize part of the basic theory of definably amenable NIP groups to $\text{NTP}_{2}$ theories, and finally, we use all this machinery to study groups with f-generic types definable in bounded pseudo real closed fields.
In this paper we develop three different subjects. We study and prove alternative versions of Hrushovski's "Stabilizer Theorem", we generalize part of the basic theory of definably amenable NIP groups …
In this paper we develop three different subjects. We study and prove alternative versions of Hrushovski's "Stabilizer Theorem", we generalize part of the basic theory of definably amenable NIP groups to NTP2 theories, and finally, we use all this machinery to study groups with f-generic types definable in bounded PRC fields.
The main result of this paper is a positive answer to the Conjecture by A. Chernikov, I. Kaplan and P. Simon: If M is a PRC-field, then Th(M) is NTP2 …
The main result of this paper is a positive answer to the Conjecture by A. Chernikov, I. Kaplan and P. Simon: If M is a PRC-field, then Th(M) is NTP2 if and only if M is bounded. In the case of PpC fields, we prove that if M is a bounded PpC field, then Th(M) is NTP2. We also generalize this result to obtain that, if M is a bounded PRC(PpC)-field, then Th(M) is strong.
In this paper, we give a general criterion for elimination of imaginaries using an abstract independence relation. We also study germs of definable functions at certain well-behaved invariant types. Finally …
In this paper, we give a general criterion for elimination of imaginaries using an abstract independence relation. We also study germs of definable functions at certain well-behaved invariant types. Finally we apply these results to prove the elimination of imaginaries in bounded pseudo $p$-adically closed fields.
In this paper, we give a very general criterion for elimination of imaginaries using an abstract independent relation. We also study germs of definable functions at certain well-behaved invariant types. …
In this paper, we give a very general criterion for elimination of imaginaries using an abstract independent relation. We also study germs of definable functions at certain well-behaved invariant types. Finally we apply these tools to the elimination of imaginaries in bounded pseudo-p-adically closed fields.
Pseudo algebraically closed, pseudo real closed, and pseudo $p$-adically closed fields are examples of unstable fields that share many similarities, but have mostly been studied separately. In this text, we …
Pseudo algebraically closed, pseudo real closed, and pseudo $p$-adically closed fields are examples of unstable fields that share many similarities, but have mostly been studied separately. In this text, we propose a unified framework for studying them: the class of pseudo $T$-closed fields, where $T$ is an enriched theory of fields. These fields verify a "local-global" principle for the existence of points on varieties with respect to models of $T$. This approach also enables a good description of some fields equipped with multiple $V$-topologies, particularly pseudo algebraically closed fields with a finite number of valuations. One important result is a (model theoretic) classification result for bounded pseudo $T$-closed fields, in particular we show that under specific hypotheses on $T$, these fields are NTP$_2$ of finite burden.
The main result of this paper is that if M is a bounded PRC field then Th(M) eliminates imaginaries in the language of rings expanded by constant symbols.
The main result of this paper is that if M is a bounded PRC field then Th(M) eliminates imaginaries in the language of rings expanded by constant symbols.
In this paper, we give a very general criterion for elimination of imaginaries using an abstract independent relation. We also study germs of definable functions at certain well-behaved invariant types. …
In this paper, we give a very general criterion for elimination of imaginaries using an abstract independent relation. We also study germs of definable functions at certain well-behaved invariant types. Finally we apply these tools to the elimination of imaginaries in bounded pseudo-p-adically closed fields.
In this paper we study groups definable in existentially closed partial differential fields of characteristic 0 with an automorphism which commutes with the derivations. In particular, we study Zariski dense …
In this paper we study groups definable in existentially closed partial differential fields of characteristic 0 with an automorphism which commutes with the derivations. In particular, we study Zariski dense definable subgroups of simple algebraic groups, and show an analogue of Phyllis Cassidy's result for partial differential fields. We also show that these groups have a smallest definable subgroup of finite index.
The main result of this paper is that if M is a bounded PRC field then Th(M) eliminates imaginaries in the language of rings expanded by constant symbols.
The main result of this paper is that if M is a bounded PRC field then Th(M) eliminates imaginaries in the language of rings expanded by constant symbols.
Pseudo algebraically closed, pseudo real closed, and pseudo p-adically closed fields are examples of unstable fields that share many similarities, but have mostly been studied separately.We propose a unified framework …
Pseudo algebraically closed, pseudo real closed, and pseudo p-adically closed fields are examples of unstable fields that share many similarities, but have mostly been studied separately.We propose a unified framework for studying them: the class of pseudo T -closed fields, where T is an enriched theory of fields.These fields verify a "local-global" principle for the existence of points on varieties with respect to models of T. This approach also enables a good description of some fields equipped with multiple V -topologies, particularly pseudo algebraically closed fields with a finite number of valuations.One important result is a (model theoretic) classification result for bounded pseudo T -closed fields, in particular we show that under specific hypotheses on T, these fields are NTP 2 of finite burden.
Pseudo algebraically closed, pseudo real closed, and pseudo p-adically closed fields are examples of unstable fields that share many similarities, but have mostly been studied separately.We propose a unified framework …
Pseudo algebraically closed, pseudo real closed, and pseudo p-adically closed fields are examples of unstable fields that share many similarities, but have mostly been studied separately.We propose a unified framework for studying them: the class of pseudo T -closed fields, where T is an enriched theory of fields.These fields verify a "local-global" principle for the existence of points on varieties with respect to models of T. This approach also enables a good description of some fields equipped with multiple V -topologies, particularly pseudo algebraically closed fields with a finite number of valuations.One important result is a (model theoretic) classification result for bounded pseudo T -closed fields, in particular we show that under specific hypotheses on T, these fields are NTP 2 of finite burden.
Pseudo algebraically closed, pseudo real closed, and pseudo $p$-adically closed fields are examples of unstable fields that share many similarities, but have mostly been studied separately. In this text, we …
Pseudo algebraically closed, pseudo real closed, and pseudo $p$-adically closed fields are examples of unstable fields that share many similarities, but have mostly been studied separately. In this text, we propose a unified framework for studying them: the class of pseudo $T$-closed fields, where $T$ is an enriched theory of fields. These fields verify a "local-global" principle for the existence of points on varieties with respect to models of $T$. This approach also enables a good description of some fields equipped with multiple $V$-topologies, particularly pseudo algebraically closed fields with a finite number of valuations. One important result is a (model theoretic) classification result for bounded pseudo $T$-closed fields, in particular we show that under specific hypotheses on $T$, these fields are NTP$_2$ of finite burden.
In this paper we study groups definable in existentially closed partial differential fields of characteristic 0 with an automorphism which commutes with the derivations. In particular, we study Zariski dense …
In this paper we study groups definable in existentially closed partial differential fields of characteristic 0 with an automorphism which commutes with the derivations. In particular, we study Zariski dense definable subgroups of simple algebraic groups, and show an analogue of Phyllis Cassidy's result for partial differential fields. We also show that these groups have a smallest definable subgroup of finite index.
In this paper, we give a general criterion for elimination of imaginaries using an abstract independence relation. We also study germs of definable functions at certain well-behaved invariant types. Finally …
In this paper, we give a general criterion for elimination of imaginaries using an abstract independence relation. We also study germs of definable functions at certain well-behaved invariant types. Finally we apply these results to prove the elimination of imaginaries in bounded pseudo $p$-adically closed fields.
In this paper, we develop three different subjects. We study and prove alternative versions of Hrushovski’s ‘stabilizer theorem’, we generalize part of the basic theory of definably amenable NIP groups …
In this paper, we develop three different subjects. We study and prove alternative versions of Hrushovski’s ‘stabilizer theorem’, we generalize part of the basic theory of definably amenable NIP groups to $\text{NTP}_{2}$ theories, and finally, we use all this machinery to study groups with f-generic types definable in bounded pseudo real closed fields.
In this paper, we give a very general criterion for elimination of imaginaries using an abstract independent relation. We also study germs of definable functions at certain well-behaved invariant types. …
In this paper, we give a very general criterion for elimination of imaginaries using an abstract independent relation. We also study germs of definable functions at certain well-behaved invariant types. Finally we apply these tools to the elimination of imaginaries in bounded pseudo-p-adically closed fields.
In this paper, we give a very general criterion for elimination of imaginaries using an abstract independent relation. We also study germs of definable functions at certain well-behaved invariant types. …
In this paper, we give a very general criterion for elimination of imaginaries using an abstract independent relation. We also study germs of definable functions at certain well-behaved invariant types. Finally we apply these tools to the elimination of imaginaries in bounded pseudo-p-adically closed fields.
In this paper we develop three different subjects. We study and prove alternative versions of Hrushovski's "Stabilizer Theorem", we generalize part of the basic theory of definably amenable NIP groups …
In this paper we develop three different subjects. We study and prove alternative versions of Hrushovski's "Stabilizer Theorem", we generalize part of the basic theory of definably amenable NIP groups to NTP2 theories, and finally, we use all this machinery to study groups with f-generic types definable in bounded PRC fields.
The main result of this paper is a positive answer to the Conjecture by A. Chernikov, I. Kaplan and P. Simon: If M is a PRC-field, then Th(M) is NTP2 …
The main result of this paper is a positive answer to the Conjecture by A. Chernikov, I. Kaplan and P. Simon: If M is a PRC-field, then Th(M) is NTP2 if and only if M is bounded. In the case of PpC fields, we prove that if M is a bounded PpC field, then Th(M) is NTP2. We also generalize this result to obtain that, if M is a bounded PRC(PpC)-field, then Th(M) is strong.
The main result of this paper is that if M is a bounded PRC field then Th(M) eliminates imaginaries in the language of rings expanded by constant symbols.
The main result of this paper is that if M is a bounded PRC field then Th(M) eliminates imaginaries in the language of rings expanded by constant symbols.
The main result of this paper is that if M is a bounded PRC field then Th(M) eliminates imaginaries in the language of rings expanded by constant symbols.
The main result of this paper is that if M is a bounded PRC field then Th(M) eliminates imaginaries in the language of rings expanded by constant symbols.
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are …
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = \mathrm{tp}(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over \mathrm{bdd}(A) , (ii) analogous statements for Keisler measures and definable groups, including the fact that G^{000} = G^{00} for G definably amenable, (iii) definitions, characterizations and properties of “generically stable” types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in o -minimal expansions of real closed fields.
The study of pseudo-algebraically closed fields (henceforth called PAC) started with the work of J. Ax on finite and pseudo-finite fields [1]. He showed that the infinite models of the …
The study of pseudo-algebraically closed fields (henceforth called PAC) started with the work of J. Ax on finite and pseudo-finite fields [1]. He showed that the infinite models of the theory of finite fields are exactly the perfect PAC fields with absolute Galois group isomorphic to , and gave elementary invariants for their first order theory, thereby proving the decidability of the theory of finite fields. Ax's results were then extended to a larger class of PAC fields by M. Jarden and U. Kiehne [21], and Jarden [19]. The final word on theories of PAC fields was given by G. Cherlin, L. van den Dries and A. Macintyre [10], see also results by Ju. Ershov [13], [14]. Let K be a PAC field. Then the elementary theory of K is entirely determined by the following data: • The isomorphism type of the field of absolute numbers of K (the subfield of K of elements algebraic over the prime field). • The degree of imperfection of K . • The first-order theory, in a suitable ω -sorted language, of the inverse system of Galois groups al(L/K) where L runs over all finite Galois extensions of K . They also showed that the theory of PAC fields is undecidable, by showing that any graph can be encoded in the absolute Galois group of some PAC field. It turns out that the absolute Galois group controls much of the behaviour of the PAC fields. I will give below some examples illustrating this phenomenon.
Abstract We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in …
Abstract We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in fact over any extension base. As an application we show that dependence is equivalent to bounded non-forking assuming NTP 2 .
The brilliant work of Ax-Kochen [1], [2], [3] and Ersov [6], and later work by Kochen [7] have made clear very striking resemblances between real closed fields and p -adically …
The brilliant work of Ax-Kochen [1], [2], [3] and Ersov [6], and later work by Kochen [7] have made clear very striking resemblances between real closed fields and p -adically closed fields, from the model-theoretical point of view. Cohen [5], from a standpoint less model-theoretic, also contributed much to this analogy. In this paper we shall point out a feature of all the above treatments which obscures one important resemblance between real and p -adic fields. We shall outline a new treatment of the p -adic case (not far removed from the classical treatments cited above), and establish an new analogy between real closed and p -adically closed fields. We want to describe the definable subsets of p -adically closed fields. Tarski [9] in his pioneering work described the first-order definable subsets of real closed fields. Namely, if K is a real-closed field and X is a subset of K first-order definable on K using parameters from K then X is a finite union of nonoverlapping intervals (open, closed, half-open, empty or all of K ). In particular, if X is infinite, X has nonempty interior. Now, there is an analogous question for p -adically closed fields. If K is p -adically closed, what are the definable subsets of K ? To the best of our knowledge, this question has not been answered until now. What is the difference between the two cases? Tarski's analysis rests on elimination of quantifiers for real closed fields. Elimination of quantifiers for p -adically closed fields has been achieved [3], but only when we take a cross-section π as part of our basic data. The problem is that in the presence of π it becomes very difficult to figure out what sort of set is definable by a quantifier free formula. We shall see later that use of the cross-section increases the class of definable sets.
It is shown that if K is an algebraically closed valued field with valuation ring R, then Th(K) has elimination of imaginaries if sorts are added whose elements are certain …
It is shown that if K is an algebraically closed valued field with valuation ring R, then Th(K) has elimination of imaginaries if sorts are added whose elements are certain cosets in Kn of certain definable R-submodules of Kn (for all ). The proof involves the development of a theory of independence for unary types, which play the role of 1-types, followed by an analysis of germs of definable functions from unary sets to the sorts.
Abstract We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences …
Abstract We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences of each of these conditions towards the rosiness of the theory. In particular we show that the existence of an ordinal valued equivalence relation rank is a (necessary and) sufficient condition for rosiness.
In this paper we study Galois theoretic properties of a large class of fields, a class which includes all fields satisfying a universal local-global principle for the existence of rational …
In this paper we study Galois theoretic properties of a large class of fields, a class which includes all fields satisfying a universal local-global principle for the existence of rational points on varieties. As applications, we give a positive answer to a long standing conjecture which originates in an unpublished note of Roquette, and asserts that the absolute Galois group of a countable, hilbertian, PAC field is profinite free; see [F-J, Problem 24.41]. Secondly, we give new evidence for the conjecture of Shafarevich which asserts that the absolute Galois group of Qab is profinite free. Finally, one of the most interesting applications of the theory we develop here is the insight in the Galois structure of the totally EG-adic numbers.
We show that there is a theory UC of differential fields (in several commuting derivatives) of characteristic $0$, which serves as a model companion for every theory of large and …
We show that there is a theory UC of differential fields (in several commuting derivatives) of characteristic $0$, which serves as a model companion for every theory of large and differential fields extending a model complete theory of pure fields. As an application, we introduce differentially closed ordered fields, differentially closed p-adic fields and differentially closed pseudo-finite fields.
Simple theories were introduced by Shelah in 1980 in [S]. Recently, Kim and Pillay proved several important results on simple theories, which revived interest in them. Several people are now …
Simple theories were introduced by Shelah in 1980 in [S]. Recently, Kim and Pillay proved several important results on simple theories, which revived interest in them. Several people are now actively studying these theories, and finding analogs of classical results from the stable context.
We consider existentially closed fields with several orderings, valuations, and [Formula: see text]-valuations. We show that these structures are NTP 2 of finite burden, but usually have the independence property. …
We consider existentially closed fields with several orderings, valuations, and [Formula: see text]-valuations. We show that these structures are NTP 2 of finite burden, but usually have the independence property. Moreover, forking agrees with dividing, and forking can be characterized in terms of forking in ACVF, RCF, and [Formula: see text]CF.
Abstract We prove two results about generically stable types p in arbitrary theories. The first, on existence of strong germs, generalizes results from [2] on stably dominated types. The second …
Abstract We prove two results about generically stable types p in arbitrary theories. The first, on existence of strong germs, generalizes results from [2] on stably dominated types. The second is an equivalence of forking and dividing, assuming generic stability of p ( m ) for all m . We use the latter result to answer in full generality a question posed by Hasson and Onshuus: If P ( x ) ε S ( B ) is stable and does not fork over A then prestrictionA is stable. (They had solved some special cases.)
Abstract We develop a new notion of independence (ϸ-independence, read “thorn”-independence) that arises from a family of ranks suggested by Scanlon (ϸ-ranks). We prove that in a large class of …
Abstract We develop a new notion of independence (ϸ-independence, read “thorn”-independence) that arises from a family of ranks suggested by Scanlon (ϸ-ranks). We prove that in a large class of theories (including simple theories and o-minimal theories) this notion has many of the properties needed for an adequate geometric structure. We prove that ϸ-independence agrees with the usual independence notions in stable, supersimple and o-minimal theories. Furthermore, we give some evidence that the equivalence between forking and ϸ-forking in simple theories might be closely related to one of the main open conjectures in simplicity theory, the stable forking conjecture. In particular, we prove that in any simple theory where the stable forking conjecture holds, ϸ-independence and forking independence agree.
JARDEN over Q and R is a real closure of Q, this means that there exist (71 ..... o e E G(Q) such that E=RO'N...NR ae.In general, if Ol ..... …
JARDEN over Q and R is a real closure of Q, this means that there exist (71 ..... o e E G(Q) such that E=RO'N...NR ae.In general, if Ol ..... ~reEG(Q) we write Qo=R~ ~ and denote by Po~ the ordering of Qo induced by the unique ordering of the real field R a'.Thus we have a family ~o=(Q~,,Pal ..... P,,e) of e-fold ordered fields indexed by G(Q) e.The absolute Galois group G(Qo) is still generated by e involutions but it is not necessarily a pro-2-group.Indeed, Geyer proved in [5] that for almost all ~ E G(Q) e the group G(Qo) is isomorphic to the free profinite product,/)~, of e copies of Z/2Z.Here "almost all" is used in the sense of the Haar measure/~ of the compact group G(Q) ~.The models ~a of OFe appear therefore to be 'rare' among all the models ~o.We therefore concentrate in this work on the theory Te of sentences that are true in 5fo for almost all o E G(Q) e.We prove the following theorems:THEOREM A. Almost all o ~ G(Q) e have the following properties:(y) If V is an absolutely irreducible variety defined over Qo and if P~l ..... Pae can be extended to orderings of the function field of V, then the set V(Qo) of all Qorational points of V is not empty.Moreover, for every l <.i<~e and every simple point qi of V which is rational over R ~ can be Prapproximated by points in V(Qa).(6) The orderings Pal ..... Poe induce distinct topologies on Q,, THEOREM B. If ~=(E, P1 ..... Pe) and F=(F, Q1 ..... Qe) are two e-fold ordered fields that satisfy (y) and (6), if G(E)-~G(F)-~s and if (~ N ~(~ N ~ then ~ is elementarily equivalent to ~.Here ON ~=(0NE,0NP1 .... , (~NPe
Abstract We establish several results regarding dividing and forking in NTP 2 theories. We show that dividing is the same as array-dividing. Combining it with existence of strictly invariant sequences …
Abstract We establish several results regarding dividing and forking in NTP 2 theories. We show that dividing is the same as array-dividing. Combining it with existence of strictly invariant sequences we deduce that forking satisfies the chain condition over extension bases (namely, the forking ideal is S1, in Hrushovski’s terminology). Using it we prove an independence theorem over extension bases (which, in the case of simple theories, specializes to the ordinary independence theorem). As an application we show that Lascar strong type and compact strong type coincide over extension bases in an NTP 2 theory. We also define the dividing order of a theory—a generalization of Poizat’s fundamental order from stable theories—and give some equivalent characterizations under the assumption of NTP 2 . The last section is devoted to a refinement of the class of strong theories and its place in the classification hierarchy.
We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group <inline-formula content-type="math/mathml"> <mml:math …
We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we show that a finite subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper X upper X Superscript negative 1 Baseline upper X EndAbsoluteValue slash StartAbsoluteValue upper X EndAbsoluteValue"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>X</mml:mi> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">|X X ^{-1}X |/ |X|</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also find a connection with Lie groups, and use it to obtain some consequences suggestive of topological nilpotence. Model-theoretically we prove the independence theorem and the stabilizer theorem in a general first-order setting.
We prove in particular that, in a large class of dp-minimal theories including the p-adics, definable types are dense amongst non-forking types.
We prove in particular that, in a large class of dp-minimal theories including the p-adics, definable types are dense amongst non-forking types.
We prove that the theory of the p -adics {\mathbb Q}_p admits elimination of imaginaries provided we add a sort for {\mathrm GL}_n({\mathbb Q}_p)/{\mathrm GL}_n({\mathbb Z}_p) for each n . …
We prove that the theory of the p -adics {\mathbb Q}_p admits elimination of imaginaries provided we add a sort for {\mathrm GL}_n({\mathbb Q}_p)/{\mathrm GL}_n({\mathbb Z}_p) for each n . We also prove that the elimination of imaginaries is uniform in p . Using p -adic and motivic integration, we deduce the uniform rationality of certain formal zeta functions arising from definable equivalence relations. This also yields analogous results for definable equivalence relations over local fields of positive characteristic. The appendix contains an alternative proof, using cell decomposition, of the rationality (for fixed p ) of these formal zeta functions that extends to the subanalytic context. As an application, we prove rationality and uniformity results for zeta functions obtained by counting twist isomorphism classes of irreducible representations of finitely generated nilpotent groups; these are analogous to similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald for subgroup zeta functions of finitely generated nilpotent groups.
The text is based on notes from a class entitled {\em Model Theory of Berkovich Spaces}, given at the Hebrew University in the fall term of 2009, and retains the …
The text is based on notes from a class entitled {\em Model Theory of Berkovich Spaces}, given at the Hebrew University in the fall term of 2009, and retains the flavor of class notes. It includes an exposition of material from \cite{hhmcrelle}, \cite{hhm} and \cite{HL}, regarding definable types in the model completion of the theory of valued fields, and the classification of imaginary sorts. The latter is given a new proof, based on definable types rather than invariant types, and on the notion of {\em generic reparametrization}. I also try to bring out the relation to the geometry of \cite{HL} - stably dominated definable types as the model theoretic incarnation of a Berkovich point.
In this paper we solve some problems posed by Kolchin about differential algebraic groups.The main result (from which the others follow) is the embeddability of any differential algebraic group in …
In this paper we solve some problems posed by Kolchin about differential algebraic groups.The main result (from which the others follow) is the embeddability of any differential algebraic group in an algebraic group.A crucial intermediate result, and one of independent interest, is a generalisation of Weil's theorem on recovering an algebraic group from birational data, to pro-algebraic groups.
First, an example of a 2‐dependent group without a minimal subgroup of bounded index is given. Second, all infinite n ‐dependent fields are shown to be Artin‐Schreier closed. Furthermore, the …
First, an example of a 2‐dependent group without a minimal subgroup of bounded index is given. Second, all infinite n ‐dependent fields are shown to be Artin‐Schreier closed. Furthermore, the theory of any non separably closed PAC field has the IP n property for all natural numbers n and certain properties of dependent (NIP) valued fields extend to the n ‐dependent context.
By Lou van den Dries: 180 pp., £24.95 (US$39.95, LMS Members' price £18.70), isbn 0 521 59838 9 (Cambridge University Press 1998).
By Lou van den Dries: 180 pp., £24.95 (US$39.95, LMS Members' price £18.70), isbn 0 521 59838 9 (Cambridge University Press 1998).
We study definably compact definably connected groups definable in a sufficiently saturated real closed field $R$. We introduce the notion of group-generic point for $\bigvee$-definable groups and show the existence …
We study definably compact definably connected groups definable in a sufficiently saturated real closed field $R$. We introduce the notion of group-generic point for $\bigvee$-definable groups and show the existence of group-generic points for definably compact groups definable in a sufficiently saturated o-minimal expansion of a real closed field. We use this notion along with some properties of generic sets to prove that for every definably compact definably connected group $G$ definable in $R$ there are a connected $R$-algebraic group $H$, a definable injective map $\phi$ from a generic definable neighborhood of the identity of $G$ into the group $H\left(R\right)$ of $R$-points of $H$ such that $\phi$ acts as a group homomorphism inside its domain. This result is used in [2] to prove that the o-minimal universal covering group of an abelian connected definably compact group definable in a sufficiently saturated real closed field $R$ is, up to locally definable isomorphisms, an open connected locally definable subgroup of the o-minimal universal covering group of the $R$-points of some $R$-algebraic group.
We prove the elimination of field quantifiers for strongly dependent henselian fields in the Denef-Pas language. This is achieved by proving the result for a class of fields generalizing algebraically …
We prove the elimination of field quantifiers for strongly dependent henselian fields in the Denef-Pas language. This is achieved by proving the result for a class of fields generalizing algebraically maximal Kaplansky fields. We deduce that if $(K,v)$ is strongly dependent, then so is its henselization.
$\operatorname {NTP}_{2}$ is a large class of first-order theories defined by Shelah generalizing simple and NIP theories. Algebraic examples of $\operatorname {NTP}_{2}$ structures are given by ultra-products of $p$-adics and …
$\operatorname {NTP}_{2}$ is a large class of first-order theories defined by Shelah generalizing simple and NIP theories. Algebraic examples of $\operatorname {NTP}_{2}$ structures are given by ultra-products of $p$-adics and certain valued difference fields (such as a non-standard Frobenius automorphism living on an algebraically closed valued field of characteristic 0). In this note we present some results on groups and fields definable in $\operatorname {NTP}_{2}$ structures. Most importantly, we isolate a chain condition for definable normal subgroups and use it to show that any $\operatorname {NTP}_{2}$ field has only finitely many Artin-Schreier extensions. We also discuss a stronger chain condition coming from imposing bounds on burden of the theory (an appropriate analogue of weight) and show that every strongly dependent valued field is Kaplansky.
Adjoining to the language of rings the function symbols for splitting coefficients, the function symbols for relative $p$-coordinate functions, and the division predicate for a valuation, some theories of pseudo-algebraically …
Adjoining to the language of rings the function symbols for splitting coefficients, the function symbols for relative $p$-coordinate functions, and the division predicate for a valuation, some theories of pseudo-algebraically closed non-trivially valued fields admit quantifier elimination. It is also shown that in the same language the theory of pseudo-algebraically closed non-trivially valued fields of a given exponent of imperfection does not admit quantifier elimination, due to Galois theoretic obstructions.
This text is an introduction to the study of NIP (or dependent) theories. It is meant to serve two purposes. The first is to present various aspects of NIP theories …
This text is an introduction to the study of NIP (or dependent) theories. It is meant to serve two purposes. The first is to present various aspects of NIP theories and give the reader the background material needed to understand almost any paper on the subject. The second is to advertise the use of honest definitions, in particular in establishing basic results, such as the so-called shrinking of indiscernibles.
In this paper, we develop three different subjects. We study and prove alternative versions of Hrushovski’s ‘stabilizer theorem’, we generalize part of the basic theory of definably amenable NIP groups …
In this paper, we develop three different subjects. We study and prove alternative versions of Hrushovski’s ‘stabilizer theorem’, we generalize part of the basic theory of definably amenable NIP groups to $\text{NTP}_{2}$ theories, and finally, we use all this machinery to study groups with f-generic types definable in bounded pseudo real closed fields.
We define the interpolative fusion $T^*_\cup$ of a family $(T_i)_{i \in I}$ of first-order theories over a common reduct $T_\cap$, a notion that generalizes many examples of random or generic …
We define the interpolative fusion $T^*_\cup$ of a family $(T_i)_{i \in I}$ of first-order theories over a common reduct $T_\cap$, a notion that generalizes many examples of random or generic structures in the model-theoretic literature. When each $T_i$ is model-complete, $T^*_\cup$ coincides with the model companion of $T_\cup = \bigcup_{i \in I} T_i$. By obtaining sufficient conditions for the existence of $T^*_\cup$, we develop new tools to show that theories of interest have model companions.
Abstract We show basic facts about dp-minimal ordered structures. The main results are: dp-minimal groups are abelian-by-finite-exponent, in a divisible ordered dp-minimal group, any infinite set has nonempty interior, and …
Abstract We show basic facts about dp-minimal ordered structures. The main results are: dp-minimal groups are abelian-by-finite-exponent, in a divisible ordered dp-minimal group, any infinite set has nonempty interior, and any theory of pure tree is dp-minimal.
We study definably amenable NIP groups. We develop a theory of generics showing that various definitions considered previously coincide, and we study invariant measures. As applications, we characterize ergodic measures, …
We study definably amenable NIP groups. We develop a theory of generics showing that various definitions considered previously coincide, and we study invariant measures. As applications, we characterize ergodic measures, give a proof of the conjecture of Petrykowski connecting existence of bounded orbits with definable amenability in the NIP case, and prove the Ellis group conjecture of Newelski and Pillay connecting the model-theoretic connected component of an NIP group with the ideal subgroup of its Ellis enveloping semigroup.
Abstract We answer three related open questions about the model theory of valued differential fields introduced by Scanlon. We show that they eliminate imaginaries in the geometric language introduced by …
Abstract We answer three related open questions about the model theory of valued differential fields introduced by Scanlon. We show that they eliminate imaginaries in the geometric language introduced by Haskell, Hrushovski and Macpherson and that they have the invariant extension property. These two results follow from an abstract criterion for the density of definable types in enrichments of algebraically closed valued fields. Finally, we show that this theory is metastable.
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 …
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 (respectively, [Formula: see text]-henselian), then [Formula: see text] and [Formula: see text] are comparable (respectively, dependent). 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.
In this paper, we give a general criterion for elimination of imaginaries using an abstract independence relation. We also study germs of definable functions at certain well-behaved invariant types. Finally …
In this paper, we give a general criterion for elimination of imaginaries using an abstract independence relation. We also study germs of definable functions at certain well-behaved invariant types. Finally we apply these results to prove the elimination of imaginaries in bounded pseudo $p$-adically closed fields.