scholarly journals CELL DECOMPOSITION AND CLASSIFICATION OF DEFINABLE SETS INp-OPTIMAL FIELDS

2017 ◽  
Vol 82 (1) ◽  
pp. 120-136 ◽  
Author(s):  
LUCK DARNIÈRE ◽  
IMMANUEL HALPUCZOK
Keyword(s):  
Decomposition Theorem ◽  
Cell Decomposition ◽  
Extreme Value ◽  
Property A ◽  
Definable Subset ◽  
Skolem Functions ◽  
Definable Sets ◽  
A Cell ◽  
Preparation Theorem ◽  
Definable Functions

AbstractWe prove that forp-optimal fields (a very large subclass ofp-minimal fields containing all the known examples) a cell decomposition theorem follows from methods going back to Denef’s paper [7]. We derive from it the existence of definable Skolem functions and strongp-minimality. Then we turn to stronglyp-minimal fields satisfying the Extreme Value Property—a property which in particular holds in fields which are elementarily equivalent to ap-adic one. For such fieldsK, we prove that every definable subset ofK×Kdwhose fibers overKare inverse images by the valuation of subsets of the value group is semialgebraic. Combining the two we get a preparation theorem for definable functions onp-optimal fields satisfying the Extreme Value Property, from which it follows that infinite sets definable over such fields are in definable bijection iff they have the same dimension.

scholarly journals Presburger sets and p-minimal fields

2003 ◽  
Vol 68 (1) ◽  
pp. 153-162 ◽  
Author(s):  
Raf Cluckers
Keyword(s):  
Decomposition Theorem ◽  
Cell Decomposition ◽  
Dimension Theory ◽  
Definable Sets ◽  
A Cell ◽  
Tight Connection ◽  
Full Classification

AbstractWe prove a cell decomposition theorem for Presburger sets and introduce a dimension theory for Z-groups with the Presburger structure. Using the cell decomposition theorem we obtain a full classification of Presburger sets up to definable bijection. We also exhibit a tight connection between the definable sets in an arbitrary p-minimal field and Presburger sets in its value group. We give a negative result about expansions of Presburger structures and prove uniform elimination of imaginaries for Presburger structures within the Presburger language.

A version of p-adic minimality

2012 ◽  
Vol 77 (2) ◽  
pp. 621-630 ◽  
Author(s):  
Raf Cluckers ◽  
Eva Leenknegt
Keyword(s):  
Positive Characteristic ◽  
Quantifier Elimination ◽  
Cell Decomposition ◽  
Real Numbers ◽  
Valued Fields ◽  
The Real ◽  
Definable Sets ◽  
A Cell ◽  
Definable Functions

AbstractWe introduce a very weak language 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 are trivial functions. We also give a definitional expansion of in which K has quantifier elimination, and we obtain a cell decomposition result for -definable sets.Our language 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.

Sheaves of continuous definable functions

1988 ◽  
Vol 53 (4) ◽  
pp. 1165-1169 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Real Closed Field ◽  
Spectral Space ◽  
Real Spectrum ◽  
Closed Field ◽  
Elementary Extension ◽  
Definable Set ◽  
Definable Subset ◽  
Skolem Functions ◽  
Minimal Structure ◽  
Definable Functions

Let M be an o-minimal structure or a p-adically closed field. Let be the space of complete n-types over M equipped with the following topology: The basic open sets of are of the form Ũ = {p ∈ Sn (M): U ∈ p} for U an open definable subset of Mn. is a spectral space. (For M = K a real closed field, is precisely the real spectrum of K[X1, …, Xn]; see [CR].) We will equip with a sheaf of LM-structures (where LM is a suitable language). Again for M a real closed field this corresponds to the structure sheaf on (see [S]). Our main point is that when Th(M) has definable Skolem functions, then if p ∈ , it follows that M(p), the definable ultrapower of M at p, can be factored through Mp, the stalk at p with respect to the above sheaf. This depends on the observation that if M ≺ N, a ∈ Nn and f is an M-definable (partial) function defined at a, then there is an open M-definable set U ⊂ Nn with a ∈ U, and a continuous M-definable function g:U → N such that g(a) = f(a).In the case that M is an o-minimal expansion of a real closed field (or M is a p-adically closed field), it turns out that M(p) can be recovered as the unique quotient of Mp which is an elementary extension of M.

Analytic cell decomposition and the closure of p-adic semianalytic sets

1997 ◽  
Vol 62 (1) ◽  
pp. 285-303
Author(s):  
Nianzheng Liu
Keyword(s):  
Analytic Function ◽  
Quantifier Elimination ◽  
Decomposition Theorem ◽  
Cell Decomposition ◽  
Closure Property ◽  
Basic Properties ◽  
Analogous Property ◽  
Subanalytic Sets ◽  
Weierstrass Preparation ◽  
Preparation Theorem

The p-adic semianalytic sets are defined, locally, as boolean combinations of sets of the form over the p-adic fields ℚp, where f is an analytic function. A well-know example due to Osgood showed the projection of a semianalytic set need not be a semianalytic set. We call those sets that are, locally, the projections of p-adic semianalytic sets p-adic subanalytic sets. The theory of p-adic subanalytic sets was presented by Denef and Van den Dries in [5]. The basic tools are the quantifier elimination techniques together with the ultrametric Weierstrass Preparation Theorem. Simultaneously with their developments of the p-adic subanalytic sets, they established some basic properties of p-adic semianalytic sets.In this paper, we prove that the closure of any p-adic semianalytic set is also a semianalytic set. The analogous property for real semianalytic sets was proved in [12] and that for rigid semianalytic sets, informed by the referee, has been proved recently by a quite different method in [14] (cf. [9]). The keys to the proof are a separation lemma (Lemma 2) and an analytic cell decomposition theorem (Theorem 2) which is an analytic version of Denef's cell decomposition theorem (see [3, 4]; A total different form of anayltic cells appeared in [13]). The analytic cell decomposition theorem allows us to partition certain kinds of basic subsets into analytic cells that possess the closure property (see §1 for the definition).

scholarly journals o-MINIMAL COHOMOLOGY: FINITENESS AND INVARIANCE RESULTS

2009 ◽  
Vol 09 (02) ◽  
pp. 167-182 ◽  
Author(s):  
ALESSANDRO BERARDUCCI ◽  
ANTONGIULIO FORNASIERO
Keyword(s):  
Betti Numbers ◽  
Decomposition Theorem ◽  
Cell Decomposition ◽  
Compact Set ◽  
Cohomology Groups ◽  
Finitely Generated ◽  
Minimal Cell ◽  
General Validity ◽  
Cw Complex ◽  
Definable Sets

The topology of definable sets in an o-minimal expansion of a group is not fully understood due to the lack of a triangulation theorem. Despite the general validity of the cell decomposition theorem, we do not know whether any definably compact set is a definable CW-complex. Moreover the closure of an o-minimal cell can have arbitrarily high Betti numbers. Nevertheless we prove that the cohomology groups of a definably compact set over an o-minimal expansion of a group are finitely generated and invariant under elementary extensions and expansions of the language.

b-MINIMALITY

2007 ◽  
Vol 07 (02) ◽  
pp. 195-227 ◽  
Author(s):  
RAF CLUCKERS ◽  
FRANÇOIS LOESER
Keyword(s):  
Characteristic Zero ◽  
Decomposition Theorem ◽  
Cell Decomposition ◽  
Dimension Theory ◽  
Minimal Cell ◽  
Euler Characteristics ◽  
Valued Fields ◽  
Henselian Valued Fields ◽  
A Cell ◽  
Abstract Notion

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].

On a cell decomposition of the Hilbert scheme of points in the plane

1988 ◽  
Vol 91 (2) ◽  
pp. 365-370 ◽  
Author(s):  
Geir Ellingsrud ◽  
Stein Arild Str�mme
Keyword(s):  
Hilbert Scheme ◽  
Cell Decomposition ◽  
Hilbert Scheme Of Points ◽  
A Cell

Algebraic theories with definable Skolem functions

1984 ◽  
Vol 49 (2) ◽  
pp. 625-629 ◽  
Author(s):  
Lou van den Dries
Keyword(s):  
Nonempty Subset ◽  
Peano Arithmetic ◽  
Real Closed Field ◽  
Closed Field ◽  
First Order ◽  
Definable Subset ◽  
Skolem Functions ◽  
Algebraic Theories ◽  
Closed Fields ◽  
Definitional Extension

(1.1) A well-known example of a theory with built-in Skolem functions is (first-order) Peano arithmetic (or rather a certain definitional extension of it). See [C-K, pp. 143, 162] for the notion of a theory with built-in Skolem functions, and for a treatment of the example just mentioned. This property of Peano arithmetic obviously comes from the fact that in each nonempty definable subset of a model we can definably choose an element, namely, its least member.(1.2) Consider now a real closed field R and a nonempty subset D of R which is definable (with parameters) in R. Again we can definably choose an element of D, as follows: D is a union of finitely many singletons and intervals (a, b) where – ∞ ≤ a < b ≤ + ∞; if D has a least element we choose that element; if not, D contains an interval (a, b) for which a ∈ R ∪ { − ∞}is minimal; for this a we choose b ∈ R ∪ {∞} maximal such that (a, b) ⊂ D. Four cases have to be distinguished:(i) a = − ∞ and b = + ∞; then we choose 0;(ii) a = − ∞ and b ∈ R; then we choose b − 1;(iii) a ∈ R and b ∈ = + ∞; then we choose a + 1;(iv) a ∈ R and b ∈ R; then we choose the midpoint (a + b)/2.It follows as in the case of Peano arithmetic that the theory RCF of real closed fields has a definitional extension with built-in Skolem functions.

scholarly journals Definable Transformation to Normal Crossings over Henselian Fields with Separated Analytic Structure

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 934
Author(s):  
Krzysztof Jan Nowak
Keyword(s):  
Quantifier Elimination ◽  
General Setting ◽  
Resolution Of Singularities ◽  
Analytic Structure ◽  
Analytic Geometry ◽  
Skolem Functions ◽  
Algebraic Properties ◽  
Rank One ◽  
Rigid Analytic Geometry ◽  
Definable Functions

We are concerned with rigid analytic geometry in the general setting of Henselian fields K with separated analytic structure, whose theory was developed by Cluckers–Lipshitz–Robinson. It unifies earlier work and approaches of numerous mathematicians. Separated analytic structures admit reasonable relative quantifier elimination in a suitable analytic language. However, the rings of global analytic functions with two kinds of variables seem not to have good algebraic properties such as Noetherianity or excellence. Therefore, the usual global resolution of singularities from rigid analytic geometry is no longer at our disposal. Our main purpose is to give a definable version of the canonical desingularization algorithm (the hypersurface case) due to Bierstone–Milman so that both of these powerful tools are available in the realm of non-Archimedean analytic geometry at the same time. It will be carried out within a category of definable, strong analytic manifolds and maps, which is more flexible than that of affinoid varieties and maps. Strong analytic objects are those definable ones that remain analytic over all fields elementarily equivalent to K. This condition may be regarded as a kind of symmetry imposed on ordinary analytic objects. The strong analytic category makes it possible to apply a model-theoretic compactness argument in the absence of the ordinary topological compactness. On the other hand, our closedness theorem enables application of resolution of singularities to topological problems involving the topology induced by valuation. Eventually, these three results will be applied to such issues as the existence of definable retractions or extending continuous definable functions. The established results remain valid for strictly convergent analytic structures, whose classical examples are complete, rank one valued fields with the Tate algebras of strictly convergent power series. The earlier techniques and approaches to the purely topological versions of those issues cannot be carried over to the definable settings because, among others, non-Archimedean geometry over non-locally compact fields suffers from lack of definable Skolem functions.

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