A version of p-adic minimality

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.

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CELL DECOMPOSITION AND CLASSIFICATION OF DEFINABLE SETS INp-OPTIMAL FIELDS

Journal of Symbolic Logic ◽

10.1017/jsl.2015.79 ◽

2017 ◽

Vol 82 (1) ◽

pp. 120-136 ◽

Cited By ~ 4

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.

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Presburger sets and p-minimal fields

Journal of Symbolic Logic ◽

10.2178/jsl/1045861509 ◽

2003 ◽

Vol 68 (1) ◽

pp. 153-162 ◽

Cited By ~ 15

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.

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b-MINIMALITY

Journal of Mathematical Logic ◽

10.1142/s0219061307000664 ◽

2007 ◽

Vol 07 (02) ◽

pp. 195-227 ◽

Cited By ~ 13

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].

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Preliminaries

10.23943/princeton/9780691161686.003.0002 ◽

2017 ◽

Author(s):

Ehud Hrushovski ◽

François Loeser

Keyword(s):

Valued Fields ◽

Definable Set ◽

Valued Field ◽

Galois Coverings ◽

Definable Type ◽

Definable Sets ◽

Background Material

This chapter provides some background material on definable sets, definable types, orthogonality to a definable set, and stable domination, especially in the valued field context. It considers more specifically these concepts in the framework of the theory ACVF of algebraically closed valued fields and describes the definable types concentrating on a stable definable V as an ind-definable set. It also proves a key result that demonstrates definable types as integrals of stably dominated types along some definable type on the value group sort. Finally, it discusses the notion of pseudo-Galois coverings. Every nonempty definable set over an algebraically closed substructure of a model of ACVF extends to a definable type.

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Shuffling of Linear Orders

Canadian Mathematical Bulletin ◽

10.4153/cmb-1995-032-1 ◽

1995 ◽

Vol 38 (2) ◽

pp. 223-229

Author(s):

John Lindsay Orr

Keyword(s):

Ordered Set ◽

Linear Orders ◽

Real Numbers ◽

The Real ◽

Linearly Ordered Set

AbstractA linearly ordered set A is said to shuffle into another linearly ordered set B if there is an order preserving surjection A —> B such that the preimage of each member of a cofinite subset of B has an arbitrary pre-defined finite cardinality. We show that every countable linearly ordered set shuffles into itself. This leads to consequences on transformations of subsets of the real numbers by order preserving maps.

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A question of van den Dries and a theorem of Lipshitz and Robinson; Not everything is standard

Journal of Symbolic Logic ◽

10.2178/jsl/1174668387 ◽

2007 ◽

Vol 72 (1) ◽

pp. 119-122 ◽

Cited By ~ 4

Author(s):

Ehud Hrushovski ◽

Ya'acov Peterzil

Keyword(s):

Real Numbers ◽

The Real ◽

First Order ◽

Real Closed Fields ◽

Minimal Structure ◽

Closed Fields ◽

New Construction ◽

The Relationship

AbstractWe use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.

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On expressible sets and p-adic numbers

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091509000091 ◽

2011 ◽

Vol 54 (2) ◽

pp. 411-422

Author(s):

Jaroslav Hančl ◽

Radhakrishnan Nair ◽

Simona Pulcerova ◽

Jan Šustek

Keyword(s):

Haar Measure ◽

Real Numbers ◽

The Real ◽

Expressible Set

AbstractContinuing earlier studies over the real numbers, we study the expressible set of a sequence A = (an)n≥1 of p-adic numbers, which we define to be the set EpA = {∑n≥1ancn: cn ∈ ℕ}. We show that in certain circumstances we can calculate the Haar measure of EpA exactly. It turns out that our results extend to sequences of matrices with p-adic entries, so this is the setting in which we work.

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