scholarly journals Defining integer-valued functions in rings of continuous definable functions over a topological field

2020 ◽  
Vol 20 (03) ◽  
pp. 2050014
Author(s):  
Luck Darnière ◽  
Marcus Tressl
Keyword(s):  
Definable Set ◽  
Ordered Field ◽  
Dimension Formula ◽  
Valued Field ◽  
Pure Dimension ◽  
Isolated Points ◽  
Topological Field ◽  
Structure Formula ◽  
Definable Functions ◽  
Ring Of Functions

Let [Formula: see text] be an expansion of either an ordered field [Formula: see text], or a valued field [Formula: see text]. Given a definable set [Formula: see text] let [Formula: see text] be the ring of continuous definable functions from [Formula: see text] to [Formula: see text]. Under very mild assumptions on the geometry of [Formula: see text] and on the structure [Formula: see text], in particular when [Formula: see text] is [Formula: see text]-minimal or [Formula: see text]-minimal, or an expansion of a local field, we prove that the ring of integers [Formula: see text] is interpretable in [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] is definably connected of pure dimension [Formula: see text], then [Formula: see text] defines the subring [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] has no isolated points, then there is a discrete ring [Formula: see text] contained in [Formula: see text] and naturally isomorphic to [Formula: see text], such that the ring of functions [Formula: see text] which take values in [Formula: see text] is definable in [Formula: see text].

Preliminaries

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.

GANZSTELLENSÄTZE IN THEORIES OF VALUED FIELDS

2008 ◽  
Vol 08 (01) ◽  
pp. 1-22 ◽  
Author(s):  
DEIRDRE HASKELL ◽  
YOAV YAFFE
Keyword(s):  
Valuation Ring ◽  
Uniform Method ◽  
Algebraic Characterization ◽  
Valued Fields ◽  
Definable Set ◽  
Valued Field ◽  
Complete Theories ◽  
The Given ◽  
Relevant Class

The purpose of this paper is to study an analogue of Hilbert's seventeenth problem for functions over a valued field which are integral definite on some definable set; that is, that map the given set into the valuation ring. We use model theory to exhibit a uniform method, on various theories of valued fields, for deriving an algebraic characterization of such functions. As part of this method we refine the concept of a function being integral at a point, and make it dependent on the relevant class of valued fields. We apply our framework to algebraically closed valued fields, model complete theories of difference and differential valued fields, and real closed valued fields.

The main theorem

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Projective Variety ◽  
Unit Vector ◽  
Definable Set ◽  
Valued Field ◽  
Definable Subset ◽  
Deformation Retraction ◽  
Relative Curve

This chapter introduces the main theorem, which states: Let V be a quasi-projective variety over a valued field F and let X be a definable subset of V x Γ‎superscript Script Small l subscript infinity over some base set V ⊂ VF ∪ Γ‎, with F = VF(A). Then there exists an A-definable deformation retraction h : I × unit vector X → unit vector X with image an iso-definable subset definably homeomorphic to a definable subset of Γ‎superscript w subscript Infinity, for some finite A-definable set w. The chapter presents several preliminary reductions to essentially reduce to a curve fibration. It then constructs a relative curve homotopy and a liftable base homotopy, along with a purely combinatorial homotopy in the Γ‎-world. It also constructs the homotopy retraction by concatenating the previous three homotopies together with an inflation homotopy. Finally, it describes a uniform version of the main theorem with respect to parameters.

Curves

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Algebraic Curve ◽  
Unit Vector ◽  
Definable Set ◽  
Valued Field ◽  
Branching Points

This chapter proves the iso-definability of unit vector C when C is a curve using Riemann-Roch. Recall that a pro-definable set is called iso-definable if it is isomorphic, as a pro-definable set, to a definable set. If C is an algebraic curve defined over a valued field F, then unit vector C is an iso-definable set. The topology on unit vector C is definably generated, that is, generated by a definable family of (iso)-definable subsets. In other words, there is a definable family giving a pre-basis of the topology. The chapter explains how definable types on C correspond to germs of paths on unit vector C. It also constructs the retraction on skeleta for curves. A key result is the finiteness of forward-branching points.

Γ‎-internal spaces

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Algebraic Variety ◽  
Topological Structure ◽  
Unit Vector ◽  
Definable Set ◽  
Valued Field ◽  
Definable Subset

This chapter describes the topological structure of Γ‎-internal spaces. Let V be an algebraic variety over a valued field. An iso-definable subset X of unit vector V is said to be Γ‎-internal if it is in pro-definable bijection with a definable set which is Γ‎-internal. A number of delicate issues arise here. A pro-definable subset X of unit vector V is Γ‎-parameterized if there exists a definable subset Y of Γ‎ⁿ, for some n, and a pro-definable map g : Y → unit vector V with image X. The chapter presents an example showing that there exists Γ‎-parameterized subsets of unit vector V which are not iso-definable, whence not Γ‎-internal. It also presents the main results about the topological structure of Γ‎-internal spaces.

A note on valuation definable expansions of fields

1998 ◽  
Vol 63 (2) ◽  
pp. 739-743 ◽  
Author(s):  
Deirdre Haskell ◽  
Dugald Macpherson
Keyword(s):  
Valuation Ring ◽  
Quantifier Elimination ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Closed Fields ◽  
Ordered Field ◽  
Valued Field ◽  
Closed Fields ◽  
Binary Predicate ◽  
Algebraically Closed Fields

In this note, we consider models of the theories of valued algebraically closed fields and convexly valued real closed fields, their reducts to the pure field or ordered field language respectively, and expansions of these by predicates which are definable in the valued field. We show that, in terms of definability, there is no structure properly between the pure (ordered) field and the valued field. Our results are analogous to several other definability results for reducts of algebraically closed and real closed fields; see [9], [10], [11] and [12]. Throughout this paper, definable will mean definable with parameters.Theorem A. Let ℱ = (F, +, ×, V) be a valued, algebraically closed field, where V denotes the valuation ring. Let A be a subset ofFndefinable in ℱv. Then either A is definable in ℱ = (F, +, ×) or V is definable in.Theorem B. Let ℛv = (R, <, +, ×, V) be a convexly valued real closed field, where V denotes the valuation ring. Let Abe a subset ofRndefinable in ℛv. Then either A is definable in ℛ = (R, <, +, ×) or V is definable in.The proofs of Theorems A and B are quite similar. Both ℱv and ℛv admit quantifier elimination if we adjoin a definable binary predicate Div (interpreted by Div(x, y) if and only if v(x) ≤ v(y)). This is proved in [14] (extending [13]) in the algebraically closed case, and in [4] in the real closed case. We show by direct combinatorial arguments that if the valuation is not definable then the expanded structure is strongly minimal or o-minimal respectively. Then we call on known results about strongly minimal and o-minimal fields to show that the expansion is not proper.

scholarly journals INTERPRETABLE SETS IN DENSE O-MINIMAL STRUCTURES

2018 ◽  
Vol 83 (04) ◽  
pp. 1477-1500
Author(s):  
WILL JOHNSON
Keyword(s):  
Negative Answer ◽  
Connected Components ◽  
Lower Dimension ◽  
Definable Set ◽  
Definable Subset ◽  
Minimal Structure ◽  
Piecewise Continuous ◽  
Definable Functions

AbstractWe give an example of a dense o-minimal structure in which there is a definable quotient that cannot be eliminated, even after naming parameters. Equivalently, there is an interpretable set which cannot be put in parametrically definable bijection with any definable set. This gives a negative answer to a question of Eleftheriou, Peterzil, and Ramakrishnan. Additionally, we show that interpretable sets in dense o-minimal structures admit definable topologies which are “tame” in several ways: (a) they are Hausdorff, (b) every point has a neighborhood which is definably homeomorphic to a definable set, (c) definable functions are piecewise continuous, (d) definable subsets have finitely many definably connected components, and (e) the frontier of a definable subset has lower dimension than the subset itself.

Specializations and ACV2F

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Definable Set ◽  
The Family ◽  
Definable Functions

This chapter introduces the theory ACV²F of iterated places and describes some algebraic criteria for v- and g-continuity. It considers the theory ACV²F of triples (K₂,K₁,K₀) of fields with surjective, non-injective places rᵢⱼ : Kᵢ → Kⱼ for i > j, r₂₀ = r₁₀ ° r₂₁, such that K₂ is algebraically closed. The chapter shows that the family of g-open sets is definable in definable families. It also presents some applications of the continuity criteria and concludes by proving that for each definable set of definable functions V → Wsuperscript Number Sign the subset of those that are g-continuous is definable.

Stable embeddedness in algebraically closed valued fields

2006 ◽  
Vol 71 (3) ◽  
pp. 831-862 ◽  
Author(s):  
E. Hrushovski ◽  
A. Tatarsky
Keyword(s):  
Valuation Ring ◽  
Algebraic Closure ◽  
Valued Fields ◽  
Definable Set ◽  
Valued Field ◽  
Definable Sets

AbstractWe give some general criteria for the stable embeddedness of a definable set. We use these criteria to establish the stable embeddedness in algebraically closed valued fields of two definable sets: The set of balls of a given radius r < 1 contained in the valuation ring and the set of balls of a given multiplicative radius r < 1. We also show that in an algebraically closed valued field a 0-definable set is stably embedded if and only if its algebraic closure is stably embedded.

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.

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