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.

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.

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.

scholarly journals NOTES ON EXTREMAL AND TAME VALUED FIELDS

2016 ◽  
Vol 81 (2) ◽  
pp. 400-416
Author(s):  
SYLVY ANSCOMBE ◽  
FRANZ-VIKTOR KUHLMANN
Keyword(s):  
Approximation Property ◽  
Elementary Theory ◽  
Residue Field ◽  
Axiom System ◽  
Complete Characterization ◽  
Valued Fields ◽  
Mixed Characteristic ◽  
Valued Field ◽  
Residue Fields

AbstractWe extend the characterization of extremal valued fields given in [2] to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that are extremal. The key to the proof is a model theoretic result about tame valued fields in mixed characteristic. Further, we prove that in an extremal valued field of finitep-degree, the images of all additive polynomials have the optimal approximation property. This fact can be used to improve the axiom system that is suggested in [8] for the elementary theory of Laurent series fields over finite fields. Finally we give examples that demonstrate the problems we are facing when we try to characterize the extremal valued fields with imperfect residue fields. To this end, we describe several ways of constructing extremal valued fields; in particular, we show that in every ℵ1saturated valued field the valuation is a composition of extremal valuations of rank 1.

scholarly journals ON THE MAIN INVARIANT OF ELEMENTS ALGEBRAIC OVER A HENSELIAN VALUED FIELD

2002 ◽  
Vol 45 (1) ◽  
pp. 219-227 ◽  
Author(s):  
Kamal Aghigh ◽  
Sudesh K. Khanduja
Keyword(s):  
Upper Bound ◽  
Algebraic Closure ◽  
Subject Classification ◽  
Unique Extension ◽  
Valued Fields ◽  
Valued Field ◽  
Henselian Valued Fields ◽  
Alpha Beta ◽  
Henselian Valuation

AbstractLet $v$ be a henselian valuation of a field $K$ with value group $G$, let $\bar{v}$ be the (unique) extension of $v$ to a fixed algebraic closure $\bar{K}$ of $K$ and let $(\tilde{K},\tilde{v})$ be a completion of $(K,v)$. For $\alpha\in\bar{K}\setminus K$, let $M(\alpha,K)$ denote the set $\{\bar{v}(\alpha-\beta):\beta\in\bar{K},\ [K(\beta):K] \lt [K(\alpha):K]\}$. It is known that $M(\alpha,K)$ has an upper bound in $\bar{G}$ if and only if $[K(\alpha):K]=[\tilde{K}(\alpha):\tilde{K}]$, and that the supremum of $M(\alpha,K)$, which is denoted by $\delta_{K}(\alpha)$ (usually referred to as the main invariant of $\alpha$), satisfies a principle similar to the Krasner principle. Moreover, each complete discrete rank 1 valued field $(K,v)$ has the property that $\delta_{K}(\alpha)\in M(\alpha,K)$ for every $\alpha\in\bar{K}\setminus K$. In this paper the authors give a characterization of all those henselian valued fields $(K,v)$ which have the property mentioned above.AMS 2000 Mathematics subject classification: Primary 12J10; 12J25; 13A18

Grothendieck Rings of ℤ-Valued Fields

2001 ◽  
Vol 7 (2) ◽  
pp. 262-269 ◽  
Author(s):  
Raf Cluckers ◽  
Deirdre Haskell
Keyword(s):  
Valuation Ring ◽  
Residue Field ◽  
Local Fields ◽  
Grothendieck Ring ◽  
Valued Fields ◽  
Logical Language ◽  
Valued Field ◽  
Grothendieck Rings

AbstractWe prove the triviality of the Grothendieck ring of a ℤ-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the plane K2 to itself minus a point. When we specialize to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point.

scholarly journals The Ohm–Rush content function

2015 ◽  
Vol 15 (01) ◽  
pp. 1650009 ◽  
Author(s):  
Neil Epstein ◽  
Jay Shapiro
Keyword(s):  
Power Series ◽  
Valuation Ring ◽  
Finite Dimension ◽  
Ring Extension ◽  
Arbitrary Ring ◽  
Prufer Domains ◽  
Prüfer Domains ◽  
The Given ◽  
The Way

The content of a polynomial over a ring R is a well-understood notion. Ohm and Rush generalized this concept of a content map to an arbitrary ring extension of R, although it can behave quite badly. We examine five properties an algebra may have with respect to this function — content algebra, weak content algebra, semicontent algebra (our own definition), Gaussian algebra, and Ohm–Rush algebra. We show that the Gaussian, weak content, and semicontent algebra properties are all transitive. However, transitivity is unknown for the content algebra property. We then compare the Ohm–Rush notion with the more usual notion of content in the power series context. We show that many of the given properties coincide for the power series extension map over a valuation ring of finite dimension, and that they are equivalent to the value group being order-isomorphic to the integers or the reals. Along the way, we give a new characterization of Prüfer domains.

scholarly journals Characterization of Dissipative Structures for First-Order Symmetric Hyperbolic System with General Relaxation

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 728
Author(s):  
Yasunori Maekawa ◽  
Yoshihiro Ueda
Keyword(s):  
Hyperbolic System ◽  
Dissipative Structure ◽  
Dissipative Structures ◽  
Algebraic Characterization ◽  
Symmetric Hyperbolic System ◽  
Full Generality ◽  
Order Linear ◽  
First Order

In this paper, we study the dissipative structure of first-order linear symmetric hyperbolic system with general relaxation and provide the algebraic characterization for the uniform dissipativity up to order 1. Our result extends the classical Shizuta–Kawashima condition for the case of symmetric relaxation, with a full generality and optimality.

An Axiomatic Characterization of a Class of Petri Nets

1991 ◽  
Vol 14 (4) ◽  
pp. 477-491
Author(s):  
Waldemar Korczynski
Keyword(s):  
Petri Nets ◽  
Petri Net ◽  
Axiomatic Characterization ◽  
Algebraic Characterization

In this paper an algebraic characterization of a class of Petri nets is given. The nets are characterized by a kind of algebras, which can be considered as a generalization of the concept of the case graph of a (marked) Petri net.

Algebraic characterization of bridged polycyclic compounds

1981 ◽  
Vol 19 (5) ◽  
pp. 929-955 ◽  
Author(s):  
Ov. Mekenyan ◽  
D. Bonchev ◽  
N. Trinajsti?
Keyword(s):  
Algebraic Characterization ◽  
Polycyclic Compounds
Sign in / Sign up

Export Citation Format

Share Document