Quantifier elimination for the theory of algebraically closed valued fields with analytic structure

2007 ◽  
Vol 53 (3) ◽  
pp. 237-246 ◽  
Author(s):  
Yalın F. Çelikler
Keyword(s):  
Quantifier Elimination ◽  
Analytic Structure ◽  
Valued Fields

scholarly journals Real closed valued fields with analytic structure

2019 ◽  
Vol 63 (1) ◽  
pp. 249-261
Author(s):  
Pablo Cubides Kovacsics ◽  
Deirdre Haskell
Keyword(s):  
Short Proof ◽  
Quantifier Elimination ◽  
Analytic Structure ◽  
Valued Fields

AbstractWe show quantifier elimination theorems for real closed valued fields with separated analytic structure and overconvergent analytic structure in their natural one-sorted languages and deduce that such structures are weakly o-minimal. We also provide a short proof that algebraically closed valued fields with separated analytic structure (in any rank) are C-minimal.

scholarly journals SOME PROPERTIES OF ANALYTIC DIFFERENCE VALUED FIELDS

2015 ◽  
Vol 16 (3) ◽  
pp. 447-499 ◽  
Author(s):  
Silvain Rideau
Keyword(s):  
Algebraic Closure ◽  
Quantifier Elimination ◽  
Order Theory ◽  
Analytic Structure ◽  
Independence Property ◽  
Valued Fields ◽  
First Order ◽  
Witt Vectors ◽  
First Order Theory

We prove field quantifier elimination for valued fields endowed with both an analytic structure that is $\unicode[STIX]{x1D70E}$-Henselian and an automorphism that is $\unicode[STIX]{x1D70E}$-Henselian. From this result we can deduce various Ax–Kochen–Eršov type results with respect to completeness and the independence property. The main example we are interested in is the field of Witt vectors on the algebraic closure of $\mathbb{F}_{p}$ endowed with its natural analytic structure and the lifting of the Frobenius. It turns out we can give a (reasonable) axiomatization of its first-order theory and that this theory does not have the independence property.

scholarly journals Quantifier elimination for valued fields equipped with an automorphism

2015 ◽  
Vol 21 (4) ◽  
pp. 1177-1201 ◽  
Author(s):  
Salih Durhan ◽  
Gönenç Onay
Keyword(s):  
Quantifier Elimination ◽  
Valued Fields

scholarly journals Unexpected imaginaries in valued fields with analytic structure

2013 ◽  
Vol 78 (2) ◽  
pp. 523-542 ◽  
Author(s):  
Deirdre Haskell ◽  
Ehud Hrushovski ◽  
Dugald Macpherson
Keyword(s):  
Analytic Structure ◽  
Valued Fields ◽  
Algebraic Setting

AbstractWe give an example of an imaginary defined in certain valued fields with analytic structure which cannot be coded in the ‘geometric’ sorts which suffice to code all imaginaries in the corresponding algebraic setting.

scholarly journals SEPARABLY CLOSED VALUED FIELDS: QUANTIFIER ELIMINATION

2016 ◽  
Vol 81 (3) ◽  
pp. 887-900 ◽  
Author(s):  
JIZHAN HONG
Keyword(s):  
Quantifier Elimination ◽  
Valued Fields ◽  
Image Position

AbstractIt is proved in this article that the theory of separably closed nontrivially valued fields of characteristic p > 0 and imperfection degree e > 0 (e ≤ ∞) has quantifier elimination in the language ${{\cal L}_{p,{\rm{div}}}} = \{ + , - , \times ,0,1\} \cup {\{ {\lambda _{n,j}}(x;{y_1}, \ldots ,{y_n})\} _{0 \le n < \omega ,0 \le j < {p^n}}} \cup \{ |\}$; in particular, when e is finite, the corresponding theory has quantifier elimination in the language ${\cal L} = \{ + , - , \times ,0,1\} \cup \{ {b_1}, \ldots ,{b_e}\} \cup {\{ {\lambda _{e,j}}(x;{b_1}, \ldots ,{b_e})\} _{0 \le j < {p^e}}} \cup \{ |\}$.

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.

A note on definable Skolem functions

1988 ◽  
Vol 53 (3) ◽  
pp. 905-911 ◽  
Author(s):  
Philip Scowcroft
Keyword(s):  
Elementary Theory ◽  
Quantifier Elimination ◽  
First Person ◽  
Valued Fields ◽  
Skolem Functions ◽  
Image Position ◽  
Primitive Recursive ◽  
Special Case ◽  
Definable Function

This note arose out of my efforts to understand results of van den Dries, Denef, and Weispfenning on definable Skolem functions in the elementary theory of Qp. The first person to prove their existence was van den Dries, who devised and applied a model-theoretic criterion for theories, admitting elimination of quantifiers, which also admit definable Skolem functions [3]. The proof, though elegant, does not describe how one defines the Skolem functions. In the particular case of Qp, Denef found an ingenious, easily described method for writing out the definitions [2, pp. 14–15]. Unfortunately, his argument directly applies only in the following special case: ifand there is a fixed m ≥ 1 such thatfor all , then can be given as a definable function of . While this special case includes many of interest, van den Dries' theorem seems more general. Weispfenning suggested how his results on primitive-recursive quantifier elimination could produce algorithms yielding definitions of Skolem functions in the specific theories van den Dries considered [10, pp. 470–471]. Though these algorithms provide a more concrete version of van den Dries' theorem, and do not suffer the lack of generality of Denef's result, Weispfenning's argument is extremely subtle and applies only to certain theories of valued fields.

On the angular component map modulo P

1990 ◽  
Vol 55 (3) ◽  
pp. 1125-1129 ◽  
Author(s):  
Johan Pas
Keyword(s):  
Natural Language ◽  
Residue Field ◽  
Quantifier Elimination ◽  
Function Symbol ◽  
Valued Fields ◽  
Valued Field ◽  
Angular Component ◽  
Henselian Valued Fields ◽  
Almost All ◽  
Component Map

In [10] we introduced a new first order language for valued fields. This language has three sorts of variables, namely variables for elements of the valued field, variables for elements of the residue field and variables for elements of the value group. contains symbols for the standard field, residue field, and value group operations and a function symbol for the valuation. Essential in our language is a function symbol for an angular component map modulo P, which is a map from the field to the residue field (see Definition 1.2).For this language we proved a quantifier elimination theorem for Henselian valued fields of equicharacteristic zero which possess such an angular component map modulo P [10, Theorem 4.1]. In the first section of this paper we give some partial results on the existence of an angular component map modulo P on an arbitrary valued field.By applying the above quantifier elimination theorem to ultraproducts ΠQp/D, we obtained a quantifier elimination, in the language , for the p-adic field Qp; and this elimination is uniform for almost all primes p [10, Corollary 4.3]. In §2 we prove that our language is essentially stronger than the natural language for p-adic fields in the sense that the angular component map modulo P cannot be defined, uniformly for almost all p, in terms of the natural language for p-adic fields.

Quantifier elimination in valued Ore modules

2010 ◽  
Vol 75 (3) ◽  
pp. 1007-1034 ◽  
Author(s):  
Luc Bélair ◽  
Françoise Point
Keyword(s):  
Residue Field ◽  
Quantifier Elimination ◽  
Difference Operators ◽  
Independence Property ◽  
Valued Fields ◽  
A Chain

AbstractWe consider valued fields with a distinguished isometry or contractive derivation as valued modules over the Ore ring of difference operators. Under certain assumptions on the residue field, we prove quantifier elimination first in the pure module language, then in that language augmented with a chain of additive subgroups, and finally in a two-sorted language with a valuation map. We apply quantifier elimination to prove that these structures do not have the independence property.

scholarly journals SEPARABLY CLOSED FIELDS AND CONTRACTIVE ORE MODULES

2015 ◽  
Vol 80 (4) ◽  
pp. 1315-1338
Author(s):  
LUC BÉLAIR ◽  
FRANÇOISE POINT
Keyword(s):  
Quantifier Elimination ◽  
Difference Operators ◽  
Valued Fields ◽  
Contractive Map ◽  
Closed Fields ◽  
Frobenius Map ◽  
A Chain

AbstractWe consider valued fields with a distinguished contractive map as valued modules over the Ore ring of difference operators. We prove quantifier elimination for separably closed valued fields with the Frobenius map, in the pure module language augmented with functions yielding components for a p-basis and a chain of subgroups indexed by the valuation group.

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