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.

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.

0-D-valued fields

2006 ◽  
Vol 71 (2) ◽  
pp. 639-660 ◽  
Author(s):  
Nicolas Guzy
Keyword(s):  
Quantifier Elimination ◽  
Valued Fields ◽  
Angular Component ◽  
Closed Fields ◽  
Analogous Theorem ◽  
Field Setting

AbstractIn [12]. T. Scanlon proved a quantifier elimination result for valued D-fields in a three-sorted language by using angular component functions. Here we prove an analogous theorem in a different language which was introduced by F. Delon in her thesis. This language allows us to lift the quantifier elimination result to a one-sorted language by a process described in the Appendix. As a byproduct, we state and prove a “positivstellensatz” theorem for the differential analogue of the theory of real-series closed fields in the valued D-field setting.

Differentially algebraic group chunks

1990 ◽  
Vol 55 (3) ◽  
pp. 1138-1142 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Algebraic Group ◽  
Characteristic Zero ◽  
Quantifier Elimination ◽  
Closed Field ◽  
Zariski Topology ◽  
First Order ◽  
Differential Field ◽  
Closed Fields ◽  
Differential Fields ◽  
Algebraically Closed Fields

We point out that a group first order definable in a differentially closed field K of characteristic 0 can be definably equipped with the structure of a differentially algebraic group over K. This is a translation into the framework of differentially closed fields of what is known for groups definable in algebraically closed fields (Weil's theorem).I restrict myself here to showing (Theorem 20) how one can find a large “differentially algebraic group chunk” inside a group defined in a differentially closed field. The rest of the translation (Theorem 21) follows routinely, as in [B].What is, perhaps, of interest is that the proof proceeds at a completely general (soft) model theoretic level, once Facts 1–4 below are known.Fact 1. The theory of differentially closed fields of characteristic 0 is complete and has quantifier elimination in the language of differential fields (+, ·,0,1, −1,d).Fact 2. Affine n-space over a differentially closed field is a Noetherian space when equipped with the differential Zariski topology.Fact 3. If K is a differentially closed field, k ⊆ K a differential field, and a and are in k, then a is in the definable closure of k ◡ iff a ∈ ‹› (where k ‹› denotes the differential field generated by k and).Fact 4. The theory of differentially closed fields of characteristic zero is totally transcendental (in particular, stable).

The theory of modules of separably closed fields 1

2002 ◽  
Vol 67 (3) ◽  
pp. 997-1015 ◽  
Author(s):  
Pilar Dellunde ◽  
Françoise Delon ◽  
Françoise Point
Keyword(s):  
Polynomial Ring ◽  
Quantifier Elimination ◽  
Additive Functions ◽  
Skew Polynomial Ring ◽  
Closed Fields ◽  
Skew Polynomial

AbstractWe consider separably closed fields of characteristic p > 0 and fixed imperfection degree as modules over a skew polynomial ring. We axiomatize the corresponding theory and we show that it is complete and that it admits quantifier elimination in the usual module language augmented with additive functions which are the analog of the p-component functions.

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