11 research outputs found
Some definable Galois theory and examples
We make explicit certain results around the Galois correspondence in the
context of definable automorphism groups, and point out the relation to some
recent papers dealing with the Galois theory of algebraic differential
equations when the constants are not "closed" in suitable senses. We also
improve the definitions and results on generalized strongly normal extensions.Comment: arXiv admin note: text overlap with arXiv:1309.633
Un Crit{\`E}Re Simple
In this short note, we mimic the proof of the simplicity of the theory ACFA
of generic difference fields in order to provide a criterion, valid for certain
theories of pure fields and fields equipped with operators, which shows that a
complete theory is simple whenever its definable and algebraic closures are
controlled by an underlying stable theory.Comment: in Frenc
Generic derivations on o-minimal structures
Let be a complete, model complete o-minimal theory extending the theory
RCF of real closed ordered fields in some appropriate language . We study
derivations on models . We introduce the notion
of a -derivation: a derivation which is compatible with the
-definable -functions on . We show
that the theory of -models with a -derivation has a model completion
. The derivation in models
behaves "generically," it is wildly discontinuous and its kernel is a dense
elementary -substructure of . If RCF, then
is the theory of closed ordered differential fields (CODF) as introduced by
Michael Singer. We are able to recover many of the known facts about CODF in
our setting. Among other things, we show that has as its open
core, that is distal, and that eliminates
imaginaries. We also show that the theory of -models with finitely many
commuting -derivations has a model completion.Comment: 29 page
Galois Theory of Parameterized Differential Equations and Linear Differential Algebraic Groups
We present a Galois theory of parameterized linear differential equations
where the Galois groups are linear differential algebraic groups, that is,
groups of matrices whose entries are functions of the parameters and satisfy a
set of differential equations with respect to these parameters. We present the
basic constructions and results, give examples, discuss how isomonodromic
families fit into this theory and show how results from the theory of linear
differential algebraic groups may be used to classify systems of second order
linear differential equations