scholarly journals Model theory of partial differential fields: From commuting to noncommuting derivations

2007 ◽  
Vol 135 (6) ◽  
pp. 1929-1934 ◽  
Author(s):  
Michael F. Singer
Keyword(s):  
Model Theory ◽  
Partial Differential ◽  
Differential Fields

Differential forms in the model theory of differential fields

2003 ◽  
Vol 68 (3) ◽  
pp. 923-945 ◽  
Author(s):  
David Pierce
Keyword(s):  
Differential Geometry ◽  
Model Theory ◽  
Characteristic Zero ◽  
Differential Forms ◽  
Main Tool ◽  
Frobenius Theorem ◽  
Lie Bracket ◽  
Existentially Closed ◽  
Differential Fields

AbstractFields of characteristic zero with several commuting derivations can be treated as fields equipped with a space of derivations that is closed under the Lie bracket. The existentially closed instances of such structures can then be given a coordinate-free characterization in terms of differential forms. The main tool for doing this is a generalization of the Frobenius Theorem of differential geometry.

scholarly journals Imaginaries and invariant types in existentially closed valued differential fields

2019 ◽  
Vol 2019 (750) ◽  
pp. 157-196 ◽  
Author(s):  
Silvain Rideau
Keyword(s):  
Model Theory ◽  
Extension Property ◽  
Valued Fields ◽  
Invariant Extension ◽  
Open Questions ◽  
Existentially Closed ◽  
Differential Fields ◽  
Abstract Criterion

Abstract We answer three related open questions about the model theory of valued differential fields introduced by Scanlon. We show that they eliminate imaginaries in the geometric language introduced by Haskell, Hrushovski and Macpherson and that they have the invariant extension property. These two results follow from an abstract criterion for the density of definable types in enrichments of algebraically closed valued fields. Finally, we show that this theory is metastable.

scholarly journals Effective uniform bounding in partial differential fields

2016 ◽  
Vol 288 ◽  
pp. 308-336 ◽  
Author(s):  
James Freitag ◽  
Omar León Sánchez
Keyword(s):  
Partial Differential ◽  
Differential Fields

scholarly journals Real Liouvillian Extensions of Partial Differential Fields

2021 ◽  
Author(s):  
Teresa Crespo ◽  
◽  
Zbigniew Hajto ◽  
Rouzbeh Mohseni ◽  
◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Galois Theory ◽  
Uniqueness Result ◽  
Real Closed Field ◽  
Closed Field ◽  
Partial Differential ◽  
Linear Algebraic Groups ◽  
Galois Correspondence ◽  
Definition Of ◽  
Differential Fields

In this paper, we establish Galois theory for partial differential systems defined over formally real differential fields with a real closed field of constants and over formally p-adic differential fields with a p-adically closed field of constants. For an integrable partial differential system defined over such a field, we prove that there exists a formally real (resp. formally p-adic) Picard-Vessiot extension. Moreover, we obtain a uniqueness result for this Picard-Vessiot extension. We give an adequate definition of the Galois differential group and obtain a Galois fundamental theorem in this setting. We apply the obtained Galois correspondence to characterise formally real Liouvillian extensions of real partial differential fields with a real closed field of constants by means of split solvable linear algebraic groups. We present some examples of real dynamical systems and indicate some possibilities of further development of algebraic methods in real dynamical systems.

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