scholarly journals LARGE FIELDS IN DIFFERENTIAL GALOIS THEORY

2020 ◽  
pp. 1-16
Author(s):  
Annette Bachmayr ◽  
David Harbater ◽  
Julia Hartmann ◽  
Florian Pop
Keyword(s):  
Galois Theory ◽  
Transcendence Degree ◽  
Large Field ◽  
Embedding Problem ◽  
Differential Galois Theory ◽  
Differential Fields

We solve the inverse differential Galois problem over differential fields with a large field of constants of infinite transcendence degree over  $\mathbb{Q}$ . More generally, we show that over such a field, every split differential embedding problem can be solved. In particular, we solve the inverse differential Galois problem and all split differential embedding problems over $\mathbb{Q}_{p}(x)$ .

scholarly journals Iterative differential Galois theory in positive characteristic: A model theoretic approach

2011 ◽  
Vol 76 (1) ◽  
pp. 125-142 ◽  
Author(s):  
Javier Moreno
Keyword(s):  
Galois Theory ◽  
Positive Characteristic ◽  
Natural Extension ◽  
Primitive Element ◽  
Normal Extension ◽  
Theoretic Approach ◽  
Differential Galois Theory ◽  
Definition Of ◽  
Differential Fields ◽  
Element Theorem

AbstractThis paper introduces a natural extension of Kolchin's differential Galois theory to positive characteristic iterative differential fields, generalizing to the non-linear case the iterative Picard–Vessiot theory recently developed by Matzat and van der Put. We use the methods and framework provided by the model theory of iterative differential fields. We offer a definition of strongly normal extension of iterative differential fields, and then prove that these extensions have good Galois theory and that a G-primitive element theorem holds. In addition, making use of the basic theory of arc spaces of algebraic groups, we define iterative logarithmic equations, finally proving that our strongly normal extensions are Galois extensions for these equations.

scholarly journals Differential Galois Theory and Lie Symmetries

2015 ◽  
Author(s):  
David Blázquez-Sanz ◽  
◽  
Juan J. Morales-Ruiz ◽  
Jacques-Arthur Weil ◽  
◽  
...  
Keyword(s):  
Galois Theory ◽  
Lie Symmetries ◽  
Differential Galois Theory

scholarly journals Rational KdV Potentials and Differential Galois Theory

2019 ◽  
Author(s):  
Sonia Jiménez ◽  
◽  
Juan J. Morales-Ruiz ◽  
Raquel Sánchez-Cauce ◽  
María-Ángeles Zurro ◽  
...  
Keyword(s):  
Galois Theory ◽  
Differential Galois Theory

scholarly journals Second proof of the irreducibility of the first differential equation of painlevé

1990 ◽  
Vol 117 ◽  
pp. 125-171 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
General Theory ◽  
Galois Theory ◽  
Rigorous Proof ◽  
Mathematical Foundation ◽  
Differential Galois Theory ◽  
Infinite Dimensional ◽  
Finite Dimensional

In our paper [U2], we proved the irreducibility of the first differential equation y″ = 6y2 + x of Painlevé. In that paper we explained the origin of the problem and the importance of giving a rigorous proof. We can say that our method in [U2] is algebraic and finite dimensional in contrast to a prediction of Painlevé who expected a proof depending on the infinite dimensional differential Galois theory. Even nowadays the latter remains to be established. It seems that Painlevé needed an armament with the general theory (the infinite dimensional differential Galois theory) in the controversy with R. Liouville on the mathematical foundation of the proof of the irreducibility of the first differential equation (1902-03).

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