scholarly journals SOME DEFINABLE GALOIS THEORY AND EXAMPLES

2017 ◽  
Vol 23 (2) ◽  
pp. 145-159 ◽  
Author(s):  
OMAR LEÓN SÁNCHEZ ◽  
ANAND PILLAY
Keyword(s):  
Differential Equations ◽  
Galois Theory ◽  
Automorphism Groups ◽  
Differential Galois Theory ◽  
Algebraic Differential Equations ◽  
Galois Correspondence

AbstractWe 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 from [Pillay, “Differential Galois theory I”, Illinois Journal of Mathematics, 42(4), 1998], using this to give a restatement of a conjecture on almost semiabelian δ-groups from [Bertrand and Pillay, “Galois theory, functional Lindemann–Weierstrass, and Manin maps”, Pacific Journal of Mathematics, 281(1), 2016].

scholarly journals Birational automorphism groups and differential equations

1990 ◽  
Vol 119 ◽  
pp. 1-80 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Critical Point ◽  
Galois Theory ◽  
Automorphism Groups ◽  
Rigorous Proof ◽  
Differential Galois Theory ◽  
General Attitude ◽  
Transcendental Functions ◽  
Birational Automorphism

Painlevé studied the differential equations y″ = R(y′ y, x) without moving critical point, where R is a rational function of y′ y, x. Most of them are integrated by the so far known functions. There are 6 equations called Painlevé’s equations which seem to be irreducible or seem to define new transcendental functions. The simplest one among them is y″ = 6y2 + x. Painlevé declared on Comptes Rendus in 1902-03 that y″ = 6y2 + x is irreducible. It seems that R. Liouville pointed out an error in his argument. In fact there are discussions on this subject between Painlevé and Liouville on Comptes Rendus in 1902-03. In 1915 J. Drach published a new proof of the irreducibility of the differential equation y″ = 6y2 + x. The both proofs depend on the differential Galois theory developed by Drach. But the differential Galois theory of Drach contains errors and gaps and it is not easy to understand their proofs. One of our contemporaries writes in his book: the differential equation y″ = 6y2 + x seems to be irreducible dans un sens que on ne peut pas songer à préciser. This opinion illustrates well the general attitude of the nowadays mathematicians toward the irreducibility of the differential equation y″ = 6y2 + x. Therefore the irreducibility of the differential equation y″ = 6y2 + x remains to be proved. We consider that to give a rigorous proof of the irreducibility of the differential equation y″ = 6y2 + x is one of the most important problem in the theory of differential equations.

Algebraic types and automorphism groups

1993 ◽  
Vol 58 (1) ◽  
pp. 232-239 ◽  
Author(s):  
Akito Tsuboi
Keyword(s):  
General Model ◽  
Galois Extension ◽  
Galois Theory ◽  
Automorphism Groups ◽  
The Other ◽  
Closed Field ◽  
Prime Characteristic ◽  
Galois Correspondence ◽  
Definable Functions ◽  
Special Case

Galois theory states that if L is a certain algebraic extension (called a Galois extension) of a field K, then there is a one-to-one correspondence (called a Galois correspondence) between subfields M, K ⊂ M ⊂ L and subgroups of the automorphism groups of L fixing the elements in K.A subfield of a field L can be considered as a substructure of L in general model theory. However, a substructure is a subset closed under functions which are interpretations of function symbols in a given language, so the notion of substructure may change if we expand the language by adding definable notions. On the other hand a definably closed substructure is a subset which is closed under all definable functions, and it does not change by such expansions. If we are interested in subfields of an algebraically closed field of characteristic 0, these two notions are the same. But in a field of prime characteristic they are not equal. Speaking roughly, a Galois extension of a field K is an extension whose subfields are relatively definably closed. Poizat [4] showed that if a structure M has elimination of imaginaries there is a kind of Galois correspondence between definably closed substructures and subgroups of bijective elementary mappings of M.In this paper, using Poizat's result we study algebraic types. As is well known, one motivation for developing the Galois theory was to show the unsolvability of equations with degree ≥ 5. We want to take this unsolvability as a special case of general phenomena. For this purpose, we introduce several notions which are stronger than mere algebraicity and study relations between these notions and groups of bijective elementary mappings. (See Theorems 3.7 and 3.9.)

scholarly journals Differential Galois theory of infinite dimension

1996 ◽  
Vol 144 ◽  
pp. 59-135 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
19Th Century ◽  
Galois Theory ◽  
Infinite Dimension ◽  
Differential Galois Theory ◽  
Dimensional Theory ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
The 19Th Century

This paper is the second part of our work on differential Galois theory as we promised in [U3]. Differential Galois theory has a long history since Lie tried to apply the idea of Abel and Galois to differential equations in the 19th century (cf. [U3], Introduction). When we consider Galois theory of differential equation, we have to separate the finite dimensional theory from the infinite dimensional theory. As Kolchin theory shows, the first is constructed on a rigorous foundation. The latter, however, seems inachieved despite of several important contributions of Drach, Vessiot,…. We propose in this paper a differential Galois theory of infinite dimension in a rigorous and transparent framework. We explain the idea of the classical authors by one of the simplest examples and point out the problems.

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