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Recent work on differential Galois theory. (English) Zbl 0931.12008

Séminaire Bourbaki. Volume 1997/98. Exposés 835–849. Paris: Société Mathématique de France, Astérisque. 252, 341-367, Exp. No. 849 (1998).
This is an excellent review of the recent progress on the inverse problem in differential Galois theory. After a brief and informative narrative of Picard-Vessiot theory, the local and global theorems of Ramis are stated:
Local theorem: A linear algebraic group \(G\) is a differential Galois group over the field of convergent Laurent series over the differential field of complex numbers, if and only if \(G/L(G)\) is topologically generated by a single element. Here \(L(G)\) is the normal subgroup of \(G\) generated by all maximal tori in \(G\).
Global theorem: Given \(X\) a smooth irreducible complex projective curve of genus \(g\) and \(S\) a finite set of points on \(X\), a linear algebraic group \(G\) can be realized as the differential Galois group of a differential equation with singularities in \(S\) if and only if \(G/L(G)\) is topologically generated by \(2g+ m-1\) elements where \(m\) is the cardinality of \(S\).
Assuming the local theorem, under suitable conditions on a compact Riemann surface \(X\) and an algebraic subgroup \(G\) of the group \(GL(V)\) where \(V\) is a vector space of dimension \(n\) over the complex numbers, such that \(G/L(G)\) is topologically generated by \(2g+m-1\) elements where \(m\) is the number of arbitrarily given points on \(X\), the global theorem is derived.
The formal theory which is presented in section 6, gives a classification of differential equations over the field of convergent Laurent series and clarifies its connection with the differential Galois group.
In section 9, a constructive solution of the inverse problem for the differential field \(C(X)\) of rational functions in \(X\) over the complex numbers is given. It may be remarked that the general result is due to Mitschi and Singer, and it says that every reductive group over \(C\) can be realized as the differential Galois group of a suitable differential equation.
In the last section other formulations of the theorems of Ramis and of Mitschi and Singer are given.
For the entire collection see [Zbl 0911.00019].

MSC:

12H05 Differential algebra
13N10 Commutative rings of differential operators and their modules
12-02 Research exposition (monographs, survey articles) pertaining to field theory
34M40 Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain
34M50 Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain
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