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Quatre descriptions des groupes de Galois différentiels. (Four descriptions of differential Galois groups). (French) Zbl 0651.12015
Sémin. d’algèbre P. Dubreil et M.-P. Malliavin, Proc., Paris 1986, Lect. Notes Math. 1296, 28-41 (1987).
[For the entire collection see Zbl 0624.00008.]
Differential Galois groups are typically defined as the group of automorphisms of the field of solutions of a (linear, homogeneous) differential equation which commute with derivation. The author proposes an intrinsic definition.
He considers the category of finite-dimensional vector spaces with integrable connections. He fixes one such M and considers the smallest subcategory containing M and being closed under direct sums, tensor products, and symmetric and exterior powers. He defines the differential Galois group $$G_{gal}(M)$$ to be the stabilizer in $$GL_ M$$ of the objects in the subcategory under all polynomial representations of $$GL_ M$$. This is shown to be an extension of the usual definition, after scalars are extended to the completion of the field of coefficient functions. It also has direct links with a conjecture of N. Katz on groups associated to p-curvative operators, to monodromy groups and the motivic Galois theory.
This paper is a nearly faithful reproduction, without proofs, of lectures. The style is informal and reportorial. Proofs will appear in a paper entitled “Quelques points de theorie de Galois différentielle”.
Reviewer: A.R.Magid

##### MSC:
 12H20 Abstract differential equations 14L99 Algebraic groups 12H05 Differential algebra 14A20 Generalizations (algebraic spaces, stacks)
##### Keywords:
Differential Galois groups; motivic Galois theory