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Galois theory and Painlevé equations. (English) Zbl 1156.34080
Delabaere, Éric (ed.) et al., Théories asymptotiques et équations de Painlevé. Paris: Société Mathématique de France (ISBN 978-2-85629-229-7/pbk). Séminaires et Congrès 14, 299-339 (2006).
An infinite dimensional Galois theory for nonlinear differential equations is explained. The author starts with the original ideas by Galois developed further by Lie, Abel and Dedekind on the group theoretic approach to the algebraic equations. Next the author describes the Picard-Vessiot approach to the extension of the Galois theory to the case of linear differential equations and the idea of its further extension to the nonlinear case. In short, the basic idea consists in consideration of the partial differential field extension determined by the nonlinear \(n\)-th order ODE using a set of its \(n\) first integrals. The author proposes ceratin refinements to overcome difficulties the theory meets. Namely, he replaces the full Lie pseudo-group of all the coordinate transformations of the set of initial conditions by an infinitesimal automorphism group, thus replacing the partial differential field extension by a partial differential algebra extension. Also, to avoid any dependence of the functor thus obtained on the choice of the reference point in the space of initial conditions, the author utilizes the universal Taylor morphism.
In the second part of his article, the author shows the equivalence of his theory and the Galois theory of foliations by Malgrange, sets various problems concerning the general Galois theory of nonlinear differential equations as well as particular problems in exploring the Galois theory for the classical Painlevé equations.
For the entire collection see [Zbl 1117.34001].

34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
12H05 Differential algebra
13B05 Galois theory and commutative ring extensions
17B65 Infinite-dimensional Lie (super)algebras
34M15 Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain
58H05 Pseudogroups and differentiable groupoids