Local normal forms, differential Galois, resonances, and formal non-integrability

I’m not through with “fundamental maths” yet, so I’ll have to cook up a new paper in 2011 :-)

Same old things: normal forms, Galois theory, resonances, and non-integrability. Some new twists: the Galois group here will be a local one (defined in the neighborhood of a fixed point, and not a non-stationary solution). There are clear relations between Galois groups  and normal forms (one can almost determine the Galois group from the “optimal normal form” and vice versa). And if the Galois group is “bad” then no hope for integrability. Which is actually a good thing: when there are given some “noncommutative resonances”, then in the generic case with such resonances the Galois group will be “bad”. This leads us to the result of generic non-integrability on g*, when g is is a big enough simple Lie algebra.

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