Given a germ of a logarithmic analytic vector field ∂, let ∂ = ∂_ss + ∂_nilp be its unique formal Jordan decomposition, where ∂_ss is its semi-simple component and ∂_nilp is its nilpotent component. We define the Bruno ideal B(∂) as the collinearity ideal of its semi-simple and nilpotent components, that is, the vanishing locus of ∂_ss ∧ ∂_nilp.
We prove the following result originally stated by A. D. Bruno: If the eigenvalues of ∂_ss satisfy Bruno's arithmetic condition, then B(∂) is analytic and the restriction of ∂ to the variety V(B(∂)) is analytically normalizable.
We will end the talk by exposing some examples of applications of this result.
This is a joint work with Daniel Panazzolo.
