Bruno Ideal and the Variety of Centers for Singular Germs of Vector Fields

Given a germ of singular analytic vector field ∂ , we consider the locus where ∂ is formally equivalent to its linear part. In the logarithmic case, this locus is defined by the vanishing of a formal ideal B(∂ ) . We show that, under convenient arithmetic conditions on the linear part of ∂ , such ideal B(∂ ) is in fact analytic. As a consequence, its zero set V is an analytic invariant variety and the foliation defined by the restriction ∂ |V is analytically linearizable.