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

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