Finite dimensional groups of local diffeomorphisms

We are interested in classifying groups of local biholomorphisms (or even formal diffeomorphisms) that can be endowed with a canonical structure of algebraic group up to add extra formal diffeomorphisms. We show that this is the case for virtually polycyclic subgroups and in particular finitely generated virtually nilpotent groups of local biholomorphisms. We provide several methods to identify this property and build examples. As a consequence we generalize results of Arnold, Seigal-Yakovenko and Binyamini on uniform estimates of local intersection multiplicities to bigger classes of groups, including for example virtually polycyclic groups.