Finite dimensional invariant KAM tori for tame vector fields

We discuss a Nash-Moser/KAM algorithm for the construction of invariant tori for tame vector fields. Similar algorithms have been studied widely both in finite and infinite dimensional contexts: we are particularly interested in the second case where tameness properties of the vector fields become very important. We focus on the formal aspects of the algorithm and particularly on the minimal hypotheses needed for convergence. We discuss various applications where we show how our algorithm allows one to reduce to solving only linear forced equations. We remark that our algorithm works at the same time in analytic and Sobolev classes.