Normal form for the completely resonant NLS on the torus T^n

We study the completely resonant cubic Nonlinear Schrodinger equation on the torus T^n with n > 2, with the purpose of constructing quasi-periodic solutions with arbitrary m frequencies. We pass to the hamiltonian formalism and perform one step of resonant Birkoff normal form to highlight the relevant part of the non-linearity. We choose appropriately the ``tangential sites'' (from which the quasi-periodic solution bifurcates) so that the new hamiltonian is as simple as possible. This gives rise to a set of geometric and combinatorial constraints on the ``tangential sites'' v_i in Z^n.