Combinatorics of Hamiltonian Normal Forms

We discuss algebraic and combinatorial aspects of the Hamiltonian normal form theory. The main objective is to describe the normal form near a singular point purely in terms of the original Hamiltonian, avoiding the normalization procedure. In the case of one degree of freedom we compute the normal form as an explicit nonlinear functional, applied to the original Hamiltonian. We present analogous results in arbitrary dimension. The corresponding formulas are more complicated but still explicit.