Poincare renormalized forms

In Poincare Normal Form theory, one considers a series of transformations g enerated by homogeneous polynomials obtained as solution of the homological equation; such solutions are unique up to terms in the kernel of the homol ogical operator. Careful consideration of the higher order terms generated by polynomials differing for a term in this kernel leads to the possibility of further reducing the Normal Form expansion of a formal power series, in a completely algorithmic way. The algorithm is also applied to a number of concrete cases. An alternative formulation, conceptually convenient but co mputationally unpractical, is also presented, and it is shown that the disc ussion immediately extends on the one side to the Hamiltonian case and Birk hoff normal forms, and to the other to the equivariant setting, (C) Elsevie r, Paris.