<p>We investigate resonant Hamiltonians with <i>n</i> degrees of freedom to which we attach a small perturbation of polynomial type. Our analysis is based on singular reduction theory within the setting of Hamiltonian dynamics. In order to perform symplectic reduction, the system is first transformed into normal form, after which the higher-order terms of the transformation are truncated. This procedure results in a reduced Hamiltonian system defined on the corresponding orbit space, which is a manifold in the case of regular reduction and an orbifold in the case of singular reduction. We then classify the singularities of the orbit space. Continuous families of periodic solutions of the original system are characterized through the study of nondegenerate critical points on the reduced space. The local desingularization of all types of orbifold singularities is achieved by means of symplectic transformations. This framework provides an effective method for approximating the characteristic multipliers of the periodic solutions, obtained by computing the eigenvalues of the linearized reduced Hamiltonian at the critical point associated with the corresponding periodic orbit. Several examples with two, three, and four degrees of freedom are presented to illustrate the theoretical results.</p>