  In this paper we first give the explicit definition of essential
perturbation. Secondly, given a perturbation of a particular family of centers
of polynomial differential systems of arbitrary degree for which we explicitly
know its Poincar\'e--Liapunov constants, we give the structure of its $k$-th
Melnikov function. This result generalizes the result obtained by Chicone and
Jacobs for perturbations of degree at most two of any center of a quadratic
polynomial system. Moreover we study the essential perturbations for all the
centers of the differential systems \[ \dot{x} \, = \, -y + P_{\rm d}(x,y),
\quad \dot{y} \, = \, x + Q_{\rm d}(x,y), \] where $P_{\rm d}$ and $Q_{\rm d}$
are homogeneous polynomials of degree ${\rm d}$, for ${\rm d}=2$ and ${\rm
d}=3$.
