The authors consider the linear differential centre
for every positive integer
They perturb this linear centre inside the class of all polynomial differential systems of the linear plus a homogeneous nonlinearity of degree
where every component of
is a linear polynomial plus a homogeneous of degree
Then, if the displacement function of order
of the perturbed system is not identically zero, the authors study the maximum number of limit cycles that can bifurcate from the periodic orbits of the linear differential centre.