  We study the displacement map associated to small one-parameter polynomial
unfoldings of polynomial Hamiltonian vector fields on the plane. Its leading
term, the generating function $M(t)$, has an analytic continuation in the
complex plane and the real zeroes of $M(t)$ correspond to the limit cycles
bifurcating from the periodic orbits of the Hamiltonian flow. We give a
geometric description of the monodromy group of $M(t)$ and use it to formulate
sufficient conditions for $M(t)$ to satisfy a differential equation of Fuchs or
Picard-Fuchs type. As examples, we consider in more detail the Hamiltonian
vector fields $\dot{z}=i\bar{z}-i(z+\bar{z})^3$ and $\dot{z}=iz+\bar{z}^2$,
possessing a rotational symmetry of order two and three, respectively. In both
cases $M(t)$ satisfies a Fuchs-type equation but in the first example $M(t)$ is
always an Abelian integral (that is to say, the corresponding equation is of
Picard-Fuchs type) while in the second one this is not necessarily true. We
derive an explicit formula of $M(t)$ and estimate the number of its real
zeroes.}
