  We study the analogue of the infinitesimal 16th Hilbert problem in dimension
zero. Lower and upper bounds for the number of the zeros of the corresponding
Abelian integrals (which are algebraic functions) are found. We study the
relation between the vanishing of an Abelian integral $I(t)$ defined over
$\mathbb{Q}$ and its arithmetic properties. Finally, we give necessary and
sufficient conditions for an Abelian integral to be identically zero.
