Preprint A544/2007
The infinitesimal 16th Hilbert problem in dimension zero

Movasati, Hossein

**Keywords: **
Gauss-Manin connection | abelian integral

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.