The infinitesimal 16th Hilbert problem in dimension zero

Details The infinitesimal 16th Hilbert problem in dimension zero The infinitesimal 16th Hilbert problem in dimension zero

2005-07-04

arxiv.org/abs/math/0507061v1

Bull. Sci. math. 131 (2007) 242--257

Mathematics

Classical Analysis and ODEs

21 pages

Scientific paper

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.

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