Finally the two texts concerned with solution of the Infinitesimal Hilbert problem, are put into the polished form (including the publisher’s LaTeX style files). The new revisions, already uploaded to ArXiv, differ from the initial submissions only by corrected typos, a few rearrangements aimed at improving the readability of the texts, and a couple of more references added. There is absolutely no need to read the new revision if you already have read the first one.

Mostly for the reasons of “internal convenience” the complete references are repoduced here:

**G. Binyamini and S. Yakovenko**,*Polynomial Bounds for Oscillation of Solutions of Fuchsian Systems*, posted as arXiv:0808.2950v2 [math.DS], 36 p.p., submitted to*Ann. Inst. Fourier*(Dec. 2008), accepted (February, 2009)**G. Binyamini, D. Novikov and S. Yakovenko**,*On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem*, posted as arXiv:0808.2952v2 [math.DS], 57 p.p., submitted to*Inventiones Mathematicae*(Nov. 2008), accepted (Oct. 2009).