Binyamini, G., Novikov, D., Yakovenko, S. On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem. Preprint arXiv:0808.2952 (v. 1, August 21, 2008, 51 pp.)
We prove that the number of limit cycles generated by a small non-conservative perturbation of a Hamiltonian polynomial vector field on the plane, is bounded by a double exponential of the degree of the fields. This solves the long-standing infinitesimal Hilbert 16th problem. The proof uses only the fact that Abelian integrals of a given degree are horizontal sections of a a regular flat meromorphic connection (Gauss-Manin connection) with a quasiunipotent monodromy group.
Binyamini G., Yakovenko, S. Polynomial Bounds for Oscillation of Solutions of Fuchsian Systems. Preprint arXiv:0808.2950 (v. 1, August 21, 2008, 31 pp.)
We study the problem of placing effective upper bounds for the number of zeros of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of the class of Fuchsian systems of given dimension n having m singular points. As a function of n,m, this bound turns out to be double exponential in the precise sense explained in the paper. As a corollary, we obtain a solution of the so called restricted infinitesimal Hilbert 16th problem, an explicit upper bound for the number of isolated zeros of Abelian integrals which is polynomially growing as the Hamiltonian tends to the degeneracy locus. This improves the exponential bounds recently established by A. Glutsyuk and Yu. Ilyashenko.
Ilyashenko, Yu., Yakovenko, S., Lectures on Analytic Theory of Ordinary Differential Equations, Graduate Studies in Mathematics, 86, Amer. Math. Soc., 2008, xiii+625 pp.
Novikov D., Yakovenko S., Lectures on Meromorphic Flat Connections, In: Normal forms, bifurcations and finiteness problems in differential equations, 387--430, NATO Sci. Ser. II Math. Phys. Chem., 137, Kluwer Acad. Publ., Dordrecht, 2004 (Preprint math.CA/0212334).
Yakovenko, S. Quantitative theory of ordinary differential equations and tangential Hilbert 16th problem, Preprint math.DS/0104140 (2001). On finiteness in differential equations and Diophantine geometry, CRM Monogr. Ser., vol. 24, Amer. Math. Soc., Providence, RI, 2005, pp. 41–109, MR2180125 (2006g:34062)
Yakovenko, S. On functions and curves defined by ordinary differential equations. The Arnoldfest (Toronto, ON, 1997), Fields Inst. Communications, 24 (1999), 497--525, Amer. Math. Soc., Providence, RI, 1999