Polynomial Bounds for Oscillation of Solutions of Fuchsian Systems

Details Polynomial Bounds for Oscillation of Solutions of Fuchsian Systems Polynomial Bounds for Oscillation of Solutions of Fuchsian Systems

2008-08-21

arxiv.org/abs/0808.2950v3

Annales de l'institut Fourier, 59 no. 7 (2009), p. 2891-2926

Mathematics

Dynamical Systems

Will appear in Annales de l'institut Fourier vol. 60 (2010)

Scientific paper

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.

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