Polynomial Bounds for Oscillation of Solutions of Fuchsian Systems

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 a 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. 1. Zeros of solutions of Fuchsian systems and restricted infinitesimal Hilbert 16th problem 1.1. Fuchsian systems and zeros of their solutions Let Ω be a meromorphic (rational) n×n-matrix 1-form on the Riemann sphere CP 1 with a singular (polar) locus Σ = {τ1, . . . , τm} consisting of m distinct points. The linear system of Pfaffian equations (1.1) dx− Ωx = 0, Ω = ω11 · · · ω1n .. . . . .. ωn1 · · · ωnn  , x = x1 .. xn  ,