Polynomial Bounds for Oscillation of Solutions of Fuchsian Systems

TL;DR

This paper establishes explicit polynomial upper bounds on the number of zeros of solutions to Fuchsian systems, leading to improved bounds for Abelian integrals and solving a special case of the Hilbert 16th problem.

Contribution

It provides the first explicit polynomial bounds for zeros of Fuchsian system solutions, improving previous exponential bounds and addressing a case of the Hilbert 16th problem.

Findings

Explicit polynomial upper bounds for zeros of Fuchsian systems

Polynomial bounds for zeros of Abelian integrals near degeneracy

Improved bounds over previous exponential estimates

Abstract

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.