Picard–Fuchs Equations and Gauss–Manin Systems with a View Towards the Riemann–Hilbert Problem

Abstract

We study Abelian integrals associated with a tame polynomial function and their Picard---Fuchs equations using the theory of algebraic Gauss---Manin systems. Especially, we look for a basis of the Petrov module, in which the Picard---Fuchs equations become as simple as possible. As an application, we discuss the related Riemann---Hilbert problem and prove that it has a positive answer under some conditions. In this case, we compute the Jordan normal form of the residue matrices of the corresponding Fuchsian system in terms of local data.