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

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. 2000 Mathematics Subject Classification. 32S40, 34C20.