Algorithms for Pfaffian Systems and Cohomology Intersection Numbers of Hypergeometric Integrals

In the theory of special functions, a particular kind of multidimensional integral appears frequently. It is called the Euler integral. In order to understand the topological nature of the integral, twisted de Rham cohomology theory plays an important role. We propose an algorithm of computing an invariant cohomology intersection number of a given basis of the twisted cohomology group. We also develop an algorithm of computing the Paffian system that a given basis satisfies. These algorithms are based on the fact that the Euler integral satisfies GKZ system and utilizes algorithms to find rational function solutions of differential equations. We provide software to perform this algorithm.