# Integral representations of GKZ hypergeometric functions: Gauss-Manin connection, intersection theory, and quadratic relations

GKZ(Gelfand, Kapranov, Zelevinsky) system is a holonomic

system which describes classical hypergeometric systems in a unified

manner. In this talk, we realize GKZ system as a Gauss-Manin connection

where we treat Euler integral and Laplace integral at the same time.

Focusing on regular holonomic case, we give a method of reinterpreting

the combinatorics of regular triangulations to the construction of the

basis of twisted cycles at "toric infinity". This naturally gives rise

to an orthogonal decomposition of the twisted homology group with

respect to the intersection pairing. As an application, we give a

general quadratic relation of GKZ hypergeometric functions associated to

a unimodular triangulation. We also discuss an algorithm of computing

cohomology intersection numbers based on a joint work with Nobuki

Takayama. The techniques above produce several new quadratic relations

of hypergeometric functions of several variables.