Differential equations for Feynman integrals from GKZ hypergeometric systems

The study of Feynman integrals is central for computing observable quantities in high-energy physics. However, many analytic properties of Feynman integrals are still poorly understood in full generality. A key step towards uncovering analytic properties of Feynman integrals is to first understand the space of functions to which they belong. It turns out that Feynman integrals arise as special cases of Gelfand-Kapranov-Zelevinsky (GKZ) hypergeometric functions - a class of functions with rich connections to toric geometry, D-module theory, twisted cohomology, and more. In this talk, we shall employ the GKZ framework to obtain Pfaffian systems for Feynman integrals, i.e. a system of 1st-order PDEs obeyed by a vector space basis of integrals. Our algorithm for obtaining Pfaffian systems is based on Macaulay matrices, offering an efficient alternative to the traditional method based on Gröbner bases.