Macaulay Matrix for Feynman Integrals: Linear Relations and Intersection Numbers

We elaborate on the connection between Gel'fand-Kapranov-Zelevinsky systems, de Rham theory for twisted cohomology groups, and Pfaffian equations for Feynman Integrals. We propose a novel, more efficient algorithm to compute Macaulay matrices, which are used to derive Pfaffian systems of differential equations. The Pfaffian matrices are then employed to obtain linear relations for A-hypergeometric (Euler) integrals and Feynman integrals, through recurrence relations and through projections by intersection numbers.