We study shift relations between Feynman integrals via the Mellin transform through parametric annihilation operators. These contain the momentum space integration by parts relations, which are well-known in the physics literature. Applying a result of Loeser and Sabbah, we conclude that the number of master integrals is computed by the Euler characteristic of the Lee-Pomeransky polynomial. We illustrate techniques to compute this Euler characteristic in various examples and compare it with numbers of master integrals obtained in previous works. for interesting discussions and helpful comments on integration by parts for Feynman integrals, Jørgen Rennemo for suggesting literature relevant for section 3.1 and Viktor Levandovskyy for communication and explanations around D-modules and in particular their implementation in SINGULAR.Thomas Bitoun thanks Claude Sabbah for feedback and bringing the work [58-60] to our attention. Thomas acknowledges funding through EPSRC grant EP/L005190/1. Christian Bogner thanks Deutsche Forschungsgemeinschaft for support under the project BO 4500/1-1.This research was supported by the Munich Institute for Astro-and Particle Physics (MIAPP) of the DFG cluster of excellence "Origin and Structure of the Universe". Furthermore we are grateful for support from the Kolleg Mathematik Physik Berlin (KMPB) and hospitality at Humboldt-Universität Berlin.Images in this paper were created with JaxoDraw [11] (based on Axodraw [93]) and FeynArts [36].

We show that almost all Feynman integrals as well as their coefficients in a Laurent series in dimensional regularization can be written in terms of Horn hypergeometric functions. By applying the results of Gelfand-Kapranov-Zelevinsky (GKZ) we derive a formula for a class of hypergeometric series representations of Feynman integrals, which can be obtained by triangulations of the Newton polytope ∆ G corresponding to the Lee-Pomeransky polynomial G. Those series can be of higher dimension, but converge fast for convenient kinematics, which also allows numerical applications. Further, we discuss possible difficulties which can arise in a practical usage of this approach and give strategies to solve them.

We give a brief introduction to a parametric approach for the derivation of shift relations between Feynman integrals and a result on the number of master integrals. The shift relations are obtained from parametric annihilators of the Lee-Pomeransky polynomial G . By identification of Feynman integrals as multi-dimensional Mellin transforms, we show that this approach generates every shift relation. Feynman integrals of a given family form a vector space, whose finite dimension is naturally interpreted as the number of master integrals. This number is an Euler characteristic of the polynomial G .

We give a brief introduction to a parametric approach for the derivation of shift relations between Feynman integrals and a result on the number of master integrals. The shift relations are obtained from parametric annihilators of the Lee-Pomeransky polynomial G. By identification of Feynman integrals as multi-dimensional Mellin transforms, we show that this approach generates every shift relation. Feynman integrals of a given family form a vector space, whose finite dimension is naturally interpreted as the number of master integrals. This number is an Euler characteristic of the polynomial G .