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 present a theory of the b-function (or Bernstein-Sato polynomial) in positive characteristic. Let f be a non-constant polynomial with coefficients in a perfect field k of characteristic p > 0. Its b-function b f is defined to be an ideal of the algebra of continuous k-valued functions on Z p . The zero-locus of the b-function is thus naturally interpreted as a subset of Z p , which we call the set of roots of b f . We prove that b f has finitely many roots and that they are negative rational numbers. Our construction builds on an earlier work of Mustaţȃ and is in terms of D-modules, where D is the ring of Grothendieck differential operators. We use the Frobenius to obtain finiteness properties of b f and relate it to the test ideals of f.

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 investigate when a meromorphic connection on a smooth rigid analytic variety 𝑋 gives rise to a coadmissible D ⏜ X \overparen{\mathcal{D}}_{X} -module, and show that this is always the case when the roots of the corresponding 𝑏-functions are all of positive type. We also use this theory to give an example of an integrable connection on the punctured unit disk whose pushforward is not a coadmissible module.

Let f be a quasi-homogeneous polynomial with an isolated singularity in C n . We compute the length of the D-modules Df λ /Df λ+1 generated by complex powers of f in terms of the Hodge filtration on the top cohomology of the Milnor fiber. When λ = −1 we obtain one more than the reduced genus of the singularity (dim H n−2 (Z, O Z ) for Z the exceptional fiber of a resolution of singularities). We conjecture that this holds without the quasi-homogeneous assumption. We also deduce that the quotient Df λ /Df λ+1 is nonzero when λ is a root of the b-function of f (which Saito recently showed fails to hold in the inhomogeneous case). We obtain these results by comparing these Dmodules to those defined by Etingof and the second author which represent invariants under Hamiltonian flow.