MultivariateResidues: A Mathematica package for computing multivariate residues

Abstract

Multivariate residues appear in many different contexts in theoretical physics and algebraic geometry. In theoretical physics, they for example give the proper definition of generalized-unitarity cuts, and they play a central role in the Grassmannian formulation of the S-matrix by Arkani-Hamed et al. In realistic cases their evaluation can be non-trivial. In this paper we provide a Mathematica package for efficient evaluation of multidimensional residues based on methods from computational algebraic geometry. The package moreover contains an implementation of the global residue theorem, which produces relations between residues at finite locations and residues at infinity.

Constructing d-log integrands and computing master integrals for three-loop four-particle scattering

“…It is based on a suitable parametrization of the loop integrand, and analyzes the resulting rational function by taking residues iteratively. This approach is complementary to the algorithm implemented in [14] that uses methods from computational algebraic geometry to compute multivariate residues. Algorithms that can be applied to Feynman integrals (in contrast to integrands) in conjunction with the methods of differential equations [15][16][17][18][19] in order to find uniform weight integrals were discussed in refs.…”

Refined swampland distance conjecture and exotic hybrid Calabi-Yaus

“…In this section we outline some details of the calculation of the multi-dimensional residue (4.60) in the ζ 0-phase of the non-abelian model. We follow the references [44][45][46][47]. Finding the contributing poles in a multidimensional residue can be translated into a geometric problem of finding intersections of divisors associated to the poles of the integrand.…”

“…To obtain this result, one has to compute a multi-dimensional residue, which is considerably harder that the one-dimensional case that we had to deal with for the abelian GLSMs. Prescriptions for evaluating such integrals can be found for instance in [44][45][46][47]. We outline some of the steps in appendix A.3.…”