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. New version program summaryProgram Title:MultivariateResiduesProgram Files doi:http://dx.doi.org/10.17632/7z87sf98xs.2Licensing provisions: GNU General Public License v3Programming language: Wolfram Mathematica version 7.0 or higherSupplementary material: Additional pdf included with program filesJournal reference of previous version: Comput.Phys.Commun. 222 (2018) 250-262Does the new version supersede the previous version?: YesReasons for the new version: Bugfix and added new functionSummary of revisions: In the previous version of MultivariateResidues, the function ▪ (using the method ▪ i.e. the default method) would occasionally yield the residue with an incorrect sign, due to a numerical prefactor being inadvertently dropped from the integrand. In this version of MultivariateResidues, this prefactor is retained, and many tests have been carried out to verify that residues are now computed with correct signs.In addition, a new function ▪ has been added. This function lists the linear relations among the residues at finite locations and residues at infinity produced by the global residue theorem when the input differential form is extended to CPn. We refer to the Supplementary Material for background material on the global residue theorem.Here we give an example of the usage of the function ▪ . The syntax is [Display omitted] where ▪ denotes the numerator of ω, ▪ the denominator factors and ▪ the variables. As an example, let us consider the differential form studied in section 1.1 of the Supplementary Material, [Display omitted] There are seven linear relations that arise from the global residue theorem, [Display omitted] The relations are recorded in the form {P,{V,R}}, where P denotes the denominator partition, V the set of poles involved in the relation and R their respective residues. For the denominator partition computed in detail in section 1.1 of the Supplementary Material we have [Display omitted] It is easy to check that the residues indeed sum to zero, [Display omitted] ▪ only records non-vanishing residues and their respective poles. In cases where all the residues that appear in a global residue theorem vanish, the empty set is returned as output. [Display omitted] Nature of problem: Evaluation of multivariate complex residuesSolution method: Mathematica implementation