Short introduction to multivariate residues(Math background and mathematica code)

This program is to calculate the multi-variable residue around zero-dimension intersection point. The main function is called "MultiResidue" which is used to calculate $$\text{Res}_{h_1=\cdots=h_r=0} ~ {f dz_1\wedge dz_2\wedge \cdots \wedge dz_r\over h_1 h_2\cdots h_r},$$ where $h_i$ is homogeneous polynomial function and the solution of $h_1=\cdots=h_r=0$ is an isolated point $p$. The $f$ function is regular at $p$.

MultiResidue[f, {h1, h2,..., hr}, {z1,z2,..., zr}]

More examples is included in the mathematica code. A julia package is still in development. This need to use CAS in julia, e.g. Symbolics.jl and/or SymEngine.

For general in-homogeneous polynomial function, it is easy to transform to the homogeneous cases by a trick using the global residue theorem. For more details and citations, please see and cite the origine papers arxiv 1609.07621 and arxiv 1709.08503.

Citation

If you use multiResidue.nb, please cite the two papers arxiv 1609.07621 and arxiv 1709.08503 as following

@article{Chen:2016fgi,
    author = "Chen, Gang and Cheung, Yeuk-Kwan E. and Wang, Tianheng and Xu, Feng",
    title = "{A differential operator for integrating one-loop scattering equations}",
    eprint = "1609.07621",
    archivePrefix = "arXiv",
    primaryClass = "hep-th",
    doi = "10.1007/JHEP01(2017)028",
    journal = "JHEP",
    volume = "01",
    pages = "028",
    year = "2017"
}

@article{Chen:2017bug,
    author = "Chen, Gang and Wang, Tianheng",
    title = "{BCJ Numerators from Differential Operator of Multidimensional Residue}",
    eprint = "1709.08503",
    archivePrefix = "arXiv",
    primaryClass = "hep-th",
    doi = "10.1140/epjc/s10052-019-7604-8",
    journal = "Eur. Phys. J. C",
    volume = "80",
    number = "1",
    pages = "37",
    year = "2020"
}

AmplitudeGravity/Multi-variable-Residue

Short introduction to multivariate residues(Math background and mathematica code)

Citation