MultiDimResidue

This is the Julia package for Multidimensional Residues 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 $p$.

To use the package, you need install SymEngine and InvertedIndices

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, please see the origine papers arxiv 1609.07621 and arxiv 1709.08503.

Examples for the julia code

The package use the CAS SymEngine

using MultiResidue
using SymEngine
@vars x y z

Example 1

A example is consider the residue around the intersection of three divisors

h1=x;
h2=y*(x + 2y);
h3=x^2 + y*x + 3z*z;

and then calculate the counter intersection around the origin for the function f

f=(2x + 3y + 4z)/(z - 2)
multiResidue(f,[h1,h2,h3],[x,y,z])

The out put is

-1/8

Example 2

Another example is

h1=y - 2x + z;
h2=x^2 *y - x*z^2 + z^3;
h3=y^4 + x*y*z^2 - y^2*z^2 + z^4;
multiResidue((2x + 3y + 4z)/(z - 2),[h1,h2,h3],[x,y,z])

The out put is

93/1664

Example 3

The third example, using macro @varss to generate multi variables

@varss z 3
h1=z1*z2 + z1*z3;
h2=z1^3*z2-2z2^4;
h3=(z1^2 - z2*z1 + z3^2);
multiResidue((z2+1)/(z1^2-z3-1),[h1,h2,h3],[z1,z2,z3])

The output is

-5/3

Example 4

The fourth example, for the equations with symbolic coefficients

@varss z 3
@vars a b c
@vars x y z
h1=a*z1*z2 + z1*z3;
h2=z1^3*z2-2b*z2^4;
h3=z1^2 - z2*z1 + z3^2;
ideal=[h1,h2,h3];
vars=[z1, z2, z3];
MultiResidue.homoEqn([h1,h2,h3],[z1,z2,z3],5)

MultiResidue.inhomoEqn([h1,h2,h3],[z1,z2,z3],5)

MultiResidue.eqnAnsatz([h1,h2,h3],[z1,z2,z3],5)

res=multiResidue((2z2+1)/(z1^2-z3-1),[h1,h2,h3],[z1,z2,z3])

The output is

-16*(-6 - 6*(4 + 8*b/a^2)/(4 - 4*a^(-2)))/(96*b/a^3 + (-3/2)*(128*b - 64*b/a^2)/a + (-1/2)*(-192*b/a^3 - 3*(64*b/a^2 + 64*a^2)/a)*(4 + 8*b/a^2)/(4 - 4*a^(-2))) 
+ 32*(6*(4 + 8*b/a^2)/(a*(4 - 4*a^(-2))) + 6*a^(-1))/(96*b/a^3 + (-3/2)*(128*b - 64*b/a^2)/a 
+ (-1/2)*(-192*b/a^3 - 3*(64*b/a^2 + 64*a^2)/a)*(4 + 8*b/a^2)/(4 - 4*a^(-2))) 
+ 384*b/(a^3*(96*b/a^3 + (-3/2)*(128*b - 64*b/a^2)/a + (-1/2)*(-192*b/a^3 - 3*(64*b/a^2 + 64*a^2)/a)*(4 + 8*b/a^2)/(4 - 4*a^(-2))))
+ 576*b/(a^2*(96*b/a^3 + (-3/2)*(128*b - 64*b/a^2)/a + (-1/2)*(-192*b/a^3 - 3*(64*b/a^2 + 64*a^2)/a)*(4 + 8*b/a^2)/(4 - 4*a^(-2)))) 
- 384*(4 + 8*b/a^2)/((4 - 4*a^(-2))*(96*b/a^3 + (-3/2)*(128*b - 64*b/a^2)/a + (-1/2)*(-192*b/a^3 - 3*(64*b/a^2 + 64*a^2)/a)*(4 + 8*b/a^2)/(4 - 4*a^(-2))))
- 576*(4 + 8*b/a^2)/(a*(4 - 4*a^(-2))*(96*b/a^3 + (-3/2)*(128*b - 64*b/a^2)/a + (-1/2)*(-192*b/a^3 - 3*(64*b/a^2 + 64*a^2)/a)*(4 + 8*b/a^2)/(4 - 4*a^(-2))))

Nowadays, we still do not have any efficient simplifcation code in Julia CAS. Let's look forward to the future version of SymEngine.jl. However, by mathematica, the result can be simplified to

-((a^3 + 2 a^4 + 2 a^5 + 4 b + 6 a b + a^2 (2 + 4 b))/(
 a^6 + 6 a^2 b + 2 b (-1 + 2 b))).

Example of Non-zero dimensional residue

multiResidue((z2+1)/(z1^2-z3-1),[z1, z2,(z1 + z2)*(z1 + z2 + z3)],[z1,z2,z3])

The output is not fully determined and you will get

false

The general math meaning or definition of such residue is still an open question.

Example for using the solve for linear equations

symbolic solution of linear equations

eqns=[x-y-1, 2(x + y)]
vars=[x,y]
solve(eqns,vars)

This solve function is not quite efficient. If in future, there is efficient solve function in SymEngine . I will replace it.

Citation

If you use multiResidue.jl, 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"
}