Linear PDE with Constant Coefficients

This repository contains the code accompanying the paper Linear PDE with constant coefficients by Rida Ait El Manssour, Marc Härkönen, and Bernd Sturmfels.

Note: These functions will added to the package NotherianOperators and distributed with Macaulay2 from version 1.18 (ETA May 2021).

Quickstart

The functions in solvePDE.m2 require Macaulay2 version 1.17 or newer.

Clone the repository, and change to the newly created directory called solvePDE

git clone https://github.com/haerski/solvePDE.git
cd solvePDE

The following command will open Macaulay2 and preload the functions

M2 solvePDE.m2

Alternatively, the functions can be loaded in an existing Macaulay2 session using the command load("/path/to/solvePDE/solvePDE.m2")

A typical session

The main function is called solvePDE. Accepted inputs are ideals or modules. The output is a list of pairs whose first entry is a prime and second entry is a list of Noetherian multipliers. The total number of Noetherian multiplier will be equal to the arithmetic multiplicity, which can be computed using the command amult.

Note: The ouput can be fed to the command netList to make the output more human-readable.

Example 1: ideals

$ M2 solvePDE.m2
Macaulay2, version 1.17.2.1
with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems, LLLBases, MinimalPrimes, PrimaryDecomposition, ReesAlgebra, Saturation, TangentCone

i1 : R = QQ[x,y,z]

o1 = R

o1 : PolynomialRing

i2 : I = ideal(x^2*y,x^2*z,x*y^2,x*y*z^2)

             2    2      2       2
o2 = ideal (x y, x z, x*y , x*y*z )

o2 : Ideal of R

i3 : amult I

o3 = 5

o3 : QQ

i4 : solvePDE I

o4 = {{ideal x, {| 1 |}}, {ideal (y, x), {| dx |}}, {ideal (z, y), {| 1 |}}, {ideal (z, y, x), {| dxdy |, | dxdydz |}}}

o4 : List

i5 : netList solvePDE I

     +---------------+----------------------+
o5 = |ideal x        |{| 1 |}               |
     +---------------+----------------------+
     |ideal (y, x)   |{| dx |}              |
     +---------------+----------------------+
     |ideal (z, y)   |{| 1 |}               |
     +---------------+----------------------+
     |ideal (z, y, x)|{| dxdy |, | dxdydz |}|
     +---------------+----------------------+

Example 2: modules

$ M2 solvePDE.m2
Macaulay2, version 1.17.2.1
with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems, LLLBases, MinimalPrimes, PrimaryDecomposition, ReesAlgebra, Saturation, TangentCone

i1 : R = QQ[x,y,z];

i2 : U = image(matrix {
         {x^2,x*y,x*z},
         {y^2,y*z,z^2}});

i3 : netList solvePDE U

     +---------------+--------+
o3 = |ideal x        |{| 1 |} |
     |               | | 0 |  |
     +---------------+--------+
     |       2       |        |
     |ideal(y  - x*z)|{| -z |}|
     |               | |  x | |
     +---------------+--------+
     |ideal (z, y)   |{|  0 |}|
     |               | | dz | |
     +---------------+--------+