noetherianOperators(Module,Ideal) -- Noetherian operators of a primary component

Synopsis

Function: noetherianOperators
Usage:

noetherianOperators (U, P)
Inputs:
- U, a module,
- P, an ideal, a minimal prime of comodule U
Outputs:
- a list, of differential operators

Description

Compute a set of Noetherian operators for the $P$-primary component of comodule U.

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

o1 = R

o1 : PolynomialRing

i2 : U = image matrix{{x,y,z},{y,z,x}}

o2 = image | x y z |
           | y z x |

                             2
o2 : R-module, submodule of R

i3 : P = first associatedPrimes comodule U

o3 = ideal (y - z, x - z)

o3 : Ideal of R

i4 : noetherianOperators(U, P)

o4 = {| -1 |}
      |  1 |

o4 : List

If there are no embedFded primes, running this command for all associated primes is equivalent to running a differential primary decomposition.

i5 : associatedPrimes comodule U /
         (P -> {P, noetherianOperators(U,P)})

                                                            2          2
o5 = {{ideal (y - z, x - z), {| -1 |}}, {ideal (x + y + z, y  + y*z + z ), {|
                              |  1 |                                        |
     ------------------------------------------------------------------------
     -z |}}}
      y |

o5 : List

i6 : netList differentialPrimaryDecomposition U

     +--------------------------------+--------+
o6 = |ideal (y - z, x - z)            |{| -1 |}|
     |                                | |  1 | |
     +--------------------------------+--------+
     |                   2          2 |        |
     |ideal (x + y + z, y  + y*z + z )|{| -z |}|
     |                                | |  y | |
     +--------------------------------+--------+

Ways to use this method: