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makeCyclic -- finds a cyclic generator of a D-module

Usage:

makeCyclic M
Inputs:
- M, a matrix, that specifies a map such that coker M is a holonomic D-module
Outputs:
- H, a hash table, where H.Generator is a cyclic generator and H.AnnG is the annihilator ideal of this generator

Description

It is known that every holonomic module is cyclic and there is an algorithm for computing a cyclic generator.

i1 : makeWA(QQ[x])

o1 = QQ[x, dx]

o1 : PolynomialRing, 1 differential variable(s)

i2 : M = matrix {{dx,0,0},{0,dx,0},{0,0,dx}} -- coker M = QQ[x]^3

o2 = | dx 0  0  |
     | 0  dx 0  |
     | 0  0  dx |

                       3                3
o2 : Matrix (QQ[x, dx])  <-- (QQ[x, dx])

i3 : h = makeCyclic M

                               3
o3 = HashTable{AnnG => ideal dx   }
               Generator => | x2 |
                            | x  |
                            | 1  |

o3 : HashTable

Caveat

The module M must be holonomic.

See also

isHolonomic -- determines whether a D-module (or ideal in Weyl algebra) is holonomic

Ways to use makeCyclic:

makeCyclic(Matrix)

For the programmer

The object makeCyclic is a method function.

The source of this document is in /__w/M2/M2/macaulay2-1.26.05+ds/M2/Macaulay2/packages/WeylAlgebras/DOC/basics.m2:246:0.