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# singularIdeal -- ideal generated by the polynomial coefficients of a differential form or vector field

- Usage:
    `singularIdeal(e)`
- Inputs:
  - e, an instance of the type DiffAlgElement, a differential form or vector field
- Outputs:
  - an ideal, the ideal generated by the polynomial coefficients of e

## Description

Given a differential form or vector field, this routine returns the ideal generated by the polynomial coefficients of such element.

In this example we compute the singular locus of a differential form w.

    i1 : w = random newForm(2,1,2,"a")

            2             2                            2            2
    o1 = (2x  - 2x x  + 6x  - 3x x  + 6x x )dx  + (- 7x  + 5x x  - x  - 2x x  -
            0     0 1     1     0 2     1 2   0        0     0 1    1     0 2
         ------------------------------------------------------------------------
                 2            2             2                     2
         x x  + x )dx  + (- 3x  - 4x x  + 7x  + 2x x  - 7x x  - 3x )dx
          1 2    2   1        0     0 1     1     0 2     1 2     2   2

    o1 : DiffAlgForm

    i2 : singularIdeal(w)

                  2             2                      2            2
    o2 = ideal (2x  - 2x x  + 6x  - 3x x  + 6x x , - 7x  + 5x x  - x  - 2x x  -
                  0     0 1     1     0 2     1 2      0     0 1    1     0 2
         ------------------------------------------------------------------------
                 2      2             2                     2
         x x  + x , - 3x  - 4x x  + 7x  + 2x x  - 7x x  - 3x )
          1 2    2      0     0 1     1     0 2     1 2     2

                   QQ[i]
    o2 : Ideal of ------[][x ..x ]
                   2        0   2
                  i  + 1

This routine is useful to obtain the RingElement representing a 0-form

    i3 : w = random newForm(2,1,2,"a");

    i4 : r = radial 2;

    i5 : F = r_w

          3     2         2     3     2                  2         2      2    3
    o5 = x  + 6x x  + 3x x  - 4x  + 5x x  - 10x x x  + 6x x  + 3x x  - x x  + x
          0     0 1     0 1     1     0 2      0 1 2     1 2     0 2    1 2    2

    o5 : DiffAlgForm

    i6 : degree F

    o6 = {2, 0, 3}

    o6 : List

    i7 : (gens singularIdeal F)_0_0

          3     2         2     3     2                  2         2      2    3
    o7 = x  + 6x x  + 3x x  - 4x  + 5x x  - 10x x x  + 6x x  + 3x x  - x x  + x
          0     0 1     0 1     1     0 2      0 1 2     1 2     0 2    1 2    2

          QQ[i]
    o7 : ------[][x ..x ]
          2        0   2
         i  + 1

## See also

- moduliIdeal -- ideal generated by the coefficients of a differential form or vector field

## Ways to use singularIdeal:

- `singularIdeal(DiffAlgElement)`

## For the programmer

The object singularIdeal is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/DiffAlg.m2:671:0.