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

## Synopsis

• 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