extracting generators of an ideal

extracting generators of an ideal

Sections:

obtain a single generator as an element

the generators as a matrix or list of elements

number of generators

minimal generating set

obtaining the input form of an ideal

obtain a single generator as an element

Once an ideal has been constructed it is possible to obtain individual elements using _. As always in Macaulay 2, indexing starts at 0.

i1 : R = ZZ[w,x,y,z];

i2 : I = ideal(z*w-2*x*y, 3*w^3-z^3,w*x^2-4*y*z^2,x);



o2 : Ideal of R

i3 : I_0



o3 = - 2x*y + w*z



o3 : R

i4 : I_3



o4 = x



o4 : R

the generators as a matrix or list of elements

Use generators or its abbreviation gens to get the generators of an ideal I as a matrix. Applying first entries to this matrix converts it to a list.

i5 : gens I



o5 = | -2xy+wz 3w3-z3 wx2-4yz2 x |



             1       4

o5 : Matrix R  <--- R

i6 : first entries gens I



                      3    3     2       2

o6 = {- 2x*y + w*z, 3w  - z , w*x  - 4y*z , x}



o6 : List

number of generators

The command numgens gives the number of generators of an ideal I.

i7 : numgens I



o7 = 4

minimal generating set

To obtain a minimal generating set of a homogeneous ideal use mingens to get the minimal generators as a matrix and use trim to get the minimal generators as an ideal.

i8 : mingens I



o8 = | x wz -4yz2 3w3-z3 |



             1       4

o8 : Matrix R  <--- R

i9 : trim I



                         2    3    3

o9 = ideal (x, w*z, -4y*z , 3w  - z )



o9 : Ideal of R

The function mingens is only well-defined for a homogeneous ideal or in a local ring. However, one can still try to get as small a generating set as possible and when it is implemented this function will be done by trim.

obtaining the input form of an ideal

If the ideal was defined using a function like monomialCurveIdeal and the generators are desired in the usual format for input of an ideal, the function toString is very useful. (Note: We are changing rings because monomialCurveIdeal is not implemented for rings over ZZ.)

i10 : R = QQ[a..d];

i11 : I = monomialCurveIdeal(R,{1,2,3});



o11 : Ideal of R

i12 : toString I



o12 = ideal(b*c-a*d,c^2-b*d,b^2-a*c)