componentOfKernel -- Finds all minimal generators of a given degree in the kernel of a ring map

Synopsis

• Usage:

  componentOfKernel(deg, dom, F, M)

  componentOfKernel(deg, dom, F, B)

  componentOfKernel(deg, dom, F)

• Inputs:
  • deg, a list, the degree of the homogeneous component to compute
  • dom, a ring, a graded ring which is the source of a homogeneous ring map $F$
  • F, a ring map, a map whose kernel is homogeneous in the grading of dom
  • M, a matrix, a monomial basis for the homogeneous component of deg with degree deg
  • B, a mutable hash table, a mutable hashtable which contains all bases of homogeneous components which correspond to lower total degrees than deg
• Optional inputs:
  • PreviousGens => a list, default value {}, a list of generators of the kernel which have lower total degree
• Outputs:
  • a matrix, A list of minimal generators for $\ker(F)$ which are in the homogeneous component of degree deg

Description

i1 : A = matrix {{1,1,1,0,0,0}, {0,0,0,1,1,1}, {1,0,0,1,0,0}, {0,1,0,0,1,0}, {0,0,1,0,0,1}}

o1 = | 1 1 1 0 0 0 |
     | 0 0 0 1 1 1 |
     | 1 0 0 1 0 0 |
     | 0 1 0 0 1 0 |
     | 0 0 1 0 0 1 |

              5       6
o1 : Matrix ZZ  <-- ZZ

i2 : R = QQ[x_(1,1)..x_(2,3)];

i3 : S = QQ[t_1..t_2, s_1..s_3];

i4 : F = map(S, R, {t_1*s_1, t_1*s_2, t_1*s_3, t_2*s_1, t_2*s_2, t_2*s_3})

o4 = map (S, R, {t s , t s , t s , t s , t s , t s })
                  1 1   1 2   1 3   2 1   2 2   2 3

o4 : RingMap S <-- R

i5 : dom = newRing(R, Degrees => A);

i6 : componentOfKernel({1,1,0,1,1}, dom, F)

o6 = {x   x    - x   x   }
       1,3 2,2    1,2 2,3

o6 : List