pointsIdeal -- create an ideal of points

Synopsis

• Usage:

  pointsIdeal M

  pointsIdeal(R, M)

• Inputs:
  • M, a matrix, of size $m \times n$, either over the coefficient ring of $R$, or a polynomial ring $R$ with $m$ variables
  • R, a ring, either the ring of $M$, or a polynomial ring with $m$ variables with coefficient ring the ring of $M$
• Outputs:
  • an ideal, the homogeneous ideal in $R$ of the points which are the columns of $M$

Description

i1 : S = ZZ/32003[a..d]

o1 = S

o1 : PolynomialRing

i2 : M = randomPoints(S, 6, Normalize => true)

o2 = | 1 0 0 0 1 107   |
     | 0 1 0 0 1 4376  |
     | 0 0 1 0 1 -5570 |
     | 0 0 0 1 1 3187  |

             4       6
o2 : Matrix S  <--- S

i3 : I = pointsIdeal M

o3 = ideal (a*d + 13356b*d - 13357c*d, b*c + 9278b*d - 9279c*d, a*c -
     ------------------------------------------------------------------------
     15663b*d + 15662c*d, a*b - 4105b*d + 4104c*d)

o3 : Ideal of S

i4 : betti res I

            0 1 2 3
o4 = total: 1 4 5 2
         0: 1 . . .
         1: . 4 2 .
         2: . . 3 2

o4 : BettiTally

See also

• randomPoints -- create a matrix whose columns are random points