toricGraver -- calculates the Graver basis of the toric ideal; invokes "graver" from 4ti2

Synopsis

• Usage:

  toricGraver(A) or toricGraver(A,R)

• Inputs:
  • A, a matrix, whose columns parametrize the toric variety. The toric ideal $I_A$ is the kernel of the map defined by A
  • R, a ring, polynomial ring in which the toric ideal $I_A$ should live
• Outputs:
  • B, a matrix, whose rows give binomials that form the Graver basis of the toric ideal of A, or
  • I, an ideal, whose generators form the Graver basis for the toric ideal

Description

i1 : A = matrix "1,1,1,1; 1,2,3,4"

o1 = | 1 1 1 1 |
     | 1 2 3 4 |

              2        4
o1 : Matrix ZZ  <--- ZZ

i2 : toricGraver(A)

o2 = | 1 -2 1  0 |
     | 2 -3 0  1 |
     | 1 -1 -1 1 |
     | 0 1  -2 1 |
     | 1 0  -3 2 |

              5        4
o2 : Matrix ZZ  <--- ZZ

i3 : R = QQ[a..d]

o3 = R

o3 : PolynomialRing

i4 : toricGraver(A,R)

               2           3    2                   2           3      2
o4 = ideal (- b  + a*c, - b  + a d, - b*c + a*d, - c  + b*d, - c  + a*d )

o4 : Ideal of R