toricGroebner -- calculates a Groebner basis of the toric ideal I_A, given A; invokes "groebner" from 4ti2

Synopsis

• Usage:

  toricGroebner(A) or toricGroebner(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, ring with as least as many generators as the columns of A
• Optional inputs:
  • Weights => ...,
• Outputs:
  • B, a matrix, whose rows give binomials that form a Groebner basis of the toric ideal of A
  • I, an ideal, whose generators form a Groebner 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 : toricGroebner(A)

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

              3        4
o2 : Matrix ZZ  <--- ZZ

i3 : R = QQ[a..d]

o3 = R

o3 : PolynomialRing

i4 : toricGroebner(A,R)

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

o4 : Ideal of R

i5 : toricGroebner(A,Weights=>{1,2,3,4})

o5 = | -1 1  1  -1 |
     | -1 2  -1 0  |
     | 0  -1 2  -1 |

              3        4
o5 : Matrix ZZ  <--- ZZ

Caveat