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

The Graver basis for any toric ideal $I_A$ contains (properly) the union of all reduced Groebner basis of $I_A$. Any element in the Graver basis of the ideal is called a primitive binomial.

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

If we prefer to store the ideal instead, we may use:

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

Note that this last ideal equals the toric ideal $I_A$ since every Graver basis element is actually in $I_A$.

Ways to use toricGraver :

toricGraver(Matrix)
toricGraver(Matrix,Ring)

For the programmer

The object toricGraver is a method function.