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FourTiTwo :: toricGraver

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

Synopsis

Description

The Graver basis for any toric ideal IA contains (properly) the union of all reduced Groebner basis of IA. 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 IA since every Graver basis element is actually in IA.

Ways to use toricGraver :