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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)
-------------------------------------------------
4ti2 version 1.3.2, Copyright (C) 2006 4ti2 team.
4ti2 comes with ABSOLUTELY NO WARRANTY.
This is free software, and you are welcome
to redistribute it under certain conditions.
For details, see the file COPYING.
-------------------------------------------------


Final basis has 1 inhomogeneous, 10 homogeneous and 0 free elements.
Writing 5 vectors to graver file, with respect to symmetry.

4ti2 Total Time: 0.00 secs
using temporary file name /tmp/M2-52542-1

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)
-------------------------------------------------
4ti2 version 1.3.2, Copyright (C) 2006 4ti2 team.
4ti2 comes with ABSOLUTELY NO WARRANTY.
This is free software, and you are welcome
to redistribute it under certain conditions.
For details, see the file COPYING.
-------------------------------------------------


Final basis has 1 inhomogeneous, 10 homogeneous and 0 free elements.
Writing 5 vectors to graver file, with respect to symmetry.

4ti2 Total Time: 0.00 secs
using temporary file name /tmp/M2-52542-2

               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 :