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toricCircuits -- calculates the circuits of the toric ideal; invokes "circuits" from 4ti2

Synopsis

Description

The circuits are contained in the Graver basis of IA. In fact, they are precisely the primitive binomials in the ideal with minimal support.
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 : C = toricCircuits 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.
-------------------------------------------------
Using 64 bit integers.
4ti2 Total Time:  0.00 secs.
using temporary file name /tmp/M2-52426-1

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

              4        4
o2 : Matrix ZZ  <--- ZZ
The ideal generated by the circuits of A in general differs from the toric ideal of A. For example:
i3 : R = QQ[a..d]

o3 = R

o3 : PolynomialRing
i4 : Icircuit = toBinomial(toricCircuits(A), R) -- this is the circuit ideal of 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.
-------------------------------------------------
Using 64 bit integers.
4ti2 Total Time:  0.00 secs.
using temporary file name /tmp/M2-52426-2

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

o4 : Ideal of R
i5 : I = toBinomial(toricMarkov(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.
-------------------------------------------------
Using 64 bit integers.
4ti2 Total Time:  0.00 secs.
using temporary file name /tmp/M2-52426-3

               2           2
o5 = ideal (- c  + b*d, - b  + a*c, - b*c + a*d)

o5 : Ideal of R
i6 : I==Icircuit

o6 = false
The two ideals are not the same. There is a minimal generator of I which is not a circuit:
i7 : a*d-b*c % I -- this binomial is in I:

o7 = 0

o7 : R
i8 : a*d-b*c % Icircuit -- but not in Icircuit:

o8 = - b*c + a*d

o8 : R

Ways to use toricCircuits :