The circuits are contained in the Graver basis of IA. In fact, they are precisely the primitive binomials in the ideal with minimal support.
A = matrix "1,1,1,1; 1,2,3,4" |
C = toricCircuits A |
The ideal generated by the circuits of A in general differs from the toric ideal of A. For example:
R = QQ[a..d] |
Icircuit = toBinomial(toricCircuits(A), R) -- this is the circuit ideal of A |
I = toBinomial(toricMarkov(A), R) |
I==Icircuit |
The two ideals are not the same. There is a minimal generator of I which is not a circuit:
a*d-b*c % I -- this binomial is in I: |
a*d-b*c % Icircuit -- but not in Icircuit: |