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) using temporary file name /tmp/M2-30725-0/0 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) using temporary file name /tmp/M2-30725-0/1 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.