Given a $d\times n$ Matrix A and a polynomial ring in $n$ variables $R$, this method returns the toric ideal associated to $A$ in $R$. To do this, toricIdeal saturates the lattice basis ideal of the kernel of $A$ with respect to the product of the variables of $R$.
i1 : A=matrix{{1,1,1,1,1,1},{1,2,1,2,3,0},{0,2,2,0,1,1}}
o1 = | 1 1 1 1 1 1 |
| 1 2 1 2 3 0 |
| 0 2 2 0 1 1 |
3 6
o1 : Matrix ZZ <--- ZZ
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i2 : R=QQ[a..f] o2 = R o2 : PolynomialRing |
i3 : toricIdeal(A,R)
3 3 2 2 2 2
o3 = ideal (c*d - e*f, a*b - e*f, c e - b f, a*c e - b d*f, a c*e - b*d f,
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3 3 3 2 2 2 2 2 3 2 2 3 2 2 2 2
a e - d f, b*d - a e , b d - a*c*e , b d - c e , a*c - b f , a c -
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2 3 2 2
b*d*f , a c - d f )
o3 : Ideal of R
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i4 : A=matrix{{1,1,1,1,1},{0,0,1,1,0},{0,1,1,0,-2}}
o4 = | 1 1 1 1 1 |
| 0 0 1 1 0 |
| 0 1 1 0 -2 |
3 5
o4 : Matrix ZZ <--- ZZ
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i5 : R=toGradedRing(A,QQ[a..e]) o5 = R o5 : PolynomialRing |
i6 : toricIdeal(A,R)
2 2 2 3 2
o6 = ideal (a*c - b*d, a*d - c e, a d - b*c*e, a - b e)
o6 : Ideal of R
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The object toricIdeal is a method function.