So I am trying to find the toric ideal of a point configuration given by the lattice points contained within a family of polytopes I am studying to assess regularity/unimodularity of triangulations of the point configuration (I'm attempting to follow the guidelines specified in Bernd Sturmfels text Grobner Bases and Convex Polytopes). I know that toric ideals are supposed to be prime and generated by binomials, but for whatever reason, the ToricIdeal() sage function is indicating that the corresponding ideal is the entire ring, which should definitely not be the case.
For example, say that we're looking at the point configuration given by the lattice points (1,0), (0,1), (0,0), (0,-1), (-1,-1), (-1,-2), (-2,-3). Inputting these points as columns in a matrix and computing the toric ideal as indicated in the sage documentation should be as simple as:
A = matrix([[1,0,0,0,-1,-1,-2],[0,1,0,-1,-1,-2,-3]])
IA = ToricIdeal(A); IA
However, when I run this I get the following output:
Ideal (1) of Multivariate Polynomial Ring in z0, z1, z2, z3, z4, z5, z6 over Rational Field
I cannot determine why this is the case. For some context, I asked a colleague to compute the toric ideal using Macaulay2, and that program indicated that the toric ideal is generated by z2 - 1, z1*z3 - 1, z0*z1*z4 - 1, z0*z1^2*z5 - 1, z0^2*z1^3*z6 - 1, z0^4*z1^7*z2*z3*z4*z5*z6 - 1 which seems far more reasonable. Any idea if my input is incorrect or misguided? I'm having a similar issue on another project when attempting to compute toric ideals of point configurations comprised of the vertices of order polytopes. Without the correct toric ideal, I cannot compute a corresponding Grobner basis to assess regularity or unimodularity. Any help is greatly appreciated. Thank you.
