statePolytope(ZZ,Ideal) -- computes the mth state polytope of an ideal

Synopsis

Usage:
statePolytope(m,I)
Function: statePolytope
Inputs:
- an integer, specifies to compute the mth state polytope
- an ideal, the ideal
Outputs:
- a list, the list of vertices of the state polytope

Description

See Sturmfels's book Groebner bases and convex polytopes, page 14 for the definition of State_m(I). (The difference between this and State(I) is that for all sufficiently large m, State_m(I) does not distinguish between initial ideals which have the same saturation with regard to the irrelevant ideal, whereas in State(I), these are separated.)

i1 : R = QQ[a..d];

i2 : I = ideal(a*c-b^2,a*d-b*c,b*d-c^2);

o2 : Ideal of R

i3 : statePolytope(3,I)   
VERTICES
1 9 6 6 9
1 9 3 12 6
1 7 5 14 4
1 5 8 14 3
1 3 12 12 3
1 6 12 3 9
1 4 14 5 7
1 3 14 8 5


o3 = {{9, 6, 6, 9}, {9, 3, 12, 6}, {7, 5, 14, 4}, {5, 8, 14, 3}, {3, 12, 12,
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     3}, {6, 12, 3, 9}, {4, 14, 5, 7}, {3, 14, 8, 5}}

o3 : List