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statePolytope -- computes the state polytope of a homogeneous ideal



A statePolytope of an Ideal I has as normalFan the Groebner fan of the ideal. We use the construction by Sturmfels, see Algorithm 3.2 in Bernd Sturmfels' Groebner Bases and Convex Polytopes, volume 8 of University Lecture Series. American Mathematical Society, first edition, 1995.

Consider the following ideal in a ring with 3 variables:
i1 : R = QQ[a,b,c]

o1 = R

o1 : PolynomialRing
i2 : I = ideal (a-b,a-c,b-c)

o2 = ideal (a - b, a - c, b - c)

o2 : Ideal of R

The state polytope of this ideal is a triangle in 3 space, because the ideal has three initial ideals:
i3 : statePolytope I

o3 = ({| b a |, | c b |, | c a |}, Polyhedron{...1...})

o3 : Sequence

The generators of the three initial ideals are given in the first part of the result.

Ways to use statePolytope :

For the programmer

The object statePolytope is a method function.