Description
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
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i2 : I = ideal (a-b,a-c,b-c)
o2 = ideal (a - b, a - c, b - c)
o2 : Ideal of R
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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 a |, | c b |}, {ambient dimension => 3 })
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 3
number of rays => 0
number of vertices => 3
o3 : Sequence
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The generators of the three initial ideals are given in the first part of the result.