# statePolytope -- computes the state polytope of a homogeneous ideal

## Synopsis

• Usage:
P = statePolytope I
• Inputs:
• I, an ideal, which must be homogeneous
• Outputs:

## 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 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 :

• "statePolytope(Ideal)"

## For the programmer

The object statePolytope is .