# homIdealPolytope -- Compute the homogeneous ideal corresponding to the vertices of a lattice polytope in $\mathbb{R}^n$.

## Synopsis

• Usage:
homIdealPolytope W
• Inputs:
• W, a list, list ${p_1,\ldots,p_r}$, where $p_1,\ldots,p_r$ are vertices of the lattice polytope in $\mathbb{R}^n$.
• Optional inputs:
• CoefficientRing => ..., default value QQ, choose the coefficient ring of the (output) ideal
• VariableBaseName => ..., default value "X", choose a base name for variables in the created ring
• Outputs:
• an ideal, A homogeneous ideal of $k[x_1,\ldots,x_{n+1}].$

## Description

Given a list of vertices of a lattice polytope, the function outputs a homogeneous ideal of $k[x_1,\ldots,x_{n+1}]$ such that the polytope is the convex hull of the lattice points of the dehomogenization of a set of monomials that generates the ideal in $k[x_1,\ldots,x_n]$.

The following example computes the homogeneous ideal corresponding to a 2-cross polytope.

 i1 : I = homIdealPolytope {(0,1),(1,0),(0,-1),(-1,0)} 2 2 2 2 o1 = ideal (X X , X X , X X , X X ) 1 2 1 2 1 3 2 3 o1 : Ideal of QQ[X ..X ] 1 3

The output can be used to compute the mixed volume of a collection of polytopes. A list of the output ideals, corresponding to the vertices of various polytopes, can be used as an input in the mMixedVolume function to compute the mixed volume of polytopes.