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

- 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.

## See also

- mMixedVolume -- Compute the mixed volume of a collection of lattice polytopes
- mixedMultiplicity -- Compute a given mixed multiplicity of ideals

## Ways to use homIdealPolytope:

- `homIdealPolytope(List)`

## For the programmer

The object homIdealPolytope is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MixedMultiplicity.m2:703:0.