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.

Ways to use homIdealPolytope :

homIdealPolytope(List)

For the programmer

The object homIdealPolytope is a method function with options.

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

Synopsis

Description

See also

Ways to use homIdealPolytope :

For the programmer