# mMixedVolume -- Compute the mixed volume of a collection of lattice polytopes

## Synopsis

• Usage:
mMixedVolume W
• Inputs:
• W, a list, of homogeneous ideals $I_1,\ldots,I_n$ over a polynomial ring, or a list of lists of vertices of the polytopes
• Outputs:

## Description

Let $Q_1,\ldots,Q_n$ be a collection of lattice polytopes in $\mathbb{R}^n$ and let $I_1,\ldots,I_n$ be homogeneous ideals in a polynomial ring over the field of rational numbers, corresponding to the given polytopes. These ideals can be obtained using the command homIdealPolytope. The mixed volume is calculated by computing a mixed multiplicity of these ideals.

The following example computes the mixed volume of two 2-cross polytopes.

 i1 : I = homIdealPolytope {(-1,0),(0,-1),(1,0),(0,1)} 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 i2 : mMixedVolume {I,I} o2 = 4

One can also compute the mixed volume of a collection of lattice polytopes by directly entering the vertices of the polytopes. Mixed Volume in the above example can also be computed as follows.

 i3 : C = {(-1,0),(0,-1),(1,0),(0,1)} o3 = {(-1, 0), (0, -1), (1, 0), (0, 1)} o3 : List i4 : mMixedVolume {C,C} o4 = 4

The following example computes the mixed volume of a 2-dimensional hypercube $H$ and a 2-cross polytope $C$.

 i5 : H = {(1,1),(1,-1),(-1,1),(-1,-1)}; i6 : C = {(-1,0),(0,1),(1,0),(0,-1)}; i7 : mMixedVolume {H,C} o7 = 8

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