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

## Synopsis

• Usage:
mixedVolume W
• Inputs:
• W, a list, list of homogeneous ideals $I_1,...,I_n$ over a polynomial ring, or a list of lists of vertices of the polytopes
• Optional inputs:
• BasisElementLimit => ..., default value infinity, Bound the number of Groebner basis elements to compute in the saturation step
• CoefficientRing => ..., default value QQ, Choose a coefficient ring of the (output) ideal
• DegreeLimit => ..., default value {}, Bound the degrees considered in the saturation step.
• MinimalGenerators => ..., default value true, Whether the saturation step returns minimal generators
• PairLimit => ..., default value infinity, Bound the number of s-pairs considered in the saturation step
• Strategy => ..., default value null, Choose a strategy for the saturation step
• VariableBaseName => ..., default value X, Choose a base name for variables in the created ring
• Outputs:

## Description

Let $Q_1,...,Q_n$ be a collection of lattice polytopes in $\mathbb{R}^n$ and let $I_1,...,I_n$ 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 three 3-cross polytopes.

 i1 : I = homIdealPolytope {(0,1,1),(1,0,1),(1,1,0),(2,1,1),(1,2,1),(1,1,2)} 2 2 2 2 2 2 o1 = ideal (X X X , X X X , X X X , X X X , X X X , X X X ) 1 2 3 1 2 3 1 2 3 1 2 4 1 3 4 2 3 4 o1 : Ideal of QQ[X ..X ] 1 4 i2 : mixedVolume {I,I,I} o2 = 8

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 = {(0,1,1),(1,0,1),(1,1,0),(2,1,1),(1,2,1),(1,1,2)} o3 = {(0, 1, 1), (1, 0, 1), (1, 1, 0), (2, 1, 1), (1, 2, 1), (1, 1, 2)} o3 : List i4 : mixedVolume {C,C,C} o4 = 8

## Ways to use mixedVolume :

• mixedVolume(List) (missing documentation)

## For the programmer

The object mixedVolume is .