# mixedVolume -- computes the mixed volume of a list of polytope

## Synopsis

• Usage:
v = mixedVolume L
• Inputs:
• L, a list, containing n polytopes in n-space
• Outputs:

## Description

Let $P_1,...,P_n$ be polytopes in $n$-space. Then the volume of the Minkowski sum $\lambda_1 P_1 + ... + \lambda_n P_n$ is a homogeneous polynomial of degree $n$ in nonnegative variables $\lambda_1,...,\lambda_n$. The coefficient Vol$(P_1,...,P_n)$ of $\lambda_1\lambda_2 ... \lambda_n$ is called the mixed volume of $P_1,...,P_n$. For example, the number of toric solutions to a generic system of /$n$ polynomial equations on $n$-space amounts to the mixed volume of the corresponding Newton polytopes.

The function mixedVolume takes the List L with $n$ polytopes in $n$-space and computes their mixed Volume by using the algorithm by Ioannis Z. Emiris in his paper Mixed Volume Implementation. Note that this function computes an upper bound by using a random lifting. To reassure the result run the function until it returns the same result.

CAVEAT: So far the input is not checked so use the function with care!

 i1 : P = crossPolytope 2 o1 = {ambient dimension => 2 } dimension of lineality space => 0 dimension of polyhedron => 2 number of facets => 4 number of rays => 0 number of vertices => 4 o1 : Polyhedron i2 : Q = hypercube 2 o2 = {ambient dimension => 2 } dimension of lineality space => 0 dimension of polyhedron => 2 number of facets => 4 number of rays => 0 number of vertices => 4 o2 : Polyhedron i3 : mixedVolume {P,Q} o3 = 8 o3 : QQ

## Ways to use mixedVolume :

• "mixedVolume(List)"

## For the programmer

The object mixedVolume is .