Let
P1,...,Pn be polytopes in
n-space. Then the volume
of the Minkowski sum
λ1 P1 + ... + λn Pn is a homogeneous polynomial of degree
n in nonnegative variables
λ1,...,λn. The coefficient Vol
(P1,...,Pn) of
λ1λ2 ... λn is called
the mixed volume of
P1,...,Pn. 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
|