next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
OldPolyhedra :: mixedVolume

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

Synopsis

Description

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

Ways to use mixedVolume :