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

Synopsis

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

See also

Ways to use mMixedVolume :

For the programmer

The object mMixedVolume is a method function.