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 |
The object mixedVolume is a method function with options.