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MixedMultiplicity :: mixedVolume

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

Synopsis

Description

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

Ways to use mixedVolume :

For the programmer

The object mixedVolume is a method function with options.